On difference equations for orthogonal polynomials on nonuniform lattices 1

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On difference equations for orthogonal polynomials onnonuniform lattices1

Mama Foupouagnigni aa Department of Mathematics , Higher Teachers Training College, University of Yaounde I ,P.O. Box 47, Yaounde, CameroonPublished online: 06 Sep 2010.

To cite this article: Mama Foupouagnigni (2008) On difference equations for orthogonal polynomials on nonuniform lattices1 ,Journal of Difference Equations and Applications, 14:2, 127-174, DOI: 10.1080/10236190701536199

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On difference equations for orthogonalpolynomials on nonuniform lattices1

MAMA FOUPOUAGNIGNI*†

Department of Mathematics, Higher Teachers Training College, University of Yaounde I, P.O. Box 47,Yaounde, Cameroon

(Received 7 July 2006; in final form 21 June 2007)

By the study of various properties of some divided-difference equations, we simplify the definition ofclassical orthogonal polynomials given by Atakishiyev et al., 1995, On classical orthogonal polynomials,Constructive Approximation, 11, 181–226, then prove that orthogonal polynomials obtained by somemodifications of the classical orthogonal polynomials on nonuniform lattices satisfy a single fourth-orderlinear homogeneous divided-difference equation with polynomial coefficients. Moreover, we factorizeand solve explicitly these divided-difference equations. Also, we prove that the product of two functions,each of which satisfying a second-order linear homogeneous divided-difference equation is a solution of afourth-order linear homogeneous divided-difference equation. This result holds in particular when thedivided-difference operator is carefully replaced by the Askey–Wilson operator Dq, following pioneeringwork by Magnus 1988, Associated Askey–Wilson polynomials as Laguerre–Hahn orthogonalpolynomials, Lecture Notes in Mathematics (Berlin: Springer), vol. 1329, pp. 261–278, connecting Dq

and divided-difference operators. Finally, we propose a method to look for polynomial solutions of lineardivided-difference equations with polynomial coefficients.

Keywords: Classical orthogonal polynomials; Modifications of orthogonal polynomials; Divided-difference equations; quadratic and q-quadratic lattices; Functions of second kind

Msc 2000: 33D45; 33C45; 33D20

1. Introduction

Let x(s) be a function of the variable s and y(x(s)) a function of x(s) satisfying a difference

equation of hypergeometric type namely

f ðxðsÞÞD

Dx21ðsÞ

7yðxðsÞÞ

7xðsÞ

� �þ

c ðxðsÞÞ

2

DyðxðsÞÞ

DxðsÞþ

7yðxðsÞÞ

7xðsÞ

� �þ lyðxðsÞÞ ¼ 0; ð1Þ

where f and c are polynomials of degree at most 2 and 1 respectively; l is a constant, D and

7 are the forward and the backward operators

Df ðsÞ ¼ f ðsþ 1Þ2 f ðsÞ; 7f ðsÞ ¼ f ðsÞ2 f ðs2 1Þ: ð2Þ

Journal of Difference Equations and Applications

ISSN 1023-6198 print/ISSN 1563-5120 online q 2008 Taylor & Francis

http://www.tandf.co.uk/journals

DOI: 10.1080/10236190701536199

1Dedicated to the memory of my father El Hadj Mowoum Moussa 1929–2004.

†Research supported by: The Alexander von Humboldt Foundation (Bonn, Germany, under grant n8 IV KAM1070235 and its various Follow up Programs); the Abdus Salam International Centre for Theoretical Physics(Trieste, Italy, under the Junior Associateship Program: 2001–2006); the Ministry of Higher Education in Cameroon(under the Academic Mobility Programs (2005–2006)) and the University of Yaounde I (Yaounde, Cameroon, underthe FUAFF Program (2004)).

*Email: [email protected]

Journal of Difference Equations and Applications,

Vol. 14, No. 2, February 2008, 127–174

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Here the lattice xmðsÞ for a complex number m is defined as

xmðsÞ ¼ x sþm

2

� �: ð3Þ

Atakishiyev et al. [5] (see also [28]) proved that the divided-difference

DyðxðsÞÞ

DxðsÞ

satisfies an equation of the same type as (1) if and only if x(s) is a linear, a q-linear,

a quadratic or a q-quadratic lattice; that is x(s) is of the form

xðsÞ ¼c1q

2s þ c2qs þ c3 if q – 1

c4s2 þ c5sþ c6 if q ¼ 1;

(ð4Þ

where q [ C, and the ci are arbitrary constants such that ðc1; c2Þ – ð0; 0Þ and

ðc4; c5Þ – ð0; 0Þ. Lattices of the form (4) with c1c2 – 0 or c4 – 0 are called nonuniform

lattices [5,28].

Next they used this characterization to give a definition of classical orthogonal

polynomials (in the broad sense of Hahn, and consistent with the latest definition proposed by

Andrews and Askey [2]):

Definition 1. A polynomial sequence (Pn) is classical if and only if:

1. (Pn) is orthogonal on a real interval (x (a), x (b)) with respect to the weight function

r (s) i.e.

degreeðPnÞ ¼ n; n $ 0;PNi¼0 PnðxðsiÞÞPmðxðsiÞÞrðsiÞ7x1ðsiÞ ¼ kndn;m; kn – 0; n;m $ 0;

(ð5Þ

with

s0 ¼ a; siþ1 ¼ si þ 1; sNþ1 ¼ b; N [ N0 < {1} ð6Þ

for the discrete orthogonality or

degreeðPnÞ ¼ n; n $ 0;ÐCPnðxðsÞÞPmðxðsÞÞrðsÞ7x1ðsÞds ¼ kndn;m; kn – 0; n;m $ 0;

8<: ð7Þ

where C is a contour in the complex s-plane, for the continuous orthogonality;

2. Any Pn(x (s)) satisfies a difference equation of the form (1) with x (s) given by (4).

3. The weight r satisfies the Pearson-type difference equation

D

7x1ðsÞðsðsÞrðsÞÞ ¼ c ðxðsÞÞrðsÞ; ð8Þ

where c (s) is a polynomial of degree 1 in x (s) and the function f defined by

f ðxðsÞÞ ¼ sðsÞ þ1

2c ðxðsÞÞ7x1ðsÞ ð9Þ

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is a non-zero polynomial of degree at most 2 in the variable x (s), with the border

conditions

sðsÞrðsÞxk s2 12

� ��s¼a;b

¼ 0; k ¼ 0; 1; 2; . . . ;ÐCD sðsÞrðsÞxk s2 1

2

� �� ds ¼ 0; k ¼ 0; 1; 2; . . . :

8<: ð10Þ

for the discrete orthogonality and the continuous orthogonality respectively.

This definition covers the q-Racah polynomials as well as the Askey–Wilson polynomials

and their limiting and special cases. It covers also the very classical orthogonal polynomials–

the classical orthogonal polynomials of a continuous variable (Jacobi, Laguerre and Hermite

by virtue of the limiting procedure); the classical orthogonal polynomials of a discrete variable

(Hahn, Meixner, Charlier and Krawtchouk) and the classical orthogonal polynomials of a

q-discrete variable (Big q-Jacobi, . . . ).

Remark 1. It should be noticed that in case of the continuous orthogonality, for the family

(Pn(x (s))) to be classical orthogonal polynomials in the real variable x (s), it should be

possible [5] to choose a contour C in such a way that the second relation of (7) can be

expressed as a real orthogonality relationðba

PnðxÞPnðxÞrðxÞdx ¼ kndn;m; kn – 0; n; m $ 0;

with r (x) . 0, x [ (a,b).

According to this definition, for a family of polynomials (Pn) orthogonal with respect to a

certain weight function r to be classical, the weight should satisfy a Pearson-type equation

with some border conditions and any Pn should additionally satisfy an equation of type (1).

By studying various properties of the operators Dx and Sx

D x f ðxðsÞÞ ¼Df ðxðsÞÞ

DxðsÞ; Sx f ðxðsÞÞ ¼

f ðxðsþ 1ÞÞ þ f ðxðsÞÞ

2; ð11Þ

we prove that the second condition of Definition 1 is not necessary. Therefore, we get rid of

this condition and obtain a definition which is similar to that of the very classical orthogonal

polynomials. Next, we derive, factorize and solve the fourth-order divided-difference

equation satisfied by the orthogonal polynomials obtained by modifications of classical

orthogonal polynomials.

Section 2 is devoted to the study of the properties of the operators Dx and Sx. In particular,

we prove that the product of two functions, each of which satisfying a second-order linear

homogeneous divided-difference equation is a solution of a fourth-order linear homogeneous

divided-difference equation. We show how this result works when the divided-difference

operator is replaced by the Askey–Wilson operator Dq. In the third section we simplify

Definition 1 and derive the expressions of the recurrence coefficients of classical orthogonal

polynomials in terms of the polynomials f and c in the same line as for the very classical

orthogonal polynomials. The fourth section is devoted to the derivation, the factorization

as well as the solution of the fourth-order divided-difference equations for modifications

of classical orthogonal polynomials (see Refs. [11 – 13] for the cases of the

very classical orthogonal polynomials). Section 5 gives some specializations and applications.

Orthogonal polynomials on nonuniform lattices 129

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In particular, we point out some situations for which the first associated of the Askey–Wilson

(resp. the Racah) polynomials remains classical and express these in terms of the initial

families. We also point out a method to look for polynomial solutions of higher order divided-

difference equations with polynomial coefficients in the same line as the one of higher order

differential equation with polynomial coefficients.

To complete the introduction, we would like to mention that here by modifications of

orthogonal polynomials (Pn) we mean any family of orthogonal polynomials ð ~PnÞ which is

related to the initial family (Pn) by a relation of the form

~Pn ¼ In;r;kðxÞPnþrðxÞ þ Jn;r;kðxÞPð1Þnþr21ðxÞ; ð12Þ

where Pð1Þnþr21 is defined in (13) and (14), r and k are nonnegative integers, In,r,k(x)

and Jn,r,k(x) are polynomials in the variable x with the property that they do not depend on n

for n $ k, i.e.:

In;r;k U Ir;k; Jn;r;k U Jr;k – 0; n $ k:

Among these modifications are the rth associated orthogonal polynomials which are obtained

by replacing n by n þ r in the recurrence coefficients bn and gn of the three-term recurrence

relation satisfied by the initial orthogonal polynomial sequence (Pn)

Pnþ1ðxÞ ¼ ðx2 bnÞPnðxÞ2 gnPn21ðxÞ; n $ 1; P0ðxÞ ¼ 1; P21ðxÞ ¼ 0: ð13Þ

The new polynomial family obtained, which is denoted by ðPðrÞn Þ, is orthogonal thanks to

Favard’s theorem [8] since it satisfies

PðrÞnþ1ðxÞ ¼ ðx2 bnþrÞP

ðrÞn ðxÞ2 gnþrP

ðrÞn21ðxÞ; n $ 1; PðrÞ

0 ðxÞ ¼ 1; PðrÞ21ðxÞ ¼ 0: ð14Þ

The relation

PðrÞn ¼

Pr21

Gr21

Pð1Þnþr21 2

Pð1Þr22

Gr21

Pnþr; n $ 0; r $ 1; ð15Þ

linking (Pn) and its rth associated compared to relation (12) yields

k ¼ 0; Ir;0 ¼ 2Pð1Þr22

Gr21

; Jr;0 ¼Pr21

Gr21

; with Gk ¼Ykj¼1

gj; k $ 1; G0 U 1: ð16Þ

Other modifications of the three-term recurrence relation which lead to relations of the

type (12) are the co-recursive and the generalized co-recursive orthogonal polynomials;

the co-recursive associated and the generalized co-recursive associated orthogonal

polynomials; the co-dilated and the generalized co-dilated orthogonal polynomials; the

co-modified and the generalized co-modified orthogonal polynomials. Information about

these families of orthogonal polynomials as well as the relations of type (12) they satisfy can be

found in Refs. [11–13] and references therein.

M. Foupouagnigni130

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2. Properties of some divided-difference operators

2.1 Properties of the quadratic and q-quadratic lattices

Let x(s) be a lattice given by (4). Such lattice satisfies [5]

xðsþ kÞ2 xðsÞ ¼ gk7xkþ1ðsÞ; ð17Þ

xðsþ kÞ þ xðsÞ

2¼ akxkðsÞ þ bk; ð18Þ

for k ¼ 0,1, . . . , with

a0 ¼ 1; a1 ¼ a; b0 ¼ 0; b1 ¼ b; g0 ¼ 0; g1 ¼ 1; ð19Þ

where the sequences (ak), (bk), (gk) satisfy the following relations

akþ1 2 2aak þ ak21 ¼ 0; bkþ1 2 2bk þ bk21 ¼ 2bak; gkþ1 2 gk21 ¼ 2ak; ð20Þ

for k ¼ 0,1, . . . . The lattice x(s) has also the property [28]

xðsþ 1Þ2 þ xðsÞ2 ¼ 2A2ðx1ðsÞÞ ¼ a2x1ðsÞ2 þ a1x1ðsÞ þ a0; ð21Þ

where the aj are constants (to be found later in (36)), with a2 – 0.

For practical reasons, it is important to express xk(s), 2 4 # k # 4, in terms of x(s) and

x(s þ 1) ¼ x2(s) using (17) and (18). For this purpose, we consider equation (17) for k ¼ 2,

s ¼ s 2 1 and k ¼ 2, s ¼ s 2 (3/2); and equation (18) for k ¼ 1; s ¼ s, for k ¼ 1;

s ¼ s 2 (1/2) and for k ¼ 2, s ¼ s 2 1. Then we solve the system of five linear equations

obtained for the unknowns

x23ðsÞ; x21ðsÞ; x22ðsÞ; x1ðsÞ; x3ðsÞ

and obtain the expressions

2ax23ðsÞ ¼ 2ð2a2 1Þ ð2aþ 1Þx2ðsÞ þ ð4a2 þ 2a2 1Þ ð4a2 2 2a2 1ÞxðsÞ

þ 2bð2aþ 1Þ ð4a2 þ 2a2 1Þ;

x22ðsÞ ¼ 2x2ðsÞ þ ð4a2 2 2ÞxðsÞ þ 4bðaþ 1Þ;

2ax21ðsÞ ¼ 2x2ðsÞ þ ð2a2 1Þ ð2aþ 1ÞxðsÞ þ 2bð2aþ 1Þ;

2ax1ðsÞ ¼ x2ðsÞ þ xðsÞ2 2b;

2ax3ðsÞ ¼ ð2a2 1Þ ð2aþ 1Þx2ðsÞ2 xðsÞ þ 2bð2aþ 1Þ;

ð22Þ

from which we deduce

x24ðsÞ ¼ 2ð4a2 2 2Þx2ðsÞ þ ð4a2 2 1Þ ð4a2 2 3ÞxðsÞ þ 4bðaþ 1Þ ð4a2 2 1Þ;

x4ðsÞ ¼ ð4a2 2 2Þx2ðsÞ2 xðsÞ þ 4bðaþ 1Þ:ð23Þ


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2.2 The operators Dx and Sx

The operators Dx and Sx defined in (11) transform any polynomial in x(s) into a polynomial

of the variable x1(s). More precisely we have (see also [28], p. 149):

Proposition 1. If Pn(x (s)) is a polynomial of degree n $ 1 in x (s), then

DxðPnðxðsÞÞÞ ¼ qn21ðx1ðsÞÞ; SxðPnðxðsÞÞÞ ¼ rnðx1ðsÞÞ; ð24Þ

where qn21 and rn are polynomials of degree n 2 1 and n respectively.

More generally,

DxmðPnðxmðsÞÞÞ ¼ ~qn21ðxmþ1ðsÞÞ; SxmðPnðxmðsÞÞÞ ¼ ~rnðxmþ1ðsÞÞ; ð25Þ

where ~qn21 and ~rn are polynomials of degree n 2 1 and n respectively.

Proof. First, for fixed n $ 1, we write the relation

Dð f ðsÞgðsÞÞ ¼f ðsþ 1Þ þ f ðsÞ

2DgðsÞ þ

gðsþ 1Þ þ gðsÞ

2Df ðsÞ;

for f ðsÞ ¼ xn21ðsÞ and g (s) ¼ x (s) and obtain using relations (18) for k ¼ 1

DxxnðsÞ ¼ ðax1ðsÞ þ bÞDxx

n21ðsÞ þ Sxxn21ðsÞ:

In the same way using in addition (21), we obtain by taking f ðsÞ ¼ xn21ðsÞeips and

g ðsÞ ¼ x ðsÞ

SxxnðsÞ ¼ ðax1ðsÞ þ bÞSxx

n21ðsÞ þ ½A2ðx1ðsÞÞ2 ðax1ðsÞ þ bÞ2�Dxxn21ðsÞ:

From the two previous relations, it is easy to show by induction that DxxnðsÞ and Sxx

nðsÞ are

polynomials of degree at most n 2 1 and n respectively in the variable x1(s). Next, we write

DxxnðsÞ ¼

Xn21

k¼0

Dn;kxk1ðsÞ; Sxx

nðsÞ ¼Xnk¼0

Sn;kxk1ðsÞ; ð26Þ

and get from the previous equation a system of two recurrence relations in Dn,n21 and Sn,n.

Solving this system with the initial conditions D1;0 ¼ 1;D2;1 ¼ 2a; S0;0 ¼ 1; S1;1 ¼ a, one

obtains thatDn;n21 – 0; n $ 1 and Sn;n – 0; n $ 0:

This proves the first assertion of the proposition. Equation (25) is obtained by replacing s by

sþ ðm=2Þ in (24). The coefficients Dn;k21 and Sn,k for k ¼ n; n2 1 and n 2 2 will be given

explicitly later. A

2.2.1 The product and quotient rules for Dx and Sx. Next, we state and prove product and

quotient rules for the companion operators Dx and Sx.

Theorem 1. The following statements hold.

1. The operators Dx and Sx obey the following product rules:

Dxð f ðxðsÞÞgðxðsÞÞÞ ¼ Sx f ðxðsÞÞDxgðxðsÞÞ þDx f ðxðsÞÞSxgðxðsÞÞ; ð27Þ

Sxð f ðxðsÞÞgðxðsÞÞÞ ¼ Q2ðx1ðsÞÞDx f ðxðsÞÞDxgðxðsÞÞ þ Sx f ðxðsÞÞSxgðxðsÞÞ; ð28Þ

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where Q2 is a polynomial of degree 2

Q2ðx1ðsÞÞ ¼ ða2 2 1Þx21ðsÞ þ 2bðaþ 1Þx1ðsÞ þ dx; ð29Þ

and dx is a constant depending on a, b and the initial values x (0) and x (1) of x (s):

dx ¼x2ð0Þ þ x2ð1Þ

4a22

ð2a2 2 1Þ

2a2xð0Þxð1Þ2

bðaþ 1Þ

a2ðxð0Þ þ xð1ÞÞ þ

b2ðaþ 1Þ2

a2: ð30Þ

2. The operators Dx and Sx also satisfy the quotient rules

Dx

f ðxðsÞÞ

gðxðsÞÞ

�¼

Sx f ðxðsÞÞDxgðxðsÞÞ2Dx f ðxðsÞÞSxgðxðsÞÞ

Q2ðx1ðsÞÞ½DxgðxðsÞÞ�2 2 ½SxgðxðsÞÞ�

2; ð31Þ

Sx

f ðxðsÞÞ

gðxðsÞÞ

�¼

Q2ðx1ðsÞÞDx f ðxðsÞÞDxgðxðsÞÞ2 Sx f ðxðsÞÞSxgðxðsÞÞ


2; ð32Þ

provided that g ðx ðsÞÞ – 0; s [ ða; bÞ.

3. More generally, relations (27)– (32) remain valid if we replace x and x1 by xm and xmþ1

respectively, m [ C. In particular, the constant dk remains unchanged if we replace x in

(30) by xk; k [ Z, i.e.dxk ¼ dx U d; k [ Z: ð33Þ

Proof. First, we solve the equations

Dx f ðxðsÞÞ ¼f ðxðsþ 1ÞÞ2 f ðxðsÞÞ

xðsþ 1Þ2 xðsÞ; Sx f ðxðsÞÞ ¼

f ðxðsþ 1ÞÞ þ f ðxðsÞÞ

2;

in terms of f (x (s þ 1)) and f (x (s)). Then, we substitute this result for f and g in the equations

Dxð f ðxðsÞÞgðxðsÞÞÞ ¼f ðxðsþ 1ÞÞgðxðsþ 1ÞÞ2 f ðxðsÞÞgðxðsÞÞ

xðsþ 1Þ2 xðsÞ;

Sxð f ðxðsÞÞgðxðsÞÞÞ ¼f ðxðsþ 1ÞÞgðxðsþ 1ÞÞ þ f ðxðsÞÞgðxðsÞÞ

2

and obtain respectively (27) and

Sxð f ðxðsÞÞgðxðsÞÞÞ ¼ðxðsþ 1Þ2 xðsÞÞ2

4Dx f ðxðsÞÞDxgðxðsÞÞ þ Sx f ðxðsÞÞSxgðxðsÞÞ:

By taking f ðx ðsÞÞ ¼ g ðx ðsÞÞ ¼ x ðsÞ in the previous equation, we get

Q2ðx1ðsÞÞ Uðxðsþ 1Þ2 xðsÞÞ2

4¼ Sxx

2ðsÞ2 ðSxxðsÞÞ2:

By means of Proposition 1, Q2(x1(s)) is a polynomial of degree at most 2 in the variable x1(s).

Hence,

ðxðsþ 1Þ2 xðsÞÞ2 ¼ d2x21ðsÞ þ d1x1ðsÞ þ d0;

where dj are constants. Application of the operator Dx1on both sides of the previous equation

and use of equations (17) and (18) for k ¼ 2 produce

d2ðx1ðsþ 1Þ þ x1ðsÞÞ þ d1 ¼ðxðsþ 2Þ2 2xðsþ 1Þ þ xðsÞÞ ðxðsþ 2Þ2 xðsÞÞ

x1ðsþ 1Þ2 x1ðsÞ

¼ 2g2½ða2 2 1Þxðsþ 1Þ þ b2�:


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The previous equation gives by means of (18) for k ¼ 1 and s replaced by s þ (1/2)

2d2ðaxðsþ 1Þ þ bÞ þ d1 ¼ 2g2½ða2 2 1Þxðsþ 1Þ þ b2�:

Therefore,

d2 ¼g2ða2 2 1Þ

a¼ 4ða2 2 1Þ; d1 ¼ 8bðaþ 1Þ

and

Q2ðx1ðsÞÞ ¼ðxðsþ 1Þ2 xðsÞÞ2

4¼ ða2 2 1Þx2

1ðsÞ þ 2bðaþ 1Þx1ðsÞ þ dx: ð34Þ

Equation (30) is obtained by taking s ¼ 0 in the previous equation and using (22).

To prove the second statement, we take f ðx ðsÞÞ ¼ ð1=g ðx ðsÞÞÞ in (27) and (28) to get

DxgðxðsÞÞSx

1

gðxðsÞÞþ SxgðxðsÞÞDx

1

gðxðsÞÞ¼ 0;

SxgðxðsÞÞSx

1

gðxðsÞÞþ Q2ðx1ðsÞÞDxgðxðsÞÞDx

1

gðxðsÞÞ¼ 1:

The determinant of the previous system with respect to the unknowns

Sx

1

gðxðsÞÞand Dx

1

gðxðsÞÞ

is


2 ¼ 4gðxðsÞÞgðxðsþ 1ÞÞ – 0: ð35Þ

Hence,

Sx

1

gðxðsÞÞ¼

2SxgðxðsÞÞ


2;

Dx

1

gðxðsÞÞ¼

DxgðxðsÞÞ


2:

Application of the product rules (27)–(28) to the product f ðx ðsÞÞ £ 1=g ðx ðsÞÞ produces (31)

and (32). The third statement of the theorem is proved by replacing s by s þ (m/2) in

(27)– (32). Also, equation (33) is obtained by direct computation using (22) and (30). A

From now on we denote dx by d, i.e. d U dx.

Remark 2. The operatorsDx andSx appeared already in the works of Magnus [23–25]. In Ref.

[23] Magnus gave also the product and quotient rules for a more general divided-difference

operator. Theorem 1 is more specific to quadratic lattices and gives in detail the coefficients

appearing in the product and quotient rules, in terms of the parameters a, b and d of the lattice.

Corollary 1. As direct consequence of the previous theorem we have:

1. From the quotient rules (31) and (32), one observes that the operators Dx and Sx

transform a rational function of the variable x (s) into a rational function of the variable

x1(s).

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2. If we square both members of (18) for k ¼ 1 and combine with (34), we obtain

x2ðsþ 1Þ þ x2ðsÞ ¼ 2ð2a2 2 1Þx21ðsÞ þ 4bð2aþ 1Þx1ðsÞ þ 2ðb2 þ dÞ; ð36Þ

xðsÞxðsþ 1Þ ¼ x21ðsÞ2 2bx1ðsÞ þ b2 2 d: ð37Þ

Remark 3. The parameters a, b and d are the ingredients for the classical orthogonal

polynomials. As will be shown later, the recurrence coefficients of the classical orthogonal

polynomials are expressed explicitly in terms of these three parameters and the coefficients

of the polynomials f and c involved in the Pearson-type equation satisfied by the

orthogonality weight function (see (8)).

2.2.2 Consequences of the product and quotient rules. The product rules for Dx and Sx

provide the recurrence relations for the coefficients Dn,k and Sn,k.

Proposition 2. The coefficients Dn,k and Sn,k of the expansions (26) satisfy

Sn;k ¼ 2aDn;k21 2 bDn;k þ Dnþ1;k; 0 # k # n; ð38Þ

Snþ1;k ¼ ða2 2 1ÞDn;k22 þ 2ðaþ 1ÞbDn;k21 þ dDn;k þ aSn;k21 þ bSn;k;

0 # k # nþ 1;ð39Þ

with the convention

Dn;n ¼ Dn;nþ1 ¼ Sn;nþ1 ¼ Dn;21 ¼ Dn;22 ¼ Sn;21 ¼ 0; n $ 0: ð40Þ

Proof. Equations (38) and (39) are obtained from the expansion formulaes (26) and the

product rules (27) and (28) for f ðx ðsÞÞ ¼ xnðsÞ and g ðx ðsÞÞ ¼ x ðsÞ. A

Coefficients Dn,k and Sn,k

Substitution of (38) in (39) reads

Dnþ2;k 2 2aDnþ1;k21 þ Dn;k22 ¼ 2bDnþ1;k þ ðd2 b2ÞDn;k þ 2bDn;k21: ð41Þ

The previous equation for k ¼ n þ 1 gives a second-order homogenous linear difference

equation with constant coefficients

Dnþ2;nþ1 2 2aDnþ1;n þ Dn;n21 ¼ 0;

whose solution with the initial conditions D0;21 ¼ 0;D1;0 ¼ 1 is

Dn;n21 ¼

n if a ¼ 1;

q ðn=2Þ2q2ðn=2Þ

q ð1=2Þ2q ð21=2Þ if a ¼ q 1=2þq21=2

2;

8<: ð42Þ

where the parameter q is the one appearing in (4).

Sn,n is therefore deduced using (38) for k ¼ n

Sn;n ¼

1 if a ¼ 1;

q ðn=2Þþq2ðn=2Þ

2if a ¼ q 1=2þq21=2

2:

8<: ð43Þ


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The coefficients Dn;n22 and Sn;n21 are deduced by solving (41) for k ¼ n 2 1 with the initial

conditions D0;22 ¼ D1;21 ¼ 0 and using (42) and (43). Using the computer algebra software

Maple 9 [27] we get with p ¼ q 2

Dn;n22 ¼

13bnðn2 1Þ ð2n2 1Þ if a ¼ 1;

2pbnðp 12nþp nÞ

ðpþ1Þ ðp21Þ22 2p 2bðp n2p2nÞ

ðpþ1Þ ðp21Þ3if a ¼ q 1=2þq21=2

2;

8<: ð44Þ

Sn;n21 ¼

bnð2n2 1Þ if a ¼ 1;

2 bnðp 12n2p nÞp21

if a ¼ q 1=2þq21=2

2:

8<: ð45Þ

The coefficients Dn;n23 and Sn;n22 are obtained likewise using equations (38)–(45) with the

initial conditions D1;22 ¼ D2;21 ¼ 0

Dn;n23

¼

130nðn2 1Þ ðn2 2Þ ð4b2n2 2 8b2nþ 5dþ 3b2Þ; for a ¼ 1;

p 22nþp n

ðp 221Þ2pn2 p n2p2n

ðp 221Þ32p3

� �d

þ p n2p 22n

ðp21Þ3ðpþ1Þ2pn2 þ p 32n27p 22nþ7p nþ1þp n

ðp21Þ4ðpþ1Þpnþ p n2p2n

ðp21Þ5ðpþ1Þ6p3

� �b2; for a ¼ q 1=2þq21=2

2:

8>>>>>>><>>>>>>>:

ð46Þ

and

Sn;n22 ¼

16nðn2 1Þ ð4b2n2 2 8b2nþ 3dþ 3b2Þ; for a ¼ 1;

p n2p 22n

2ðp 221Þdnþ p nþp 22n

ðp21Þ2n2 þ p n23p nþ1þ3p 22n2p 32n

2ðp21Þ3n

� �b2; for a ¼ q 1=2þq21=2

2:

8><>:

ð47Þ

The remaining coefficients Dn;k21 and Sn;k; k ¼ 1 . . . n2 3 can be computed by following the

same procedure.

Additional properties for Dx and Sx

The product rules for Dx and Sx can also be used to write the expressions DxxnðsÞ and

SxxnðsÞ in terms of the polynomials Q1 and Q2 and also in matrix form:

Theorem 2. For any integer n, the following relations hold

DxxnðsÞ ¼

ðQ1ðx1ðsÞÞ þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiQ2ðx1ðsÞÞ

pÞn 2 ðQ1ðx1ðsÞÞ2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiQ2ðx1ðsÞÞ

pÞn

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiQ2ðx1ðsÞÞ

p ; ð48Þ

SxxnðsÞ ¼

ðQ1ðx1ðsÞÞ þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiQ2ðx1ðsÞÞ

pÞn þ ðQ1ðx1ðsÞÞ2


pÞn

2; ð49Þ

where Q1ðx1ðsÞÞ ¼ Sxx ðsÞ ¼ ax1ðsÞ þ b and Q2ðx1ðsÞ given by (29).

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Proof. Let n be a nonnegative integer. From (27) and (28) for f ðx ðsÞÞ ¼ x ðsÞ and

g ðx ðsÞÞ ¼ xnðsÞ, we obtain the relation

Dxxnþ1ðsÞ

Sxxnþ1ðsÞ

" #¼

Q1ðx1ðsÞÞ 1

Q2ðx1ðsÞÞ Q1ðx1ðsÞÞ

" #Dxx

nðsÞ

SxxnðsÞ

" #

from which we deduce

DxxnðsÞ

SxxnðsÞ

" #¼

Q1ðx1ðsÞÞ 1

Q2ðx1ðsÞÞ Q1ðx1ðsÞÞ

" #n0

1

" #:

Since the matrix involved in the previous equation is invertible for its determinant being

different from zero, we perform linear algebra calculus to compute its nth power in terms of

Q1,Q2 and n and deduce (48) and (49) for any nonnegative integer n.

Taking into account that relations (48) and (49) are satisfied for nonnegative integers, these

relations for negative integers are obtained by using (31) and (32) for f ðx ðsÞÞ ¼ 1 and

g ðx ðsÞÞ ¼ xnðsÞ

Dxx2nðxÞ ¼ 2

DxxnðsÞ

Q2ðx1ðsÞÞ½DxxnðsÞ�2 2 ½SxxnðsÞ�

2

¼ðQ1ðx1ðsÞÞ þ


pÞ2n 2 ðQ1ðx1ðsÞÞ2


pÞ2n

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiQ2ðx1ðsÞÞ

p ;

Sxx2nðxÞ ¼

SxxnðsÞ

Q2ðx1ðsÞÞ½DxxnðsÞ�2 2 ½SxxnðsÞ�

2

¼ðQ1ðx1ðsÞÞ þ


pÞ2n þ ðQ1ðx1ðsÞÞ2


pÞ2n

2:

The proof is therefore complete. Notice however that (48) and (49) can also be obtained

directly using the equations

Q1ðx1ðsÞÞ ¼ ax1ðsÞ þ b ¼xðsþ 1Þ þ xðsÞ

2;

Q2ðx1ðsÞÞ ¼ ða2 2 1Þx21ðsÞ þ 2bðaþ 1Þx1ðsÞ þ d ¼

ðxðsþ 1Þ2 xðsÞÞ2

4

to express x (s) and x (s þ 1) in terms of Q1ðx1ðsÞÞ and Q2ðx1ðsÞÞ

xðsþ 1Þ ¼ Q1ðx1ðsÞÞ þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiQ2ðx1ðsÞÞ

p; xðsÞ ¼ Q1ðx1ðsÞÞ2


p:

A

The following theorem gives relations between the products of the operators Dx and Sx

Dx21Dx22

; Dx21Sx22

; Sx21Dx22

; Sx21Sx22

:

These relations happen to be very important in the next part of this work as well as in the

characterization of the classical orthogonal polynomials [14].


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Theorem 3. The following relations hold:

Dx21Sx22

T22 ¼ aSx21Dx22

T22 þ U1ðsÞDx21Dx22

T22; ð50Þ

Sx21Sx22

T22 ¼ U1ðsÞSx21Dx22

T22 þ aU2ðsÞDx21Dx22

T22 þ I; ð51Þ

where

U1ðsÞ U U1ðxðsÞÞ ¼ ða2 2 1ÞxðsÞ þ bðaþ 1Þ;

U2ðsÞ U U2ðxðsÞÞ ¼ Q2ðxðsÞÞ ¼ ða2 2 1Þx2ðsÞ þ 2bðaþ 1ÞxðsÞ þ d: ð52Þ

Here, the operator Tm which acts on the variable s is defined by

T m f ðsÞ ¼ f sþm

2

� �; or Tm f ðxðsÞÞ ¼ f ðxmðsÞÞ; ð53Þ

while I is the identity operator If ðsÞ ¼ f ðsÞ.

Proof. We write

Dx21Sx22

f ðx22ðsÞÞ ¼ B1Sx21Dx22

f ðx22ðsÞÞ þ B2Dx21Dx22

f ðx22ðsÞÞ þ B3 f ðxðsÞÞ;

Sx21Sx22

f ðx22ðsÞÞ ¼ C1Sx21Dx22

f ðx22ðsÞÞ þ C2Dx21Dx22

f ðx22ðsÞÞ þ C3 f ðxðsÞÞ;

and use (11) with x replaced by x21 or x22 to transform the previous equations into linear

combinations of f ðx ðs2 1ÞÞ; f ðx ðsÞÞ and f ðx ðsþ 1ÞÞ. Then we equate the coefficients of

f ðx ðs2 1ÞÞ; f ðx ðsÞÞ and f (x (s þ 1)) in the resulting equations and obtain for each equation a

system of three linear equations in terms of the unknowns Bi (respectively (Ci)). Solving

these two systems, we obtain

B1 ¼1

2

xðsþ 1Þ2 xðs2 1Þ

xðsþ ð1=2ÞÞ2 xðs2 ð1=2ÞÞ; B2 ¼

xðsþ 1Þ2 2xðsÞ þ xðs2 1Þ

4; B3 ¼ 0;

and

C1 ¼xðsþ 1Þ2 2xðsÞ þ xðs2 1Þ

4; C3 ¼ 1;

C2 ¼ 21

8xðs2 1Þx sþ

1

2

�þ

1

8xðs2 1Þx s2

1

2

�þ

1

8xðsþ 1Þx sþ

1

2

�

21

8xðsþ 1Þx s2

1

2

�:

The previous equation combined with (22) and (49) yields

B1 ¼ a; B2 ¼ U1ðsÞ; B3 ¼ 0; C1 ¼ U1ðsÞ; C2 ¼ aU2ðsÞ; C3 ¼ 1:

A

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2.3 The operators Fx and Mx

For the manipulation of the difference equations for orthogonal polynomials, it is sometimes

more convenient to work with the operators Fx and Mx

Fx f ðsÞ ¼D

Dxðs2 12Þ

7f ðsÞ

7xðsÞ; Mx f ðsÞ ¼

1

2

Df ðsÞ

DxðsÞþ

7f ðsÞ

7xðsÞ

�; ð54Þ

for they share the property of transforming any polynomial in the variable x(s) into a

polynomial of lower degree but in the same variable x(s). This property is similar to the one

of the usual derivative.

2.3.1 The product and quotient rules.

Theorem 4. The following properties hold:

1. If Pn(x (s)) is a polynomial of degree n in the variable x (s), then FxðPnðx ðsÞÞÞ and

MxðPnðx ðsÞÞÞ are polynomials of degree n 2 2 and n 2 1 respectively in x (s).

2. Fx and Mx obey the product rules

FxMx f ðsÞ ¼ ð2a2 2 1ÞMxFx f ðsÞ þ 2aU1ðsÞFxFx f ðsÞ; ð55Þ

MxMx f ðsÞ ¼ aFx f ðsÞ þ 2aU1ðsÞMxFx f ðsÞ þ ð2a2 2 1ÞU2ðsÞFxFx f ðsÞ; ð56Þ

where the expression FxMx f ðsÞ refers to FxðMx f ðsÞÞ and

U1ðsÞ ¼ ðaþ 1Þ½ða2 1ÞxðsÞ þ b�;

U2ðsÞ ¼ ða2 2 1Þx2ðsÞ þ 2bðaþ 1ÞxðsÞ þ d:ð57Þ

Proof. From the definition of Dx;Sx; Fx and Mx (see (11) and (54)) we have

Fxð f ðxðsÞÞ ¼D

Dxðs2 12Þ

7f ðxðsÞÞ

7xðsÞ¼ Dx21

Dx22f ðx22ðsÞÞ; ð58Þ

Mxð f ðxðsÞÞ ¼1

2

Df ðxðsÞÞ

DxðsÞþ

7f ðxðsÞÞ

7xðsÞ

�¼ Sx21

Dx22f ðx22ðsÞÞ: ð59Þ

The previous relations are equivalent to

Fx ¼ Dx21Dx22

T22; Mx ¼ Sx21Dx22

T22: ð60Þ

Let Pn(x (s)) be a polynomial of degree n in x (s). From (58) and Proposition 1, we deduce that

Dx22transforms Pnðx22ðsÞÞ into a polynomial of degree n 2 1 in x21ðsÞ and Dx21

transforms

Dx22Pnðx22ðsÞÞ into a polynomial of degree n 2 1 in x0ðsÞ ¼ x ðsÞ. Therefore, FxPnðx ðsÞÞ is a

polynomial of degree n 2 2 in x (s). Similarly, we deduce that MxPnðx ðsÞÞ is a polynomial of

degree n 2 1 in x (s).

Before starting the proof of the second statement, one should keep in mind the relations

TmDx ¼ Dxm ; TmSx ¼ Sxm : ð61Þ


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In the first step, we combine (60) with (61) to get

FxMx ¼ Dx21Dx22

T22 Sx21Dx22

T22

� ¼ Dx21

Dx22Sx23

Dx24T24

� ¼ Dx21

Dx22Sx23

� Dx24

T24 ¼ Dx21Dx22

Sx23T23

� T3 Dx24

T24��

:

Use of (50) (and later on (27)) transform the previous equation into

FxMx ¼ Dx21aSx22

Dx23T23 þ U1ðx21ðsÞÞDx22

Dx23T23

� T3Dx24

T24

¼ aDx21Sx22

Dx23Dx24

T24 þDx21U1ðx21ðsÞÞDx22

Dx23T23

� T3Dx24

T24

¼ a aSx21Dx22

T22 þ U1ðxðsÞÞDx21Dx22

T22

� T2Dx23

Dx24T24

þDx21U1ðx21ðsÞÞDx22

Dx23T23

� T3Dx24

T24

¼ ð2a2 2 1ÞMxFx þ 2aU1ðxðsÞÞFxFx;

taking into account that

Dx21½U1ðx21ðsÞÞ� ¼ a2 2 1 and Sx21

½U1ðx21ðsÞÞ� ¼ aU1ðxðsÞ:

Relation (56) is obtained in the same way using (28), (50) and (51). A

Theorem 5. The following product and quotient rules hold:

Fxð fgÞ ¼ Fxð f Þgþ fFxðgÞ þ 2aMxð f ÞMxðgÞ þ 2U1½Fxð f ÞMxðgÞ þMxð f ÞFxðgÞ�

þ 2aU2Fxð f ÞFxðgÞ; ð62Þ

Mxð fgÞ ¼ Mxð f Þgþ fMxðgÞ þ 2U1Mxð f ÞMxðgÞ þ 2aU2½Fxð f ÞMxðgÞ þMxð f ÞFxðgÞ�

þ 2U1U2Fxð f ÞFxðgÞ; ð63Þ

Fxf

g

�¼ 2af ½MxðgÞ�

2 2 2aMxð f ÞMxðgÞgþ Fxð f Þg2 2 fgFxðgÞ þ 2aU2Fxð f ÞFxðgÞg

22aU2 f ½FxðgÞ�2 þ 2U1Fxð f ÞMxðgÞg2 2U1Mxð f ÞFxðgÞg

�=

U2½U1FxðgÞ þ aMxðgÞ�2 2 ½gþ 2U1MxðgÞ þ 2aU2FxðgÞ�

2 �

g; ð64Þ

Mx

f

g

�¼ 2U1U2 f ½FxðgÞ�

222U1U2Fxðf ÞFxðgÞgþMxðf Þg22 fgMxðgÞ

þ2aU2Mxðf ÞFxðgÞg22aU2Fxðf ÞMxðgÞg

þ2U1Mxðf ÞMxðgÞg22U1 f ½MxðgÞ�2�=

U2½U1FxðgÞþaMxðgÞ�22½gþ2U1MxðgÞþ2aU2FxðgÞ�

2 �

g; ð65Þ

with

f ; f ðsÞ; Uj;UjðsÞ and g;gðsÞ–0; ;s[ ða;bÞ:

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Proof. Use of (60) gives the equation

Fxð f ðxðsÞÞgðxðsÞÞÞ ¼ Dx21Dx22

ð f ðx22ðsÞÞgðx22ðsÞÞÞ

which using (27) and (28) is transformed into

Fxð f ðxðsÞÞgðxðsÞÞÞ ¼ Dx21Dx22

f ðx22ðsÞÞSx22gðx22ðsÞÞ þ Sx22

f ðx22ðsÞÞDx22gðx22ðsÞÞ

� ¼ Dx21

Dx22f ðx22ðsÞÞSx21

Sx22gðx22ðsÞÞ

þ Sx21Dx22

f ðx22ðsÞÞDx21Sx22

gðx22ðsÞÞ

þDx21Sx22

f ðx22ðsÞÞSx21Dx22

gðx22ðsÞÞ

þ Sx21Sx22

f ðx22ðsÞÞDx21Dx22

gðx22ðsÞÞ:

Elimination of the products of the form Sx21Sx22

f ðx22ðsÞÞ and Dx21Sx22

f ðx22ðsÞÞ in the

previous equation using (50) and (51) produces

Fxð f ðxðsÞÞgðxðsÞÞÞ ¼ Fxð f Þ½U1ðsÞMxðgÞ þ aU2ðsÞFxðgÞ þ g� þMxð f Þ½aMxðgÞ þ U1ðsÞFxðgÞ�

þ FxðgÞ½U1ðsÞMxð f Þ þ aU2ðsÞFxð f Þ þ f � þMxðgÞ½aMxð f Þ

þ U1ðsÞFxð f Þ�

¼ Fxð f Þgþ fFxðgÞ þ 2aMxð f ÞMxðgÞ þ 2U1½Fxð f ÞMxðgÞ þMxð f ÞFxðgÞ�

þ 2aU2Fxð f ÞFxðgÞ:

Relation (63) is derived in the same way.

The quotient rules (64) and (65) are derived by applying the product rules (62) and (63)

to f ðsÞ £ ð1=g ðsÞÞ. In fact, we first express Fxð1=g ðsÞÞ and Mxð1=g ðsÞÞ for

g ðsÞ – 0;;s [ ða; bÞ, in terms of g ðsÞ; Fxg ðsÞ and Mxg ðsÞ. For this purpose, we take

f ðsÞ ¼ ð1=g ðsÞÞ in (62) and (63) and get the linear system in Fxð1=gÞ and Mxð1=gÞ

½gþ 2U1MxðgÞ þ 2aU2FxðgÞ�Fx1

g

�þ 2½U1FxðgÞ þ aMxðgÞ�Mx

1

g

�¼ 2

1

gFxðgÞ

2U2½U1FxðgÞ þ aMxðgÞ�Fx1

g

�þ ½gþ 2U1MxðgÞ þ 2aU2FxðgÞ�Mx

1

g

�¼ 2

1

gMxðgÞ

whose determinant is

½gþ 2U1MxðgÞ þ 2aU2FxðgÞ�2 2 4U2½U1FxðgÞ þ aMxðgÞ�

2 ¼ gðs2 1Þgðsþ 1Þ

– 0;;s [ ða; bÞ

Therefore, Fxð1=gÞ and Mxð1=gÞ are uniquely determined from the previous linear system,

and quotient rules Fxðf=gÞ and Mxðf=gÞ are deduced by application of the product rules (62)

and (63) to f ðsÞ £ ð1=g ðsÞÞ. A


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2.3.2 Consequences of the product and quotient rules. The product rules provide the

recurrence relation for the coefficients Fn,k and Mn,k of the expansion

FxxnðsÞ ¼

Xn22

k¼0

Fn;kxkðsÞ; Mxx

nðsÞ ¼Xn21

k¼0

Mn;kxkðsÞ: ð66Þ

Proposition 3. The coefficients Fn,k and Mn,k satisfy

Fnþ1;k 2 2aMn;k 2 2bðaþ 1ÞFn;k þ ð1 2 2a2ÞFn;k21 ¼ 0; 0 # k # n2 1 ð67Þ

Mnþ1;k 2 2bðaþ 1ÞMn;k 2 2aða2 2 1ÞFn;k22 2 2adFn;k 2 4abðaþ 1ÞFn;k21

þ ð1 2 2a2ÞMn;k21 2 dn;k ¼ 0; 0 # k # n; ð68Þ

where dn,k is the Kronecker symbol, with the convention

Fnþ1;n ¼ Fn;nþ1 ¼ Mn;nþ1 ¼ Fn;n ¼ Mn;n ¼ Fn;21 ¼ Fn;22 ¼ Mn;21 ¼ 0; n $ 0:

Proof. The proof is obtained using (66) and the product rules (62) and (63) for

f ðsÞ ¼ xnðsÞ; g ðsÞ ¼ x ðsÞ. A

The previous proposition allows to compute recurrently the coefficients Fn,k and Mn,k.

Furthermore, these coefficients can also be computed via the following relations in terms of

the coefficients Dn,k and Sn,k.

Proposition 4. The coefficients Dn,k and Sn,k of the expansions (26) are related to the

coefficients Fn,k and Mn,k of the expansions (66) by

Fn;k ¼Xn21

j¼kþ1

Dn;jDj;k; 0 # k # n2 2; ð69Þ

Mn;k ¼Xn21

j¼k

Dn;jSj;k; 0 # k # n2 1: ð70Þ

Proof. The proof follows from equations (26), (58), (59) and (66). Notice that the

coefficients Fn;n222k;Mn;n212k; 0 # k # 2 are computed explicitly in (97) and (98). A

As another consequence of the product rule, we state the following:

Proposition 5. For any nonnegative integer n, the following relations hold

FxxnðsÞ ¼

Xn21

k¼0

xkðsÞ

2ffiffiffiffiffiffiffiffiffiffiffiU2ðsÞ

p ðV1ðsÞ þ 2affiffiffiffiffiffiffiffiffiffiffiU2ðsÞ

pÞn212k 2 ðV1ðsÞ2 2a

ffiffiffiffiffiffiffiffiffiffiffiU2ðsÞ

pÞn212k

h i; ð71Þ

MxxnðsÞ ¼

Xn21

k¼0

xkðsÞ

2ðV1ðsÞ þ 2a


pÞn212k þ ðV1ðsÞ2 2a


pÞn212k

h i; ð72Þ

Fx1

xnðsÞ¼2

Xnk¼1

x2kðsÞ

2ffiffiffiffiffiffiffiffiffiffiffiU2ðsÞ

p ðV1ðsÞþ2affiffiffiffiffiffiffiffiffiffiffiU2ðsÞ

pÞk2n21 2 ðV1ðsÞ22a


pÞk2n21

h i; ð73Þ

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Mx

1

xnðsÞ¼ 2

Xnk¼1

x2kðsÞ

2ðV1ðsÞ þ 2a


pÞk2n21 þ ðV1ðsÞ2 2a


pÞk2n21

h i; ð74Þ

with V1ðsÞ ¼ x ðsÞ þ 2U1ðsÞ where U1(s) and U2(s) are given by (57).

Proof. Using the product rules (62) and (63) for f ðx ðsÞÞ ¼ xnðsÞ; g ðx ðsÞÞ ¼ x ðsÞ

(respectively f ðx ðsÞÞ ¼ ð1=xnðsÞÞ; g ðx ðsÞÞ ¼ x ðsÞ), we obtain

FxxnðsÞ

MxxnðsÞ

" #¼

V1ðxðsÞÞ 2a

2aU2ðxðsÞÞ V1ðxðsÞÞ

" #Fxx

n21ðsÞ

Mxxn21ðsÞ

" #þ

0

xn21ðsÞ

" #

and

Fxð1=xn21ðsÞÞ

Mxð1=xn21ðsÞÞ

24

35 ¼

V1ðxðsÞÞ 2a

2aU2ðxðsÞÞ V1ðxðsÞÞ

" #Fxð1=x

nðsÞÞ

Mxð1=xnðsÞÞ

" #þ

0

ð1=xnðsÞÞ

" #

respectively.

The changes of variables

Wn

!

¼

FxxnðsÞ

MxxnðsÞ

24

35; Xn

!

¼

0

xnðsÞ

24

35;

Yn

!

¼

Fxð1=xnðsÞÞ

Mxð1=xnðsÞÞ

24

35; Zn

!

¼

0

ð1=xnðsÞÞ

24

35; A ¼

V1 2a

2aU2 V1

24

35

transform the previous two equations into relations

Wn

!

¼ AWn21

��!

þ Xn21

��!

; Yn21

��!

¼ AYn

!

þ Zn

!

;

whose iteration taking into account that the matrix A is invertible gives

Wn

�!

¼Xn21

k¼0

Anþ12k Xk

�!

; Yn

!

¼ 2Xnk¼1

Ak2n21Zk

!

:

Equations (71)– (74) are obtained from the previous ones by computing the kth power of the

matrix A in terms of its eigenvalues V1ðsÞ þ 2affiffiffiffiffiffiffiffiffiffiffiU2ðsÞ

pand V1ðsÞ2 2a


p. A

As a fourth consequence of the product rules, we shall prove another important result

concerning the linear divided-difference equation of higher order satisfied by products of

functions each of them satisfying a linear divided-difference equation. For this purpose, we

start with the following preliminaries.

Lemma 1. Let f (s) be a function of the variable x (s) satisfying a second-order divided-

difference equation

Fx f ðsÞ þ a1ðsÞMx f ðsÞ þ a0ðsÞ f ðsÞ ¼ 0; ð75Þ

where a0 and a1 are given functions of x (s).


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Then the expressions FxFx f ðsÞ;MxFx f ðsÞ; FxMx f ðsÞ;MxMx f ðsÞ and Fx f ðsÞ can be

written uniquely in the form

c1ðsÞ f ðsÞ þ c2ðsÞMx f ðsÞ;

where c1(s) and c2(s) are functions of a0(s) and a1(s).

Proof. Assuming that x ðsþ 1Þ – x ðsÞ for s [ ða; bÞ, equation (75) is equivalent to

ð7x1ðsÞa1ðsÞ þ 2Þ f ðsþ 1Þ þ C0ðsÞ f ðsÞ þ ð7x1ðsÞa1ðsÞ2 2Þ f ðs2 1Þ ¼ 0; ð76Þ

where C0(s) is a function of a1ðsÞ; a0ðsÞ and x (s).

1. If a1ðsÞ ¼ ^ð2=7x1ðsÞÞ, then the previous equation is equivalent to

( f (s þ 1))/( f (s)) ¼ C1(s); therefore, f ðsþ jÞ; j ¼ 1 . . . is proportional to f (s).

2. If a1ðsÞ – ^ð2=7x1ðsÞÞ, then from (76), we get

f ðs2 2Þ

¼ð½24 þ 2a1ðsÞ7x1ðsÞ2 2a1ðs2 1Þ7x1ðs2 1Þ þ a1ðs2 1Þ7x1ðs2 1Þa1ðsÞ7x1ðsÞ� f ðsÞ

ð22 þ a1ðsÞ7x1ðsÞÞ ð22 þ a1ðs2 1Þ7x1ðs2 1ÞÞ

þðC0ðs2 1ÞC0ðsÞÞ f ðsÞ

ð22 þ a1ðsÞ7x1ðsÞÞ ð22 þ a1ðs2 1Þ7x1ðs2 1ÞÞ

þC0ðs2 1Þ ð2 þ a1ðsÞ7x1ðsÞÞ f ðsþ 1Þ

ð22 þ a1ðsÞ7x1ðsÞÞ ð22 þ a1ðs2 1Þ7x1ðs2 1ÞÞ ð77Þ

f ðs2 1Þ ¼ð7x1ðsÞa1ðsÞ þ 2Þ f ðsþ 1Þ þ C0ðsÞ f ðsÞ

7x1ðsÞa1ðsÞ2 2;

f ðsþ 2Þ ¼ðDx1ðsÞa1ðsþ 1Þ2 2Þ f ðsÞ2 C0ðsþ 1Þ f ðsþ 1Þ

Dx1ðsÞa1ðsþ 1Þ þ 2:

Then, we use the definition of the operators Fx and Mx of (54) to write FxFx f ðsÞ;MxFx f ðsÞ;

FxMx f ðsÞ and MxMx f ðsÞ as linear combination of f ðsþ jÞ;22 # j # 2.

Next, we use the previous equations to write the expressions FxFx f ðsÞ;MxFx f ðsÞ,

FxMx f ðsÞ and MxMx f ðsÞ as linear combination of f (s) and f (s þ 1) only. Finally we use

(75) and the following equation linking the operators Dx; Fx and Mx

f ðsþ 1Þ ¼ f ðsÞ þ DxðsÞMx f ðsÞ þ1

27x1ðsÞDxðsÞFx f ðsÞ

to convert the expressions FxFx f ðsÞ;MxFx f ðsÞ, FxMx f ðsÞ and MxMx f ðsÞ and MxMx f ðsÞ

from the linear combination of f (s) and f (s þ 1) to the linear combination of f (s) and

Mx f ðsÞ. A

Theorem 6. Let f (s) and g (s) be two functions of the variable x (s) satisfying respectively

Fx f ðsÞ þ a1ðsÞMx f ðsÞ þ a0ðsÞ f ðsÞ ¼ 0; FxgðsÞ þ b1ðsÞMxgðsÞ þ b0ðsÞgðsÞ ¼ 0; ð78Þ

where aj and bj are given functions of x (s).

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Then, the product f (s) g (s) is a solution of a fourth-order divided-difference equation of

the form

I4ðsÞFxFxyðsÞ þ I3ðsÞMxFxyðsÞ þ I2ðsÞFxyðsÞ þ I1ðsÞMxyðsÞ þ I0ðsÞyðsÞ ¼ 0

where Ij are functions of aj and bj. If the ajðsÞ; j ¼ 0; 1 and the bj(s),j ¼ 0,1 are rational

functions in x (s), then the coefficients IjðsÞ; j ¼ 0 . . . 4 can be chosen to be polynomials in the

variable x (s).

Proof. We apply the identity operator as well as the operators FxFx;MxFx; Fx and Mx to the

equationyðsÞ ¼ f ðsÞgðsÞ;

and use the product rules (62) and (63) to get five equations whose right-hand sides are linear

combinations of expressions of the form p1ðsÞp2ðsÞ with

pjðsÞ [ {FxFxhjðsÞ; MxFxhjðsÞ; FxMxhjðsÞ; MxMxhjðsÞ; FxhjðsÞ}; j ¼ 1; 2;

with h1 ¼ f ; h2 ¼ g. These right-hand sides are transformed by means of the previous lemma

into linear combinations of

f ðsÞgðsÞ; f ðsÞMxgðsÞ; ½Mx f ðsÞ�gðsÞ and ½Mx f ðsÞ�MxgðsÞ:

Thus, these five equations can be written as

X0;0 ¼ yðsÞ;

c2;1ðsÞX0;0ðsÞ þ c2;2ðsÞX0;1ðsÞ þ c2;3ðsÞX1;0ðsÞ þ c2;4ðsÞX1;1ðsÞ ¼ MxyðsÞ;

c3;1ðsÞX0;0ðsÞ þ c3;2ðsÞX0;1ðsÞ þ c3;3ðsÞX1;0ðsÞ þ c3;4ðsÞX1;1ðsÞ ¼ FxyðsÞ;

c4;1ðsÞX0;0ðsÞ þ c4;2ðsÞX0;1ðsÞ þ c4;3ðsÞX1;0ðsÞ þ c4;4ðsÞX1;1ðsÞ ¼ MxFxyðsÞ;

c5;1ðsÞX0;0ðsÞ þ c5;2ðsÞX0;1ðsÞ þ c5;3ðsÞX1;0ðsÞ þ c5;4ðsÞX1;1ðsÞ ¼ FxFxyðsÞ;

ð79Þ

with the notations

X0;0 ¼ f ðsÞgðsÞ; X0;1 ¼ f ðsÞMxgðsÞ; X1;0 ¼ ½Mx f ðsÞ�gðsÞ; X1;1 ¼ ½Mx f ðsÞ�MxgðsÞ

where cj,k are functions of aj and bj. The system (79) contains 5 linear equations for

4 unknowns, namely Xj;kðsÞ; j; k ¼ 0; 1. For the solutions of this system to exist, it is

necessary for y (s) to satisfy the equation

1 0 0 0 yðsÞ

c2;1ðsÞ c2;2ðsÞ c2;3ðsÞ c2;4ðsÞ MxyðsÞ

c3;1ðsÞ c3;2ðsÞ c3;3ðsÞ c3;4ðsÞ FxyðsÞ

c4;1ðsÞ c4;2ðsÞ c4;3ðsÞ c4;4ðsÞ MxFxyðsÞ

c5;1ðsÞ c5;2ðsÞ c5;3ðsÞ c5;4ðsÞ FxFxyðsÞ

��

��¼ 0; ð80Þ

which is the fourth-order divided-difference equation desired. A

As consequence of this theorem, we claim the following:

Corollary 2. If f j; j ¼ 1; . . . n, are functions of the variable x (s) such that any fj satisfies

a linear divided-difference equation of order rj involving only the operators Fx andMx, then


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the product f ¼Qn

j¼1f j satisfies a divided-difference equation of order r ¼Qn

j¼1rj involving

only (at most) the operators

MjxF

kx; j ¼ 0; 1; and 0 # 2k þ j # r ¼

Ynj¼0

rj:

3. Recurrence coefficients for classical orthogonal polynomials

3.1 From orthogonality to second-order difference equation

The definition of classical orthogonal polynomials given in [5] is not similar to those of the

very classical orthogonal polynomials, because, according to this definition, for a family of

polynomials to be classical, it should be orthogonal with respect to a weight function

satisfying a Pearson-type equation; and, should in addition, satisfy a second-order difference

equation. The requirement for Pn to satisfy a second-order difference equation, which in the

case of the very classical orthogonal polynomials is a consequence of the orthogonality, is

redundant. This condition can be omitted.

Let us remind that Atakishiyev et al. [5], using the second-order difference equation (1)

satisfied by an orthogonal family (Pn), the Pearson-type equation (8) satisfied by the

orthogonality weight and the border conditions (10), obtained the orthogonality relation (5).

Here, we prove the converse: The orthogonality relation plus the Pearson-type equation and

the border conditions (all provided in Definition 1) lead to the second-order divided-

difference equation of form (1).

Theorem 7. Let (Pn) be a sequence of polynomials orthogonal with respect to a weight

function r (see (5)) satisfying (8) and (10). Then, each Pn satisfies

f ðxðsÞÞFxPnðxðsÞÞ þ c ðxðsÞÞMxPnðxðsÞÞ þ lnPnðxðsÞÞ ¼ 0; ð81Þ

where ln is a constant term given by

ln ¼ 2Dn;n21ðf2Dn21;n22 þ c1Sn21;n21Þ; ð82Þ

with

f ðxðsÞÞ ¼ f2x2ðsÞ þ f1xðsÞ þ f0; c ðxðsÞÞ ¼ c1xðsÞ þ c0: ð83Þ

In order to simplify the proof, we state and prove the following lemma.

Lemma 2. Under the hypothesis of the previous theorem, the following identities hold:

D sðsÞrðsÞ7PnðxðsÞÞ

7xðsÞÞ

� �PmðxðsÞÞ2 D sðsÞrðsÞ

7PmðxðsÞÞ

7xðsÞÞ

� �PnðxðsÞÞ

¼ D{sðsÞrðsÞWðPnðxðsÞÞ;PmðxðsÞÞÞ}; ð84Þ

D sðsÞrðsÞ7PmðxðsÞÞ

7xðsÞÞ

� �¼ ½f ðxðsÞFxPmðxðsÞÞ þ c ðxðsÞÞMxPmðxðsÞ�rðsÞ7x1ðsÞ; ð85Þ

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where

WðPnðxðsÞÞ; PmðxðsÞÞÞ ¼ PnðxðsÞÞ7PmðxðsÞÞ

7xðsÞÞ2 PmðxðsÞÞ

7PnðxðsÞÞ

7xðsÞð86Þ

is the discrete analog of the Wronskian.

The Wronskian WðPnðxðsÞÞ;PmðxðsÞÞÞ is a polynomial of degree at most n þ m 2 1 in the

variable x21ðsÞ ¼ xðs2 ð1=2ÞÞ.

Proof. First, we observe that the Pearson-type equation (8) is equivalent to

rðsþ 1Þ

rðsÞ¼

sðsÞ þ f ðsÞ7x1ðsÞ

sðsþ 1Þ; with f ðxðsÞÞ ; f ðsÞ: ð87Þ

Using the relation

Dð f ðsÞgðsÞÞ ¼ f ðsÞDgðsÞ þ gðsþ 1ÞDf ðsÞ ¼ f ðsþ 1ÞDgðsÞ þ gðsÞDf ðsÞ; ð88Þ

we obtain for given n;m [ N,

D sðsÞrðsÞ7PnðxðsÞÞ

7xðsÞÞ

� �PmðxðsÞÞ2 D sðsÞrðsÞ

7PmðxðsÞÞ

7xðsÞÞ

� �PnðxðsÞÞ

¼ D½sðsÞrðsÞWðPnðxðsÞÞ;PmðxðsÞÞÞ�;

where

WðPnðxðsÞÞ;PmðxðsÞÞÞ ¼ PnðxðsÞÞ7PmðxðsÞÞ

7xðsÞÞ2 PmðxðsÞÞ

7PnðxðsÞÞ

7xðsÞÞ: ð89Þ

The Wronskian W ðPnðx ðsÞÞ;Pmðx ðsÞÞÞ thanks to (88) can also be written as

WðPnðxðsÞÞ;PmðxðsÞÞÞ ¼ Sx22Pnðx22ðsÞÞDx22

Pmðx22ðsÞÞ

2 Sx22Pmðx22ðsÞÞDx22

Pnðx22ðsÞÞ: ð90Þ

Therefore, from (25), we remark that W ðPnðx ðsÞÞ;Pmðx ðsÞÞÞ is a polynomial of degree at

most n þ m 2 1 in the variable x21ðsÞ ¼ x ðs2 ð1=2ÞÞ.

For the second identity, we use again (88), then (8) together with (87) and finally the

identity (easy to obtain)

1

2

DyðsÞ

DxðsÞþ

7yðsÞ

7xðsÞ

� �¼

7yðsÞ

7xðsÞþ

1

27

7yðsÞ

7xðsÞ

� �;

to get

D sðsÞrðsÞ7PmðxðsÞÞ

7xðsÞÞ

� �¼ D½sðsÞrðsÞ�

7PmðxðsÞÞ

7xðsÞþ sðsþ 1Þrðsþ 1ÞD

7PmðxðsÞÞ

DxðsÞ

¼ c ðsÞ7x1ðsÞrðsÞ7PmðxðsÞÞ

7xðsÞþ ðsðsÞ þ c ðsÞ7x1ðsÞÞrðsÞD

7PmðxðsÞÞ

7xðsÞ

¼ c ðsÞ7x1ðsÞrðsÞ MxPmðxðsÞ21

2D7PmðxðsÞÞ

7xðsÞÞ

� �

þ ðsðsÞ þ c ðsÞ7x1ðsÞÞrðsÞD7PmðxðsÞÞ

7xðsÞ

¼ ½fxðsÞFxPmðxðsÞÞ þ c ðxðsÞÞMxPmxðsÞ�rðsÞ7x1ðsÞ;

with f given by (9). A


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Next we give the proof of Theorem 7.

Proof. We set n;m [ N and write

VnðxðsÞÞ ¼ f ðxðsÞÞFxPnðxðsÞÞ þ c ðxðsÞÞMxPnðxðsÞ; n $ 1; V0ðxðsÞÞ ¼ 1:

We shall prove that the family (Vn) is also orthogonal with respect to the weight r (s). First we

prove that degreeðVnÞ ¼ n; n $ 1. To do this, we assume that (Pn) is monic and use the

expansions (66) of FxxnðsÞ and Mxx

nðsÞ to obtain that the leading coefficient of Vn, which we

denote by hn, is

hn ¼ f2Fn;n22 þ c1Mn;n21 ¼ Dn;n21ðf2Dn21;n22 þ c1Sn21;n21Þ; n $ 1; h0 ¼ 1; ð91Þ

with the latter identity obtained thanks to Proposition 4.

Following [5], page 204, if we define the moments of the weight function r by

Mn ¼Xb21

s¼a

½xðsÞ2 xðaþ n2 1Þ�ðnÞrðsÞ7x1ðsÞ

or

Mn ¼1

2pi

ðC

½xðsÞ2 xðaþ n2 1Þ�ðnÞrðsÞ7x1ðsÞds

for the discrete or the continuous orthogonality respectively, with s (a) ¼ 0 and the

generalized power of the lattice x (s) defined as

½xrðzÞ2 xrðsÞ�ðkÞ ¼

Yk21

j¼1

½xrðzÞ2 xrðs2 jÞ�;

it turns out that Mn satisfies (see [5], equation (68))

ðf2Dn;n21 þ c1Sn;nÞMnþ1 ¼ 2cnðaÞMn; n $ 0:

For all the moments Mn; n $ 0 to exist, it is necessary to have

f2Dn;n21 þ c1Sn;n – 0; n $ 0:

Since Dn;n21 – 0; n $ 1; we deduce that hn – 0; n $ 1 and degreeðVnÞ ¼ n; n $ 1.

Next, we assume without any loss of generality that m # n. Then, using (84) and (85),

we get

XNi¼0

VnðxðsiÞÞPmðxðsiÞÞrðsiÞ7x1ðsiÞ ¼XNi¼0

PnðxðsiÞÞVmðxðsiÞÞrðsiÞ7x1ðsiÞ

þ sðsjÞrðsjÞWðPnðxðsjÞÞ;PmðxðsjÞÞÞjNþ1j¼0 :

Finally, use of the previous relation, the orthogonality relation for (Pn) (5) and the border

conditions (10) together with the fact that W ðPnðx ðsÞÞ;Pmðx ðsÞÞÞ is a polynomial in the

variable x ðs2 ð1=2ÞÞ, gives

XNi¼0

VnðxðsiÞÞPmðxðsiÞÞrðsiÞ7x1ðsiÞ ¼XNi¼0

PnðxðsiÞÞVmðxðsiÞÞrðsiÞ7x1ðsiÞ ¼ hnkndn;m:

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The latter relation, combined with the fact that Vn is of degree n assures that the family (Vn) is

orthogonal with respect to the weight function r. Hence, (Pn) and (Vn) are proportional since

they are orthogonal with respect to the same weight; therefore, there exists a constant term lnsuch that

VnðxðsÞÞ ¼ 2lnPnðxðsÞÞ; n $ 0:

Comparison of the leading terms in both members of the previous equation yields

ln ¼ 2Dn;n21ðf2Dn21;n22 þ c1Sn21;n21Þ:

The proof using the continuous orthogonality is obtained in the same way. A

Definition 2. We, therefore, propose as definition of classical orthogonal polynomials

Definition 1 without the second condition.

3.2 Parameters a, b, d, f, c

Let x(s) be a lattice of the form (4). It satisfies (17) and (18), and therefore, x(s) depends only

on the parameters a, b and d with the latter given by (30).

The classical orthogonal polynomials (Pn(x(s))), solution of

f ðxðsÞÞFxPnðxðsÞÞ þ c ðxðsÞÞMxPnðxðsÞÞ þ lnPnðxðsÞÞ ¼ 0; ð92Þ

depend only on the parameters a, b, d and the five coefficients of the polynomials f and c.

Equation (92) is the most general second-order difference equation satisfied by classical

orthogonal polynomials. It contains the very classical orthogonal polynomials (special and

limiting cases) as well as the classical orthogonal polynomials of quadratic and q-quadratic

lattices.

In the following, we shall discuss special values of the parameters a, b, d and the

corresponding polynomial families.

First, computations using relations (17) and (18) show that x(s) satisfies a second-order

difference equation of the form [5]

xðsþ 1Þ2 2ð2a2 2 1ÞxðsÞ þ xðx2 1Þ ¼ 4bðaþ 1Þ;

whose solution is

xðsÞ ¼C1q

s þ C2q2s þ b

12a; a – 1;

4bs2 þ C3sþ C4; a ¼ 1:

8<: ð93Þ

3.2.1 Case I: a 5 1. When a ¼ 1, the lattice xðsÞ ¼ 4bs2 þ C3sþ C4 is quadratic.

Case I.1. a ¼ 1, b ¼ C3 ¼ 0

In this case, the lattice x ðsÞ ¼ C4 is constant and from (30), dx ¼ 0. Therefore,

Dx f ðxðsÞÞ ¼ limxðsþ1Þ!xðsÞ

f ðxðsþ 1ÞÞ2 f ðxðsÞÞ

xðsþ 1Þ2 xðsÞ;

d

dxf ðxÞ;


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and Fx, Mx correspond to

Fx ;d2

dx2; Mx ;

d

dx:

Equation (92) reads

f ðxÞP00nðxÞ þ c ðxÞP

0

nðxÞ þ lnPnðxÞ ¼ 0:

Therefore, the case a ¼ 1, b ¼ d ¼ 0 corresponds to the classical orthogonal polynomials of

a continuous variable [19] (Jacobi, Hermite, Laguerre and Bessel).

Case I.2. a ¼ 1, C3 ¼ 1, b ¼ C4 ¼ 0

The lattice reads x (s) ¼ s from which one gets dx ¼ ð1=4Þ. Equation (92) reduces to

~fðsÞD7PnðsÞ þ c ðsÞDPnðsÞ þ lnPnðsÞ ¼ 0;

with ~f ðsÞ ¼ f ðsÞ2 ð1=2Þc ðsÞ. Thus the case a ¼ 1, b ¼ 0, dx ¼ (1/4) corresponds to the

classical orthogonal polynomials of a discrete variable on a linear lattice [19] (Hahn,

Meixner, Charlier and Krawtchouk).

Case I.3. a ¼ 1, b – 0

Here, the lattice x ðsÞ ¼ 4bs2 þ C3sþ C4 is quadratic and the corresponding polynomials

are called classical orthogonal polynomials of a discrete variable on a quadratic lattice [19]

(Wilson, the continuous dual Hahn, the continuous Hahn, the Meixner–Pollaczek, the Racah

and the dual Hahn polynomials).

3.3 Case II: a ¼ q1=2 þ q21=2, q – 0, q – 1

In this case, it can be seen from (93) that the coefficient b is involved only in the

constant term of the lattice. This constant can therefore be omitted without any loss of

generality.

Case II.1. b ¼ 0, d ¼ 0

For b ¼ C2 ¼ 0 and C1 ¼ 1, then x (s) ¼ q s and one gets from (30) d ¼ 0. The operators

Dx, Fx and Mx in this case read

Dx ¼ Dq; Fx ¼ DqD1q

; Mx ¼1

2Dq þ D1

q

� �;

where Dq is the Hahn operator [17] (also called Jackson derivative [18])

Dq f ðxÞ ¼f ðqxÞ2 f ðxÞ

ðq2 1Þx:

Equation (92) is therefore equivalent to

~fðxÞDqD1q

PnðxÞ þ c ðxÞDqPnðxÞ þ lnPnðxÞ ¼ 0; x ¼ qs;

with ~f ðxÞ ¼ q2f ðxÞ2 ð1=2Þ ðq2 1Þxc ðxÞ. Thus the case a ¼ q1=2 þ q21=2, b ¼ d ¼ 0

corresponds to the q-classical orthogonal polynomials [19,26]: The Big q-Jacobi,

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Big q-Laguerre, Little q-Jacobi, Little q-Laguerre (Wall), q-Laguerre, Alternative q-Charlier,

Al-Salam–Carlitz I, Al-Salam–Carlitz II, Stieltjes-Wigert, Discrete q-Hermite, Discrete

q 21-Hermite II, q-Hahn, q-Meixner, Quantum q-Krawtchouk, q-Krawtchouk, Affine q-

Krawtchouk, and q-Charlier polynomials).

Case II.2. b ¼ 0, d – 0

When C1C2 – 0, then from (30), d – 0. The lattice x ðsÞ ¼ C1qs þ C2q

2s is q-quadratic

and the corresponding orthogonal polynomials are the classical orthogonal polynomials of a

discrete variable on q-quadratic lattices [19] (The Askey–Wilson, the q-Racah, the

continuous dual q-Hahn, the continuous q-Hahn, the dual q-Hahn, the Al-Salam–Chihara,

the q-Meixner–Pollaczek, the continuous q-Jacobi, the the continuous dual q-Krawtchouk,

the continuous big q-Hermite, the continuous q-Laguerre, the continuous q-Hermite, the

Wilson, the continuous dual Hahn, the continuous Hahn and the Meixner–Pollaczek

polynomials).

To conclude this section, we give the parameters a, b, d as well as the polynomials f and c

for 14 families of classical orthogonal polynomials on nonuniform lattices out of 18 listed

above. The four remaining families, namely, the Wilson, the continuous dual Hahn, the

continuous Hahn and the Meixner–Pollaczek polynomials deal with the complex difference-

derivative which is not included in this work and will be treated later separately. However,

these families can be reached by limiting procedures from the Askey–Wilson polynomials

for which we illustrate in the following lines the method we have used to obtain the

parameters a, b, d.

Part I: Cases of the q-quadratic lattices

1. Askey–Wilson polynomials:

2nanðabcdqn21;qÞnðab;ac;ad;qÞn

~pnðx;a;b;c;djqÞ ¼ 4f3

q2n;abcdqn21;aeiuae2iu

ab;ac;ad

��q;q

!; x¼ cosu:

rðxÞU rðx;a;b;c;djqÞ ¼1ffiffiffiffiffiffiffiffiffiffiffiffiffi

12 x2p

ðe2iu;qÞ1ðaeiu;beiu;ceiu;deiu;qÞ1

��2

; x¼ cosu:

If a, b, c and d are real, or occur in complex conjugate pairs if complex, and if

maxðjaj; jbj; jcj; jdjÞ, 1;

then we have the following orthogonality relation

1

2p

ð1

21

rðxÞ~pnðx;a;b;c;djqÞ~pmðx;a;b;c;djqÞdx¼ hndn;m;

where

hn ¼2nðabcdqn21;qÞn� 22

ðabcdqn21;qÞnðabcdq2n;qÞ1

ðqnþ1;abqn;acqn;adqn;bcqn;bdqn;cdqn;qÞ1:


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For the lattice we write

x¼ cosu¼eiuþ e2iu

2¼qsþq2s

2U xðsÞ; qs U eiu:

It follows from (18) that ax ¼q

12þq

212

2, bx ¼ 0, C5 ¼ C6 ¼ (1/2) and dx ¼

2ðq21Þ2

4qby (30).

The Askey–Wilson operator

Dq f ðxÞUdq f ðxÞ

dqx; x¼ cosu;

with

dq f ðeiuÞ ¼ f ðq

12e iuÞ2 f ðq21

2e iuÞ;

can be written in terms of Dx as

Dq f ðxÞ ¼Dx21f ðx21ðsÞÞ; xU xðsÞ ¼

qsþq2s

2: ð94Þ

More details about the Askey–Wilson operator are given in Subsection 5.1.4.

In order to compute the polynomials f and c, we write the Pearson-type equation (8)

in its equivalent form (using (9))

rðsþ1Þ

rðsÞ¼

f ðsÞþ 12cðsÞ7x1ðsÞ

f ðsþ1Þ2 12c ðsþ1Þdx1ðsÞ

and use the previous expression of the Askey–Wilson weight rðxÞ; rðxðsÞÞ; rðsÞ to get

rðsþ1Þ

rðsÞ¼ðqsþ1 21Þðqsþ1 þ1Þð21þaqsÞqð21þbqsÞð21þ cqsÞð21þdqsÞ

ðqsþ1 2dÞð2qsþ1 þ cÞ ðqsþ1 2bÞð2qsþ1 þaÞð1þqsÞð21þqsÞ:

Then we combine the last two equations and use the expansion (83) with xðsÞ ¼

ðqsþq2sÞ=2 to obtain a polynomial equation in q s with coefficients depending linearly

on those of the polynomials f and c. Collection of different coefficients of the powers of

q s leads to a system of linear equations in the variables f2, f1, f0, c1 and c0. Solving this

system, we obtain the polynomials f and c:

fðsÞ ¼ 2ðdcbaþ1Þx2ðsÞ2 ðaþbþ cþdþabcþabdþacdþbcdÞxðsÞ

þabþacþadþbcþbdþ cd2abcd21;

c ðsÞ ¼4ðabcd21Þq1=2xðsÞ

q21þ

2ðaþbþ cþd2abc2abd2acd2bcdÞq1=2

q21:

By using the relations (24), (27), (58), (59) and (94), we have checked that the difference

equation (equation (3.1.6) in Ref. [19])

ð12qÞ2Dq rðx;aq12;bq

12;cq

12;dq

12; jqÞDq ~pnðxÞ

� �þlnrðx;a;b;c;djqÞ~pnðxÞ ¼ 0; x¼ cosu;

with ~pnðx;a;b;c;djqÞ; ~pnðxÞ, is equivalent to

f ðxÞD2q ~pnðxÞþc ðxÞSqDq ~pnðxÞþln ~pnðxÞ ¼ 0;

where the operator Sq is given in the next section by (130) and the constant ln by (104)

with the polynomials f (s) and c (s) given as above.

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The coefficients a,b,d (relabelled ax,bx,dx in order to avoid confusion with the

parameters bearing the same names involved in the definition) as well as the polynomials

f and c given below for the remaining families are obtained in the same way using the

notations from Ref. [19]:

2. q-Racah polynomials. The lattice x(s) and the coefficients ax, bx and dx are:

xðsÞ ¼ q2sþgdqsþ1; ax ¼q1=2 þq21=2

2; bx ¼ 0;dx ¼2ðq21Þ2gd:

Polynomials f and c are:

fðxðsÞÞ¼ ðq2abþ1Þx 2ðsÞ2 ðqbdgþqagþqabdþabqþaþbdþgdþgÞqxðsÞ

22ðq2agdb2qbdg2qabd2qg2d2qagd2qag2qgd2bþgdÞq;

cðxðsÞÞ¼2ðq2ab21Þq1=2xðsÞ

q212

2qðqbdgþqagþqabdþabq2a2bd2gd2gÞq1=2

q21:

3. Continuous dual q-Hahn polynomials. The lattice x(s) and the coefficients ax, bx and dxare:

xðsÞ¼qsþq2s

2; ax¼

q1=2þq21=2

2; bx¼0; dx¼2

ðq21Þ2

4q


fðxðsÞÞ¼2x2ðsÞ2 ðaþbþcþabcÞxðsÞþabþacþbc21;

cðxðsÞÞ¼24q1=2xðsÞ

q21þ

2ðaþbþc2abcÞq1=2

q21:

4. Continuous q-Hahn polynomials:

The lattice x(s) and the coefficients ax, bx and dx are:

xðsÞ¼eiwqsþe2iwq2s

2; ax¼

q1=2þq21=2

2; bx¼0; dx¼2

ðq21Þ2

4q; eiuUqs:


fðxðsÞÞ¼2ðdcbaþ1Þx 2ðsÞ2ðdþdcbþat 2þbt 2adþcbat 2þcþcdaþbt 2ÞxðsÞ

t

þcat 2þbt 2d2 t 2cbadþcbt 2þcdþ t 2þbt 4aþ t 2ad

t 2;

cðxðsÞÞ¼4ð21þdcbaÞq1=2xðsÞ

q212

2q1=2ð2c2dþcda2bt 22at 2þdcbþcbat 2þbt 2adÞ

ðq21Þt;

with the notation t¼ eiw.

5. Dual q-Hahn polynomials. The lattice x(s) and the coefficients ax, bx and dx are:

xðsÞ¼q2sþgdqsþ1; ax¼q1=2þq21=2

2; bx¼0; dx¼2ðq21Þ2gd:


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fðxðsÞÞ¼x2ðsÞ2ðgqþqNqgdþgqNqþ1ÞxðsÞ

2qNþgðqNqgd2dqNþdþ1Þq

2qN;

cðxðsÞÞ¼22q1=2xðsÞ

q21þðqNqgdþgqNq2gqþ1Þq1=2

ðq21ÞqN:

6. Al-Salam–Chihara polynomials. The lattice x(s) and the coefficients ax, bx and dx are:

xðsÞ¼qsþq2s

2; ax¼

q1=2þq21=2

2; bx¼0; dx¼2

ðq21Þ2

4q:

Polynomials f and c:

fðxðsÞÞ¼2x2ðsÞ2ðaþbÞxðsÞþab21; cðxðsÞÞ¼24q1=2xðsÞ

q21þ

2ðaþbÞq1=2

q21:

7. q-Meixner–Pollaczek polynomials. The lattice x(s) and the coefficients ax, bx and dx are:

xðsÞ¼eiwqsþe2iwq2s

2; ax¼

q1=2þq21=2

2; bx¼0; dx¼2

ðq21Þ2

4q; eiuUqs:


fðxðsÞÞ¼2x2ðsÞ22acoswxðsÞþa221; cðxðsÞÞ¼24q1=2xðsÞ

q21þ

4aq1=2cosw

q21:

8. Continuous q-Jacobi polynomials. The lattice x(s) and the coefficients ax, bx and dx are:

xðsÞ¼qsþq2s

2; ax¼

q1=2þq21=2

2; bx¼0; dx¼2

ðq21Þ2

4q

The coefficients of the polynomials f and c are:

f2¼p2aþ2bþ4þ1; f1¼1

2ðpþ1Þp1=2ðp2aþbþ22pa2paþ2bþ2þpbÞ;

f0¼21

2p2aþ2bþ42

1

2paþbþ3þ

1

2p2aþ22paþbþ2þ

1

2p2bþ22

1

2p1þaþb2

1

2;

c1¼2pðp2aþ2bþ421Þ

ðp21Þðpþ1Þ; c0¼2

p ð3=2Þð2p2aþbþ22paþpaþ2bþ2þpbÞ

p21;

with q¼p2.

9. Continuous Dual q-Krawtchouk polynomials. The lattice x(s) and the coefficients ax, bx

and dx are:

xðsÞ¼q2sþcqs2N ; ax¼q1=2þq21=2

2; bx¼0; dx¼2cðq21Þ2q212N :


fðxðsÞÞ¼x2ðsÞ2ðcþ1Þq2NxðsÞ22cðq2N2q22NÞ; cðxðsÞÞ¼22q1=2

q21xðsÞþ

2ðcþ1Þq1=2

ðq21ÞqN:

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10. Continuous big q-Hermite polynomials. The lattice x(s) and the coefficients ax, bx and dxare:

xðsÞ¼qsþq2s

2; ax¼

q1=2þq21=2

2; bx¼0; dx¼2

ðq21Þ2

4q:


fðxðsÞÞ¼2x2ðsÞ2axðsÞ21; cðxðsÞÞ¼24q1=2

q21xðsÞþ

2aq1=2

q21:

11. Continuous q-Laguerre polynomials. The lattice x(s) and the coefficients ax, bx and dx are:

xðsÞ¼qsþq2s

2; ax¼

q1=2þq21=2

2; bx¼0; dx¼2

ðq21Þ2

4q:


fðxðsÞÞ¼2x2ðsÞ2paþð1=2Þðpþ1ÞxðsÞþp2aþ221; cðxðsÞÞ¼24p

p221xðsÞþ

2paþð3=2Þ

p21:

12. Continuous q-Hermite polynomials. The lattice x(s) and the coefficients ax, bx and dx are:

xðsÞ¼qsþq2s

2; ax¼

q1=2þq21=2

2; bx¼0; dx¼2

ðq21Þ2

4q:


fðxðsÞÞ¼2x2ðsÞ21; cðxðsÞÞ¼24q1=2

q21xðsÞ:

Part II: Cases of quadratic lattices

13. Racah polynomials. The lattice x(s) and the coefficients ax, bx and dx are:

xðsÞ¼sðsþgþdþ1Þ; ax¼1; bx¼1

4; dx¼

ðgþdþ1Þ2

4:


fðxðsÞÞ¼2x2ðsÞþ½2bdþbgþ2gþagþ2gdþ4þad

þ3aþ3bþ2baþ2d�xðsÞþð1þgÞð1þdþgÞðdþbþ1Þðaþ1Þ;

cðxðsÞÞ¼ð2aþ2bþ4ÞxðsÞþ2ð1þgÞðdþbþ1Þðaþ1Þ:

14. Dual Hahn polynomials. The lattice x(s) and the coefficients ax, bx and dx are:

xðsÞ¼sðsþgþdþ1Þ; ax¼1; bx¼1

4; dx¼

ðgþdþ1Þ2

4:

Polynomials f and c are:fðxðsÞÞ¼ð21þ2Nþd2gÞxðsÞþNð1þgÞð1þdþgÞ; cðxðsÞÞ¼22xðsÞþ2Nð1þgÞ:

3.4 Three-term recurrence coefficients

The method we will use here to compute the recurrence coefficients is the same used for the

very classical orthogonal polynomials [20].

We assume that (Pn) is a system of monic classical orthogonal polynomials such that each

Pn satisfies the equation (1). Because of the orthogonality, (Pn) satisfies the three-term

recurrence relation (13) which we recall here:

Pnþ1ðxÞ ¼ ðx2 bnÞPnðxÞ2 gnPn21ðxÞ; n $ 1; P21 ¼ 0; P0ðxÞ ¼ 1:


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First, we write

PnðxðsÞÞ ¼Xnj¼0

Tn;n2jxjðsÞ; Tn;0 ¼ 1; ð95Þ

and substitute this expression into equation (1) (or (92)), then use the expansions (66) to get

f2x2ðsÞ þ f1xðsÞ þ f0

� Xn22

k¼0

Xnj¼kþ2

Tn;n2jFj;k

!xkðsÞ

þ c1xðsÞ þ c0

� Xn21

k¼0

Xnj¼kþ1

Tn;n2jMj;k

!xkðsÞ þ ln

Xnk¼0

Tn;n2kxkðsÞ ¼ 0:

Then we look for the coefficients of the monomials x n(s) in the previous equation and obtain

ln ¼ 2f2Fn;n22 2 c1Mn;n21: ð96Þ

Next, we collect the coefficients of the monomials x jðsÞ, j ¼ n2 1, n 2 2 to get the

following linear system with respect to the unknowns Tn;1; Tn;2

ðln þ f2Fn21;n23 þ c1Mn21;n22ÞTn;1 þ f2Fn;n23 þ f1Fn;n22 þ c1Mn;n22 þ c0Mn;n21 ¼ 0;

ðc0Mn21;n22 þ c1Mn21;n23 þ f1Fn21;n23 þ f2Fn21;n24ÞTn;1

þ ðf2Fn22;n24 þ c1Mn22;n23 þ lnÞTn;2

þ c0Mn;n22 þ c1Mn;n23 þ f1Fn;n23 þ f2Fn;n24 þ f0Fn;n22 ¼ 0:

Solving this system produces the coefficients Tn;1 and Tn;2.

Computations taking into account (42)–(47) and the relations

Mn;n21 ¼ Dn;n21Sn21;n21;

Mn;n22 ¼ Dn;n21Sn21;n22 þ Dn;n22Sn22;n22;

Mn;n23 ¼ Dn;n21Sn21;n23 þ Dn;n22Sn22;n23 þ Dn;n23Sn23;n23;

Fn;n22 ¼ Dn;n21Dn21;n22;

Fn;n23 ¼ Dn;n21Dn21;n23 þ Dn;n22Dn22;n23;

Fn;n24 ¼ Dn;n21Dn21;n24 þ Dn;n22Dn22;n24 þ Dn;n23Dn23;n24;

obtained from (69) and (70), give the following expressions for the coefficients Fn;k21 and

Mn;k for k ¼ n, n 2 1 and n 2 3.

First case: ax ¼ 1:

Fn;n22 ¼ nðn2 1Þ; Fn;n23 ¼4

3bxnðn2 2Þ ðn2 1Þ2;

Fn;n24 ¼1

45nðn2 1Þ ðn2 2Þ ðn2 3Þ 32b2

xn2 2 96b2

xnþ 52b2x þ 15dx

� �;

Mn;n21 ¼ n;

Mn;n22 ¼2

3bxnðn2 1Þ ð4n2 5Þ;

Mn;n23 ¼2

15nðn2 1Þ ðn2 2Þ 16b2

xn2 2 52b2

xnþ 32b2x þ 5dx

� �:

ð97Þ

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Second case: ax ¼q 1=2þq21=2

2, bx ¼ 0:

Fn;n22 ¼ðqn 2 1Þ ðqn 2 qÞq1=2

ðq2 1Þ2qn;

Fn;n23 ¼ 0;

Fn;n24 ¼ðqþ 1Þ ðq2n 2 q3Þn

ðq2 1Þ3qnþ1=22

ðqn 2 1Þ ð3qnþ1 þ qn 2 q3 2 3q2Þq1=2

ðq2 1Þ4qn

�dx;

Mn;n21 ¼ðqn 2 1Þ ðqn þ qÞ

2ðq2 1Þqn;

Mn;n22 ¼ 0;

Mn;n23 ¼dx

2

ðqþ 1Þ ðq2n þ q3Þn

ðq2 1Þ2qnþ12

ðqþ 1Þ ðqn 2 1Þ ðqn þ q2Þ

ðq2 1Þ3qn

�:

ð98Þ

The coefficients bn and gn are deduced from Tn,1 and Tn,2 using the following relations

between bn; gn and the Tn;i [8,9]

Tnþ1;1 ¼ Tn;1 2 bn; Tnþ1;2 ¼ Tn;2 2 bnTn;1 2 gn;

and the previous expressions of Fn;k and Mn;k. We can now state the following explicit results

obtained with the help of Maple [27]:

Theorem 8. The coefficients bn and gn of the recurrence coefficients (13) of the classical

orthogonal polynomials satisfying (92) are given explicitly by:

Case 1. ax ¼ 1:

bn ¼ 24n 2c1n2 c1 2 2f2nþ 2f2n

2� �

ðc1 þ f2n2 f2Þ

ð2f2nþ c1Þ ð2f2ðn2 1Þ þ c1Þbx

222c0f2 þ c1c0 2 2f1nf2 þ 2f1c1nþ 2f1n

2f2

ð2f2nþ c1Þ ð2f2ðn2 1Þ þ c1Þ;

ð99Þ

gn ¼ 2nðn2 1Þ ðc1 þ f2n2 f2Þ ðc1 þ f2n2 2f2Þ

ð2f2n2 f2 þ c1Þ ðð2f2n2 3f2 þ c1Þdx

þ16ðn2 1Þ3ðc1 þ f2n2 2f2Þ ðc1 þ f2n2 f2Þ

3n

ð2f2n2 3f2 þ c1Þ ð2f2n2 f2 þ c1Þ ð2f2n2 2f2 þ c1Þ2b2x

þ 2f1n2f2 2 4f1nf2 þ 2f1c1nþ 2f2f1 2 2f1c1 þ c1c0

� �£

4ðn2 1Þ ðc1 þ f2n2 f2Þ ðc1 þ f2n2 2f2Þn

ð2f2n2 3f2 þ c1Þ ð2f2n2 f2 þ c1Þ ð2f2n2 2f2 þ c1Þ2bx

2ðc1 þ f2n2 2f2Þn

ð2f2n2 3f2 þ c1Þ ð2f2n2 f2 þ c1Þ ð2f2n2 2f2 þ c1Þ2

£ 4f0f22n

2 2 8f0f22nþ 4f0f

22 2 f2

1n2f2 þ 2f2

1nf2 þ 4f0f2nc1 þ c20f2

�24f0f2c1 2 f2

1f2 2 f21nc1 þ f2

1c1 þ f0c21 2 c0f1c1

�; ð100Þ


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Case 2. For ax ¼ ðpþ p21Þ=2 and bx ¼ 0, with the notations p ¼ q1=2; q –0; 1; and z ¼ pn;

bn ¼ 22f1p3 þ c0p

4 2 2c0p2 þ c0 2 2pf1

� �c1p

2 þ 2f2p2 c1

� �z6

�2ðp2 þ 1Þ p6c1c0 þ 2p5f1c1 2 2p5c0f2 2 4p4f2f1 2 p4c1c0

�þ4p3c0f2 2 4p3c1f1 2 4p2f1f2 2 p2c1c0 þ 2pf1c1 2 2c0f2pþ c1c0

�z4

2p2 22f1p3 þ c0p

4 2 2c0p2 þ c0 2 2pf1

� �c1p

2 2 2f2p2 c1

� �z2=

c1p2 þ 2f2p2 c1

� �z4 þ c1p

2 2 2f2p2 c1

� � £ c1p

2 þ 2f2p2 c1

� �z4 þ p4 c1p

2 2 2f2p2 c1

� �� ; ð101Þ

gn¼ 22 2c1p42c1z

2þc1p6þc1p

2z2þ2f2pz222f2p

5� �

£ðz221Þp3z2ðp221Þ2 2c1p42c1z

4þc1p6þc1z

4p2þ2f2pz422f2p

5� �2

f0

þ 2c1p42c1z

2þc1p6þc1p

2z2þ2f2pz222f2p

5� �

ðz221Þp2ðp22z2Þ

£ 22f2p3þ2f2pz

22c1p2þc1p

42c1z2þc1p

2z2� �

£ 2c1p42c1z

4þc1p6þc1z

4p2þ2f2pz422f2p

5� �2

dx

þ 2c1p42c1z

2þc1p6þc1p

2z2þ2f2pz222f2p

5� �

£ðz221Þp4z4ðp221Þ2 c0p422f1p

32c0p2þz2c0p

2þ2f1z2p2z2c0

� �£ p6c1c022p5c0f2þ2p5f1c12p4c1z

2c022p4c1c024p4f2f1

�22p3z2f2c0þ2p3z2f1c1þ2p3c0f222p3c1f1þ2p2c1z

2c0þ4p2z2f1f2

þp2c1c0þ2pz2f2c022pz2f1c12c1z2c0

��=

ðp221Þ2 c1p2þ2f2p2c1

� �z4þp6 c1p

222f2p2c1

� �� £ c1p

2þ2f2p2c1

� �z4þp2 c1p

222f2p2c1

� �� £ c1p

2þ2f2p2c1

� �z4þp4 c1p

222f2p2c1

� �� 2o: ð102Þ

The coefficient ln of (92) is given by

ln¼2nðc1þðn21Þf2Þ; ð103Þ

for ax ¼ 1 and

ln ¼2c1p

2 2 2f2pþc1

� �z 4 þ 2c1p

2 þ 2f2p2c1 þ 2f2p3 2c1p

4� �

z 2 þ p 2 c1p2 2 2f2p2c1

� �2ðp 2 2 1Þ2z 2

ð104Þ

for ax ¼ ðpþ p21Þ=2, with p ¼ q 1/2 and z ¼ p n.

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Remark 4.

1. By selecting specific values of the parameters ax, bx and dx (as indicated in Subsection

3.2) in the previous theorem, we recover after appropriate changes in f and c the

coefficients bn and gn of the very classical orthogonal polynomials. The coefficients bn

and gn of the very classical orthogonal polynomials were given explicitly by Koepf and

Schmersau [20,21] using a similar approach.

2. The relations (99)–(102) are important tools for computer algebra since they provide in 4

relations the recurrence coefficients of all classical orthogonal polynomials [19].

4. Fourth-order divided-difference equations for OP

4.1 Functions of the second kind

In connection with the classical orthogonal polynomials (Pn), there is an important function

called function of second kind denoted Qn and connected to Pn in the following way [31]

QnðxðzÞÞ ¼1

rðzÞ

Xb21

s¼a

PnðxðsÞÞrðsÞ7x1ðsÞ

xðsÞ2 xðzÞ; z – a; aþ 1; . . . ; b2 1: ð105Þ

Also, the first associated Pð1Þn of Pn, defined by the recurrence relation (14) for r ¼ 1 is related

to Pn by

Pð1Þn ðxðzÞÞ ¼

Xb21

s¼a

½Pnþ1ðxðsÞÞ2 Pnþ1ðxðzÞÞ�rðsÞ7x1ðsÞ

xðsÞ2 xðzÞ; z – a; aþ 1; . . . ; b2 1; ð106Þ

for the discrete orthogonality.

The following properties [31] will be used in the next subsection.

Theorem 9. The function Qn obeys:

1. ;n [ N0, Pn and Qn are two linearly independent solutions of (92).

2. Pn and Qn are the two linearly independent solutions of the recurrence relation

Xnþ1ðxðsÞÞ ¼ ðxðsÞ2 bnÞXnðxðsÞÞ2 gnXn21ðxðsÞÞ; n $ 1:

3. Pn and Qn and the first associated Pð1Þn are related from (105) and (106) by

QnðxðsÞÞ ¼ PnðxðsÞÞQ0ðxðsÞÞ þg0

rðsÞPð1Þn21ðxðsÞÞ;

n $ 1; s – a; aþ 1; . . . ; b2 1:

ð107Þ

4. Qn fulfils the following asymptotic property

QnðxðsÞÞ ¼ 2

Qni¼0 gi

rðsÞxnþ1ðsÞ1 þ O

1

xðsÞ

�� ; xðsÞ!1; ð108Þ

with

g0 ¼Xns¼0

rðsÞ7x1ðsÞ or g0 ¼

ðC

rðsÞ7x1ðsÞds;

for the discrete and the continuous orthogonality respectively (see definition 1).


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4.2 Intermediate relations

To derive the fourth-order difference equations for the modifications of the classical

orthogonal polynomials, we shall start with the following intermediate result.

Theorem 10. Let (Pn) be a sequence of classical orthogonal polynomials satisfying the

second-order difference equation (92) with the corresponding orthogonality weight

satisfying the Pearson-type equation (8) and the border conditions (10) where the functions

s (s) and the polynomials f (s) and c (s) are related by (9). Then, the first associated ðPð1Þn Þ of

(Pn) satisfies

f ðsÞ½A1ðsÞFx þ B1ðsÞMx þ C1ðnÞ�Pð1Þn21ðxðsÞÞ ¼ 2h½D1ðsÞMx þ E1ðs; nÞ�PnðsÞ; ð109Þ

with

A1ðsÞ ¼ 2f ðsÞ2 2a2U2ðsÞf00 þ 2aU1ðsÞc ðsÞ2 2U1ðsÞMxf ðsÞ

þ 4aU21ðsÞc

0 þ 2að2a2 2 1ÞU2ðsÞc0;

B1ðsÞ ¼ ð2a2 2 1Þc ðsÞ2 2aMxf ðsÞ þ 2ð4a2 2 1ÞU1ðsÞc0 2 2aU1ðsÞf

00;

C1ðnÞ ¼ ð2a2 2 1Þc0 2 af00 2 ln;

D1ðsÞ ¼ af ðsÞ2 U1ðsÞc ðsÞ;

E1ðs; nÞ ¼ 2lnU1ðsÞ;

ð110Þ

where the polynomials U1 and U2 are defined by (57),

h ¼ axc0 2

f00

2

�g0 ¼ ðaxc1 2 f2Þg0; ð111Þ

and

g0 ¼XNs¼0

rðsÞ7x1ðsÞ or g0 ¼

ðC

rðsÞ7x1ðsÞds ð112Þ

for discrete and continuous orthogonality respectively.

The proof of the theorem will use the following lemma.

Lemma 3. The function of second kind Qnðx ðsÞÞ defined by (105) satisfies:

f ðsÞ21

2c ðsÞ7x1ðsÞ

�rðsÞ

7Q0ðxðsÞÞ

7xðsÞ¼ h; ;s[ ða;bÞ; s– a; aþ 1; . . . ;b2 1; ð113Þ

rðsÞMxQ0xðsÞ ¼hf ðsÞ

f2ðsÞ2U2ðsÞc2ðsÞ; ð114Þ

where h is given by (111).

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Proof. Since Qn is a solution of (92), taking into account the fact that l0 ¼ 0 from (96),

we have

f ðsÞFxQ0ðxðsÞÞ þ tðsÞMxQ0ðxðsÞÞ ¼ 0 , D ðf ðsÞ21

2c ðsÞ7x1ðsÞÞrðsÞ

7Q0ðxðsÞÞ

7xðsÞ

� �¼ 0:

Therefore, the left-hand side of (113) is a periodic function of period 1. This combined with

the asymptotic behavior obtained using (108)

f ðsÞ21

2c ðsÞ7x1ðsÞ

�rðsÞ

7Q0ðxðsÞÞ

7xðsÞ¼ h 1 þ O

1

xðsÞ

�� ; xðsÞ!1;

allows to deduce that the left-hand side of the previous equation is the constant h.

To derive equation (114), we use an equivalent form of (113)

7Q0ðxðsÞÞ

7xðsÞ¼

h

rðsÞ f ðsÞ2 12c ðsÞ7x1ðsÞ

� �and obtain

rðsÞMxQ0ðxðsÞÞ ¼rðsÞ

2

DQ0ðxðsÞÞ

DxðsÞþ

7Q0ðxðsÞÞ

7xðsÞ

�

¼1

2

hrðsÞ

rðsþ 1Þ½f ðsþ 1Þ2 12c ðsþ 1Þdx1ðsÞ�

þh

f ðsÞ2 12c ðsÞ7x1ðsÞ

!:

Next, we use an equivalent form of the Pearson-type equation (8)

rðsþ 1Þ

rðsÞ¼

f ðsÞ þ 12c ðsÞ7x1ðsÞ

f ðsþ 1Þ2 12c ðsþ 1ÞDx1ðsÞ

ð115Þ

to get

rðsÞMxQ0ðxðsÞÞ ¼1

2

h

f ðsÞ þ 12c ðsÞ7x1ðsÞ

þh

f ðsÞ2 12c ðsÞ7x1ðsÞ

!

¼hf ðsÞ

f2ðsÞ2 14½c ðsÞ7x1ðsÞ�

2¼

hf ðsÞ

f2ðsÞ2 c2ðsÞU2ðsÞ;

since from equation (34),

½7x1ðsÞ�2

4¼ QðxðsÞÞ ¼ U2ðsÞ: A

Let us now give the proof of Theorem 10.

Proof. In the first step, we use relation (107) and the fact that Qn is a solution of (92) to get

ðf ðsÞFx þ c ðsÞMx þ lnÞ PnðxðsÞÞQ0ðxðsÞÞ þ1

rðsÞPð1Þn21ðxðsÞÞ

� �¼ 0: ð116Þ

Using the product rules (62) and (63) for Fx and Mx and equation (92) in order to eliminate all

occurrences of FxPnðx ðsÞÞ and FxQ0ðx ðsÞÞ we transform the previous equation into an


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equation whose left-hand side is a linear combination of

FxPð1Þn21ðxðsÞÞ; MxP

ð1Þn21ðxðsÞÞ; Pð1Þ

n21ðxðsÞÞ; MxPnðxðsÞÞ and PnðxðsÞÞ:

The coefficients of this linear combination are functions of

xðsÞ; f ðsÞ; c ðsÞ; U1ðsÞ; U2ðsÞ; MxQ0ðxðsÞÞ; FxrðsÞ; MxrðsÞ; and rðsÞ:

The expression MxQ0ðx ðsÞÞ is eliminated thanks to (114). It remains now to express Fxr ðsÞ

and Mxr ðsÞ in terms of r (s) times rational functions of x (s).

For this aim, we use (115) to eliminate all occurrences of r (s 2 1) and r (s þ 1) in the

equations

FxrðsÞ ¼D

Dx s2 12

� �7rðsÞ7xðsÞ

; MxrðsÞ ¼1

2

DrðsÞ

DxðsÞþ

7rðsÞ

7xðsÞ

�;

and obtain

FxrðsÞ

rðsÞand

MxrðsÞ

rðsÞ

as rational functions of the two variables x (s) and x (s þ 1) whose coefficients depend

only on those of the polynomials f (s) and c (s) ((22) and (23) have been used as well).

Then, assuming a – 1, we combine (57) and the relation (which is easily deduced from (34)

using (22))

U2ðsÞ ¼½xðsþ 1Þ þ ð1 2 2a2ÞxðsÞ2 2bðaþ 1Þ�2

4a2¼ ða2 2 1Þx2ðsÞ þ 2bðaþ 1ÞxðsÞ þ d;

to get the system

U1ðsÞ ¼ ðaþ 1Þ½ða2 1ÞxðsÞ þ b�; 2uaffiffiffiffiffiffiffiffiffiffiffiU2ðsÞ

p¼ xðsþ 1Þ þ ð1 2 2a2ÞxðsÞ2 2bðaþ 1Þ;

where u ¼ ^1. Solving this system in terms of the unknowns x (s) and x (s þ 1) yields

xðsÞ ¼ U1ðsÞða21Þ ð1þaÞ

2 ba21

;

xðsþ 1Þ ¼ ð2a 221ÞU1ðsÞða21Þ ð1þaÞ

þ 2uaffiffiffiffiffiffiffiffiffiffiffiU2ðsÞ

p2 b

a21:

8><>: ð117Þ

Computations with Maple 9 [27] using the previous equation allow to express

FxrðsÞ

rðsÞand

MxrðsÞ

rðsÞ;

as rational functions of the variable x (s) depending only on integer powers of the functions

U1ðsÞ; U2ðsÞ; Fxf ðsÞ; Mxf ðsÞ; f ðsÞ; Mxc ðsÞ and c ðsÞ:

Summing up, we obtain equation (109).

Notice that if a ¼ 1, the result obtained is still valid since the singularity 1=ða2 1Þ

appearing in (117) will be automatically cancelled in the expressions of

FxrðsÞ

rðsÞand

MxrðsÞ

rðsÞ

after the computations. A

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4.3 Fourth-order difference equations for modifications of the classical orthogonal

polynomials

Theorem 10 is the key to the following fundamental result.

Theorem 11. Let (Pn) be a classical orthogonal polynomial system satisfying (92) and ð ~PnÞ

the orthogonal polynomial system related to (Pn) by

~PnðxðsÞÞ ¼ In;r;kðsÞPnþrðxðsÞÞ þ Jn;r;kðsÞPð1Þnþr21ðxðsÞÞ; ð118Þ

where r and k are nonnegative integers, In;r;kðsÞ and Jn;r;kðsÞ are polynomials in the variable

x (s) but not depending on n for n $ k, i.e.

In;r;kðsÞ U Ir;kðsÞ; Jn;r;kðsÞ U Jr;kðsÞ – 0 for n $ k: ð119Þ

Then, each ~Pn satisfies a fourth-order difference equation of the factorized form

Gn;r;kðyðxðsÞÞ ¼ ~An;r;kðsÞFx þ ~Bn;r;kðsÞMx þ ~Cn;r;kðsÞ� £ Ar;kðsÞFx þ Br;kðsÞMx þ Cn;r;kðsÞ�

yðsÞ ¼ 0; n $ k; ð120Þ

where Ar,k(s), Br,k(s), Cn,r,k(s), ~An;r;kðsÞ, ~Bn;r;kðsÞ and ~Cn;r;kðsÞ are polynomials in the variable

x (s) whose degree does not depend on n.

This fourth-order difference equation can also be written as

Gn;r;kðyðxðsÞÞ ¼ ½G4ðs; n; r; kÞFxFx þ G3ðs; n; r; kÞMxFx þ G2ðs; n; r; kÞFx

þ G1ðs; n; r; kÞMx þ G0ðs; n; r; kÞyðsÞ ¼ 0; n $ k; ð121Þ

where Gjðs; n; r; kÞ; j ¼ 0 . . . 4 are polynomials in the variable x (s) whose degree does not

depend on n.

Proof. First, we use equation (109) for n ¼ n þ r and (118) to get

f ðsÞ½A1ðsÞFx þ B1ðsÞMx þ C1ðnþ rÞ� £~PnðxðsÞÞ

Jr;kðsÞ2

Ir;kðsÞ

Jr;kðsÞPnþrðxðsÞÞ

� �

¼ 2h½D1ðsÞMx þ E1ðs; nþ rÞ�PnþrðsÞ; n $ k: ð122Þ

Next, we combine the previous equation, the quotient rules (64)–(65) and equation (92) (in

order to eliminate FxPnþrðx ðsÞÞ) to obtain

Ar;kðsÞFx þ Br;kðsÞMx þ Cn;r;kðsÞ�

~PnðxðsÞÞ

¼ ½Dn;r;kðsÞMx þ En;r;kðsÞ�PnþrðxðsÞÞ; n $ k; ð123Þ

where the coefficients Ar,k(s), Br,k(s), Cn,r,k(s), Dn,r,k(s) and En,r,k(s) are polynomials, and

functions of the polynomials Ir,k(s), Jr,k(s), A1(s), B1(s), C1(n), D1(s) and E1(s,n þ r).

If we write

~QnþrðxðsÞÞ ¼ Dn;r;kðsÞMx þ En;r;kðsÞ�

PnþrðxðsÞÞ; n $ k; ð124Þ


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then using the result of Lemma 1 and the fact that Pnþr satisfies (109) for n ¼ n þ r, we get

Mx~QnþrðxðsÞÞ ¼ Gn;r;kðsÞMx þ Hn;r;kðsÞ

� PnþrðxðsÞÞ; n $ k; ð125Þ

Fx ~QnþrðxðsÞÞ ¼ ~Gn;r;kðsÞMx þ ~Hn;r;kðsÞ�

PnþrðxðsÞÞ; n $ k; ð126Þ

where Gn;r;kðsÞ, Gn;r;kðsÞ, ~Gn;r;kðsÞ and ~Hn;r;kðsÞ are rational functions of the variable x (s).

Equations (124)–(126) which can be seen as a system of three linear equations with respect

to the unknowns MxPnþrðx ðsÞÞ and Pnþrðx ðsÞÞ produce the second-order difference equation

for ~Qnþrðx ðsÞÞ

Dn;r;kðsÞ En;r;kðsÞ ~QnþrðxðsÞÞ

Gn;r;kðsÞ Hn;r;kðsÞ Mx~QnþrðxðsÞÞ

~Gn;r;kðsÞ ~Hn;r;kðsÞ Fx ~QnþrðxðsÞÞ

��

��¼ 0; n $ k:

The previous equation after cancellation of the common denominator can be brought into the

form~An;r;kðsÞFx þ ~Bn;r;kðsÞMx þ ~Cn;r;kðsÞ�

~QnþrðxðsÞÞ ¼ 0; n $ k;

where ~An;r;kðsÞ, ~Bn;r;kðsÞ and ~Cn;r;kðsÞ are polynomials in the variable x (s) whose degree does

not depend on n. Combination of (123) and the previous equation produces equation (120).

Finally, (121) is derived from (120) by simultaneous application of the product rules (55)–

(56) and (62)–(63). A

4.4 Solutions of the fourth-order difference equations

In the following, we solve the fourth-order difference equation satisfied by the modifications

of classical orthogonal polynomials in terms of the polynomials Pn and its corresponding

function of second kind Qn.

Theorem 12. Under the hypothesis of Theorem 11, we have: The four linearly independent

solutions of the difference equation (120)

Gn;r;kyðsÞ ¼ 0; n $ k;

are

S1ðs; n; r; kÞ ¼ rðsÞJr;kðsÞPnþrðxðsÞÞ;

S2ðs; n; r; kÞ ¼ rðsÞJr;kðsÞQnþrðxðsÞÞ;

S3ðs; n; r; kÞ ¼ Ir;kðsÞ2 g210 rðsÞQ0ðxðsÞÞJr;kðsÞ

� PnþrðxðsÞÞ;

S4ðs; n; r; kÞ ¼ Ir;kðsÞ2 g210 rðsÞQ0ðxðsÞÞJr;kðsÞ

� QnþrðxðsÞÞ:

Proof. In the first step, we observe that because of the factorization in equation (120), any

solution of the equation


yðsÞ ¼ 0; n $ k ð127Þ

is also solution of (120).

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In the second step, we also observe from the procedure we have used to construct equation

(120) that (using (116) and (122))


yðsÞ ¼ 0

m

A1ðsÞFx þ B1ðsÞMx þ C1ðnþ rÞ½ � yðsÞJr;kðsÞ

n o¼ 0; n $ k

m

f ðsÞFx þ c ðsÞMx þ lnþr

� yðsÞ

rðsÞJr;kðsÞ

n o¼ 0; n $ k:

Therefore, S1ðs; n; r; kÞ and S2ðs; n; r; kÞ are solutions of (127) and, therefore, of (120).

In the third step, we use (118) and the relation

PðrÞn ðxðsÞÞ ¼

rðsÞ½Pr21ðxðsÞÞQnþrðxðsÞÞ2 Qr21ðxðsÞÞPnþrðxðsÞÞ�

g0Gr21

obtained from the fact that Pnþr, Qnþr and PðrÞn satisfy the second-order recurrence relation

(see Theorem 9 and also [12]) to get

~PnðxðsÞÞ ¼ S3ðs; n; r; kÞ þ g210 S2ðs; n; r; kÞ:

Here, Gr21 and g0 are given respectively by (16) and (112). Then, since ~PnðxðsÞÞ and

S2(s;n,r,k) are both solutions of (120), it turns out from the previous equation that S3ðs; n; r; kÞ

is also solution of (120). Also, S4ðs; n; r; kÞ is another solution of (120) because it is obtained

by replacing Pnþr by Qnþr in the expression of S3ðs; n; r; kÞ and the functions Pnþr and Qnþr

satisfy the same second-order difference equation, namely (92) for n ¼ nþ r. We complete

the proof by observing that the four solutions are linearly independent since by means of

(108), they enjoy different asymptotic behavior. A

For any modification of the classical orthogonal polynomials leading to a relation of type

(12), we get explicit expressions of the functions Sjðs; n; r; kÞ; j ¼ 1 . . . 4 in terms of rðsÞ,

PnðxðsÞÞ and QnðxðsÞÞ. As can be seen from the previous theorem, these solutions have the

same structure for the difference equations satisfied by modifications of all classical

orthogonal polynomials. These solutions were given explicitly in our previous works for the

modifications of the very classical orthogonal polynomials. We, therefore, refer to these three

papers [11–13].

It should be mentioned that the fourth-order difference equation satisfied by the

Laguerre–Hahn polynomials orthogonal on special nonuniform lattices was derived in

Ref. [7]. This result which is based on the properties of the formal Stieltjes function of the

corresponding functional [23] covers the modifications of the classical orthogonal

polynomials. However, it does not deal with factorization nor with the solutions of

the difference equation derived. Our approach, which uses the operators Fx, Mx, the Pearson-

type equation for the orthogonality weight and the second-order divided difference equation

satisfied by the initial polynomials allows us to factorize and solve the difference equations

obtained and constitute a natural extension of the results obtained for the very classical

orthogonal polynomials [11–13].


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Corollary 3. The four linearly independent solutions of the fourth-order divided-

difference equations satisfied by the rth associated classical orthogonal polynomials are

S1ðs; n; r; 0Þ ¼ rðsÞPr21ðxðsÞPnþrðxðsÞÞ;

S2ðs; n; r; 0Þ ¼ rðsÞQr21ðxðsÞPnþrðxðsÞÞ;

S3ðs; n; r; 0Þ ¼ rðsÞPr21ðxðsÞQnþrðxðsÞÞ;

S4ðs; n; r; 0Þ ¼ rðsÞQr21ðxðsÞQnþrðxðsÞÞ:

This is obtained by combining the previous theorem, (15), (16) and (107).

Remark 5. The fourth-order divided-difference equation for the rth associated

classical orthogonal polynomials of a discrete variable on a nonuniform lattice given in

Theorem 11 for nonnegative integer r is valid if r is a positive real number. The four linearly

independent solutions given in the previous corollary are still valid but one should keep in

mind that Pr and Qr in this cases represent the two linearly independent solutions of

f ðxðsÞÞFxyðxðsÞÞ þ c ðxðsÞÞMxyðxðsÞÞ þ lryðxðsÞÞ ¼ 0

for the real number r [31]. This extension can be deduced following the method used for

the rth associated classical orthogonal polynomials of a continuous variable (see [12],

Theorem 8).

5. Specializations and applications

In this section, we mainly investigate the results obtained for specific values of the

parameters ax, bx and dx of the lattice x(s), namely those leading to the very classical

orthogonal polynomials as already mentioned in Subsection 3.2. We also give some

applications.

5.1 Specializations

5.1.1 The classical orthogonal polynomials of a continuous variable. For

ax ¼ 1; bx ¼ 0 and dx ¼ 0;

we have

Mx ¼d

dx; Fx ¼

d2

dx2and U1ðsÞ ¼ U2ðsÞ ¼ 0:

Therefore, Relation (109) reads

f ðxÞd2

dx2þ ð2f0ðxÞ2 c ðsÞÞ

d

dxþ ln þ f00 2 c0

� �Pð1Þn21ðxÞ ¼ ðf00 2 2c0ÞP0

nðxÞ:

This relation, which was first derived by Ronveaux [29], is the key to the derivation

and solutions of the fourth-order differential equations satisfied by modifications of

the classical orthogonal polynomials of a continuous variable. For more details, we refer to

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the paper [12] which appears now to be a particular case of the results obtained in the

framework of this work.

5.1.2 The classical orthogonal polynomials of a discrete variable on a linear lattice. For

ax ¼ 1; bx ¼ 0 and dx ¼1

4;

we have

xðsÞ ¼ s; Mx ¼1

2ðDþ 7Þ; U1ðsÞ ¼ 0; Fx ¼ D7; and U2ðsÞ ¼

1

4:

Therefore, Relation (109) reads

ð2f ðsÞ þ f00 2 c0ÞD7þ ðc ðsÞ þ ½Dþ 7�f ðsÞÞ½Dþ 7� þ ln þ f00 þ c0 �

Pð1Þn21ðxÞ

¼ ðf00 2 2c0ÞD7PnðxÞ:

The previous relation, due to Atakishiyev et al. [6], has been used to derive the fourth-order

difference equations satisfied by the modifications of classical orthogonal polynomials of

a discrete variable on a linear lattice. We refer to the paper [11] for details about these

equations as well as their solutions.

5.1.3 The q-classical orthogonal polynomials. For

ax ¼q1=2 þ q ð21=2Þ

2; bx ¼ dx ¼ 0; with q – 0; 1;

we have

xðsÞ ¼ q^s; Fx ¼ q2DqD1

q; Mx ¼

1

2Dq þ D1

q

�; U1ðsÞ ¼ a2

x 2 1� �

xðsÞ; and

U2ðsÞ ¼ a2x 2 1

� �x2ðsÞ

and the corresponding families are the q-classical orthogonal polynomials.

For xðsÞ ¼ qs, the relation (109) is equivalent to equation (16) in Ref. [13] with the

polynomial f replaced by

q2f ðxðsÞÞ21

2ðq2 1ÞxðsÞc ðxðsÞÞ:

This relation which is due to Foupouagnigni et al. [10] allowed in Ref. [13] to derive and

solve the fourth-order q-difference equation satisfied by the modifications of the q-classical

orthogonal polynomials.

5.1.4 The Askey–Wilson operator. Relations between Dx, Sx and the Askey–Wilson

operator Dq

We establish relations between the operators Dx, Sx and the Askey–Wilson operator Dq and

then use these relations to state Theorems 1–3 for the special case of the Askey–Wilson

operator.


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The Askey–Wilson operator Dq is defined as [4]

Dq f ðxÞ ¼dq f ðxÞ

dqx; x ¼ cos u

with

dq f ðeiuÞ ¼ f q

12e iu

�2 f ðq

212 e iuÞ:

The relation with Dx and Sx

For x ¼ cos u ¼ ðe iu þ e2iuÞ=2, we set e iu ¼ qs and obtain x U xðsÞ ¼ ðqs þ q2sÞ=2. The

coefficients a, b and d corresponding to this lattice are

a ¼q

12 þ q

212

2; b ¼ 0; d ¼ 2

ðq2 1Þ2

4q: ð128Þ

The operators Dx and Dq are related by

Dq f ðxÞ ¼f x sþ 1

2

� �� 2 f x s2 1

2

� �� x sþ 1

2

� �2 x s2 1

2

� � ¼ Dx21f ðx21ðsÞÞ: ð129Þ

As was the case for Dx and Sx, there is a need to define the companion operator for Dq,

namely the operator Sq

Sq f ðxÞ ¼Eþq f ðxÞ þ E2

q f ðxÞ

2; x ¼ cos u; ð130Þ

with

Eþq f ðe

iuÞ ¼ f q12e iu

�; E2

q f ðeiuÞ ¼ f q

212 e iu

�:

This operator is related to Sx in the following way

Sq f ðxÞ ¼ Sx21f ðx21ðsÞÞ: ð131Þ

The operators equivalent to Fx and Mx in this case are

F q ¼ D2q; Mq ¼ SqDq: ð132Þ

Iteration of (129) and (131) give

D2qf ðxÞ ¼ Dx21

Dx22f ðx22ðsÞÞ; S2

qf ðxÞ ¼ Sx21Sx22

f ðx22ðsÞÞ;

SqDq f ðxÞ ¼ Sx21Dx22

f ðx22ðsÞÞ; DqSq f ðxÞ ¼ Dx21Sx22

f ðx22ðsÞÞ:ð133Þ

Using (129) and (131), one remarks that the operator Dq (respectively Sq) transforms a

polynomial of degree n in the variable x into a polynomial of degree n 2 1 in x (respectively

of degree n in x). Moreover, we have the following theorems.

Theorem 13. The following statements hold.

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1. The operators Dq and Sq obey the following product rules:

Dqð f ðxÞgðxÞÞ ¼ Sq f ðxÞDqgðxÞ þDq f ðxÞSqgðxÞ; ð134Þ

Sqð f ðxÞgðxÞÞ ¼ Sq f ðxÞSqgðxÞ þ V2ðxÞDq f ðxÞDqgðxÞ; ð135Þ

where

V2ðxÞ ¼ðq2 1Þ2

4qðx2 2 1Þ: ð136Þ

2. The operators Dq and Sq also satisfy the quotient rules:

Dq

f ðxÞ

gðxÞ

�¼

Sq f ðxÞDqgðxÞ2Dq f ðxÞSqgðxÞ

V2ðxÞ½DqgðxÞ�2 2 ½SqgðxÞ�

2; ð137Þ

Sq

f ðxÞ

gðxÞ

�¼

V2ðxÞDq f ðxÞDqgðxÞ2 Sq f ðxÞSqgðxÞ

V2ðxÞ½DqgðxÞ�2 2 ½SqgðxÞ�

2: ð138Þ

Remark 6. For any integer n, the following relations hold

Dqxn ¼

axþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiða2 2 1Þ ðx2 2 1Þ

p� �n2 ax2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiða2 2 1Þ ðx 2 2 1Þ

p� �n2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiða2 2 1Þ ðx2 2 1Þ

p ; ð139Þ

Sqxn ¼

axþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiða2 2 1Þ ðx2 2 1Þ

p� �nþ ax2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiða2 2 1Þ ðx 2 2 1Þ

p� �n2

; ð140Þ

where a is given by (128).

Theorem 14. The following relations hold:

DqSq ¼ aSqDq þ axDqDq; ð141Þ

SqSq ¼ axSqDq þ aða2 2 1Þ ðx2 2 1ÞDqDq þ I; ð142Þ

F qMq ¼ ð2a2 2 1ÞMqF q þ 2aða2 2 1ÞxF qF q; ð143Þ

MqMq ¼ aF q þ 2aða2 2 1ÞxMqF q þ ð2a2 2 1ÞV2ðxÞF qF q; ð144Þ

where a and V2ðxÞ given by (128) and (135) respectively.

Remark 7. Theorem 13 and Proposition 6 for the operators F q and Mq (see (132)) are

deduced from Theorem 5 and Proposition 5 by taking x (s) ¼ x, b ¼ 0,

a ¼ ðq1=2 þ q21=2Þ=2, with the operators Fx, Mx replaced respectively by F q and Mq.

Theorem 15. Let f ðxÞ and g ðxÞ be two functions of the variable x satisfying respectively

D2qf ðxÞ þ a1ðxÞSqDq f ðxÞ þ a0ðxÞ f ðxÞ ¼ 0;

D2qf ðxÞ þ b1ðxÞSqDq f ðxÞ þ b0ðxÞgðxÞ ¼ 0;

ð145Þ

where aj and bj are given functions of x.


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Then, the product f (x)g (x) is a solution of a fourth-order divided-difference equation of

the form

I4ðxÞD4qyðxÞ þ I3ðxÞSqD3

qyðxÞ þ I2ðxÞD2qyðxÞ þ I1ðxÞSqDqyðxÞ þ I0ðxÞyðxÞ ¼ 0

where Ij are functions of aj and bj. If the ajðxÞ, j ¼ 0,1 and the bjðxÞ, j ¼ 0,1 are rational

functions of x, then the coefficients IjðxÞ, j ¼ 0 . . . 4 can be chosen to be polynomials in the

variable x.

More generally, if fj, j ¼ 1; . . . ; n are functions of the variable x such that any fj satisfies a

linear divided-difference equation of order rj involving only the operatorsD2q and SqDq, then

the product f ¼Qn

j¼1f j satisfies a divided-difference equation of order r ¼Qn

j¼1rj involving

only (at most) the operators

D2iq and SqD2j21

q ; with 0 # 2i # r; and 0 # 2j2 1 # 2j # r ¼Ynk¼1

rk:

As consequence of the previous theorem we state the following.

Corollary 4. If Pn(x) is a solution of a second-order divided-difference equation of

hypergeometric form

f ðxÞD2qyðxÞ þ c ðxÞSqDqyðxÞ þ lnyðxÞ ¼ 0;

where f and c are polynomials of degree at most two and one respectively, then the product

PnðxÞPrðxÞ satisfies a fourth-order divided-difference equation

I4ðn; r; xÞD4qyðxÞ þ I3ðn; r; xÞSqD3

qyðxÞ þ I2ðn; r; xÞD2qyðxÞ

þ I1ðn; r; xÞSqDqyðxÞ þ I0ðn; r; xÞyðxÞ ¼ 0;

where the Ijðn; r; xÞ are polynomials whose degree does not depend on n. Computations using

computer algebra software can allow to find explicitly the coefficients Ijðn; r; xÞ in terms of f,

c and ln.

5.2 Applications

5.2.1 Special cases of classical orthogonal polynomials. The relation (109) allows to

observe that when h ¼ 0, the first associated Pð1Þn21 satisfies the second-order difference

equation

½A1ðsÞFx þ B1ðsÞMx þ C1ðnÞ�yðsÞ ¼ 0;

where the coefficients A1, B1 and C1 are those of (109). It turns out that the previous equation

is of hypergeometric type. Under certain conditions (on the parameters involved in the

definition), the first associated of the Askey–Wilson, the q-Racah and the Racah

polynomials, remains classical. This property is not true in general because the first

associated of classical orthogonal polynomials is in general not classical but belongs rather to

the so-called Laguerre–Hahn class [22]–[24]. Similar results exist for the very classical

orthogonal polynomials [10,29,30].

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Theorem 16. We have the following.

1. For abcd ¼ q, the first associated of the Askey–Wilson polynomials remains classical

and is related to the Askey–Wilson polynomials by

~pð1Þn xðsÞ; a; b; c;q

abc

��q� �¼ un ~pn uxðsÞ;

uq

a;uq

b;uq

c; uabc

��q� �; ð146Þ

with u ¼ ^1.

2. For abq ¼ 1, the first associated of the q-Racah polynomials remains classical and is

related to the q-Racah polynomials by

~Rð1Þ

n xðsÞ;a;1

qa; g; d

��q �¼ ðgdÞn ~Rn

xðsÞ

gd;

1

a;aq;

1

g;1

d

��q �: ð147Þ

3. For a þ b ¼ 21, the first associated of the Racah polynomials remains classical and is

related to the Racah polynomials by

~Rð1Þ

n ðxðsÞ;a;21 2 a; g; dÞ ¼ ~RnðxðsÞ þ gðaþ 1Þ; d2 a; 1 þ a2 d;2g; dÞ: ð148Þ

Proof. For the Askey–Wilson polynomials, one obtains by direct computation using (111)

and the data given in Section 3.2.3 that

h ¼ axc0 2

f00

2

�¼

4ðq2 abcdÞ

1 2 q

with ax ¼ ðq1=2 þ q21=2Þ=2 and

gnþ1 a; b; c;q

abc

��q� �¼ gn

uq

a;uq

b;uq

c; uabc

��q� �;

bnþ1 a; b; c;q

abc

��q� �¼ ubn

uq

a;uq

b;uq

c; uabc

��q� �:

The Relation (146) is obtained using the last two relations and the fact that ~pnþ1 and ~pð1Þn

satisfy the same three-term recurrence relation. The proof of the identities (147) and (148) is

obtained in the same way since

h ¼2qð1 2 abqÞ

1 2 q; ax ¼

q12 þ q

212

2;

gnþ1 a;1

qa; g; d

�¼ g2d2gn

1

a;aq;

1

g;1

djq

�;

bnþ1 a;1

qa; g; d

�¼ gdbn

1

a;aq;

1

g;1

djq

�


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and

h ¼ 2ðaþ bþ 1Þ; ax ¼ 1;

gnþ1ða;21 2 a; g; dÞ ¼ gnðd2 a; 1 þ a2 d;2g; dÞ;

bnþ1ða;21 2 a; g; dÞ ¼ gnðd2 a; 1 þ a2 d;2g; dÞ2 gðaþ 1Þ

for the q-Racah and the Racah polynomials, respectively. A

Notice that relations such as (146)–(148) exist for very classical orthogonal polynomials.

They were given in Refs. [3,10,15,16] for Jacobi, Hahn and Big-q-Jacobi (and also Little-q-

Jacobi) respectively.

5.2.2 Polynomial solutions of some difference equations. The product rules (55), (56) and

(62), (63) can be used to look for polynomial solutions of difference equations with

polynomial coefficients involving only the operators Fx and Mx. For example, if A(s), B(s),

C(s) and D(s) are polynomials in the variable x(s), then the polynomial solution of the

equation (if it exists)

AðsÞFxyðsÞ þ BðsÞMxyðsÞ þ CðsÞyðsÞ ¼ DðsÞ

can be found by writing

yðsÞ ¼Xnk¼0

an;kxkðsÞ;

and solving the linear system obtained with respect to the unknowns an,k in terms of

the coefficients of the polynomials A(s), B(s), C(s) and D(s). This in the same way as for the

usual differential equation since the operators Fx and Mx transform a polynomial in the

variable x(s) into a polynomial of the same variable. Using the quotient rules, one can look

for rational solutions of some difference equations with polynomial coefficients in the same

way. This approach can be used to look in general for analytic solutions (here we mean

solutions which can be expanded in power series in terms of the variable x(s)) of difference

equations with polynomial coefficients.

5.2.3 Steps forward towards the characterization of some classes of OP. In this work, we

have derived diverse results for classical orthogonal polynomials in the same line as for those

of the very classical orthogonal polynomials. In the sequel, we have obtained many

intermediate results such as the product and quotient rules, the coefficients

Dn;k; Sn;k;Fn;k;Mn;k;bn; Tn;1; Tn;2;bn and gn. These coefficients can be used for the

implementation of codes in computer algebra relative to the classical orthogonal

polynomials on quadratic and q-quadratic lattices. They could also be used for the complete

characterization of the classical orthogonal polynomials. As illustration, for the very classical

orthogonal polynomials, the ratio ðTn;kÞ=ðTn;k21Þ is a rational function of n or q n depending on

whether the variable is continuous, linear or q-linear respectively. This is an equivalent

characterization property for the very classical orthogonal polynomials [1,17]. But from the

results obtained here, we observe that the ratio ðTn;2Þ=ðTn;1Þ is a rational function of n and q n

(at the same time) for the q-quadratic lattice and for the basis (x n(s)). This is an indication for

the complexity for the formulation of the characterization theorems for the classical (not very

classical) orthogonal polynomials. Summing up, we can conclude that the results obtained in

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this work constitute some important steps forward towards the complete characterization of the

classical orthogonal polynomials as well as that of the semi-classical and the Laguerre–Hahn

orthogonal polynomials of a discrete variable on nonuniform lattices [23,24].

Acknowledgements

My sincere thanks to Professors Wolfram Koepf, Andre Ronveaux and Mark Van Hoeij, the

first two for their various comments which helped me to improve the manuscript and the

latter for his help in writing some of the Maple codes needed for this work. Many thanks to

Professor Andreas Ruffing (Munich University of Technology, Department of Mathematics)

for various financial supports which made possible my participation to the meetings:

International Workshop on Special Functions, Orthogonal Polynomials, Quantum Groups

and Related Topics, dedicated to Dick Askey on his 70th birthday, Bexbach, Germany,

October 18–22, 2003; International Conference on Difference Equations, Special Functions

and Applications, Munich, Germany, July 25–29, 2005. Partial results of this paper were

presented during these meetings and discussions which followed my talks were very helpful

for the improvement and the extensions of the initial results.

References

[1] Al-Salam, W.A., 1990, Characterization theorems for orthogonal polynomials. In: P.G. Nevai (Ed.) OrthogonalPolynomials: Theory and Practice, NATO ASI Series (Dordrecht: Kluwer), pp. 1–24.

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On difference equations for orthogonal polynomials on nonuniform lattices 1

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