This article was downloaded by: [Moskow State Univ Bibliote]On: 30 October 2013, At: 10:03Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registeredoffice: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK

Integral Transforms and SpecialFunctionsPublication details, including instructions for authors andsubscription information:http://www.tandfonline.com/loi/gitr20

Lowering operators associated with D-Laguerre–Hahn polynomialsFrancisco Marcellán a & Ridha Sfaxi ba Departamento de Matemáticas , Universidad Carlos III deMadrid , Avenida de la Universidad 30, 28911, Leganés, Spainb Faculté des Sciences de Gabès, Département deMathématiques , Cité Erriadh, 6072, Gabès, TunisiaPublished online: 15 Feb 2011.

To cite this article: Francisco Marcellán & Ridha Sfaxi (2011) Lowering operators associated withD-Laguerre–Hahn polynomials, Integral Transforms and Special Functions, 22:12, 879-893, DOI:10.1080/10652469.2010.542576

To link to this article: http://dx.doi.org/10.1080/10652469.2010.542576

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information (the“Content”) contained in the publications on our platform. However, Taylor & Francis,our agents, and our licensors make no representations or warranties whatsoever as tothe accuracy, completeness, or suitability for any purpose of the Content. Any opinionsand views expressed in this publication are the opinions and views of the authors,and are not the views of or endorsed by Taylor & Francis. The accuracy of the Contentshould not be relied upon and should be independently verified with primary sourcesof information. Taylor and Francis shall not be liable for any losses, actions, claims,proceedings, demands, costs, expenses, damages, and other liabilities whatsoever orhowsoever caused arising directly or indirectly in connection with, in relation to or arisingout of the use of the Content.

This article may be used for research, teaching, and private study purposes. Anysubstantial or systematic reproduction, redistribution, reselling, loan, sub-licensing,systematic supply, or distribution in any form to anyone is expressly forbidden. Terms &Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions

Integral Transforms and Special FunctionsVol. 22, No. 12, December 2011, 879–893

Lowering operators associated with D-Laguerre–Hahnpolynomials

Francisco Marcellána* and Ridha Sfaxib

aDepartamento de Matemáticas, Universidad Carlos III de Madrid, Avenida de la Universidad 30,28911 Leganés, Spain; bFaculté des Sciences de Gabès, Département de Mathématiques,

Cité Erriadh 6072 Gabès, Tunisia

(Received 9 November 2010; final version received 21 November 2010 )

In this paper, a new lowering operator Du with a linear functional u as a parameter is introduced in thelinear space P of all polynomials in one variable with complex coefficients. It is given by

Du(p)(x) = p′(x) +⟨uy,

p(x) − p(y)

x − y

⟩, p ∈ P.

The concept of Du-semiclassical polynomial sequence is defined. We show that such a sequence belongsto the family of D-Laguerre–Hahn polynomials. This allows us to provide some characterizations of a Du-semiclassical polynomial sequence in terms of a distributional equation that the linear functional satisfiesas well as a structure relation. An illustrative example is considered.

Keywords: linear functionals; quasi-orthogonality; orthogonal polynomials; Laguerre–Hahn polynomials;Appell polynomials

AMS Subject Classifications: 33C45; 42C05

1. Introduction

Let P be the linear space of polynomials with complex coefficients. Let P′ be its dual space.

(u)n := 〈u, xn〉, n ≥ 0, will denote the moments of u ∈ P′ with respect to the sequence {xn}n≥0.

In the sequel, we set P′M := {u ∈ P

′, (u)0 �= −n, n ≥ 1}. When (u)0 = 1, the linear functional u

is said to be normalized. Let us define the following operations on P′, (see [6,12]). For any c ∈ C,

p, q ∈ P and u, v ∈ P′, we have

〈qu, p〉 = 〈u, qp〉, 〈u′, p〉 = −〈u, p′〉,〈δc, p〉 = p(c), (Dirac delta at point c ∈ C), δ := δ0,

*Corresponding author. Email: pacomarc@ing.uc3m.es

ISSN 1065-2469 print/ISSN 1476-8291 online© 2011 Taylor & Francishttp://dx.doi.org/10.1080/10652469.2010.542576http://www.tandfonline.com

Dow

nloa

ded

by [

Mos

kow

Sta

te U

niv

Bib

liote

] at

10:

03 3

0 O

ctob

er 2

013

880 F. Marcellán and R. Sfaxi

〈uv, p〉 = 〈v, up〉, where (up)(x) =⟨uy,

xp(x) − yp(y)

x − y

⟩,

〈(x − c)−1u, p〉 = 〈u, θc(p)〉 =⟨u,

p(x) − p(c)

x − c

⟩.

Here, 〈uy, p(x, y)〉 means the action of u on the polynomial p(x, y) with respect to the variabley. We also denote C

∗ = C\{0}.The following lemma will be useful in the sequel.

Lemma 1.1 ( [6]) For any u, v ∈ P′, f, g ∈ P and c ∈ C, we have

(x − c)((x − c)−1u) = u, (1.1)

(f u)′ = f u′ + f ′u, (1.2)

f (x−1u) = x−1(f u) + 〈u, θ0f 〉δ, (1.3)

f (uv) = (f u)v + x(uθ0f )(x)v, (1.4)

(uθ0(fg))(x) = g(x)(uθ0f )(x) + ((f u)θ0g)(x), (1.5)

〈u2, θ0(fg)〉 = 〈u, f (uθ0g) + g(uθ0f )〉. (1.6)

A non-zero v ∈ P′ is said to be weakly regular if for a φ ∈ P such that φv = 0, then φ = 0 [10].

Clearly, δc is not a weakly regular functional for every c ∈ C, since (x − c)δc = 0.

Let {Bn}n≥0 be a monic polynomial sequence (MPS), deg Bn = n, and let {wn}n≥0 be its dualsequence defined by 〈wn, Bm〉 = δn,m, n, m ≥ 0, where δn,m is the Kronecker delta. The MPS{Bn}n≥0 is said to be orthogonal (MOPS) with respect to w ∈ P

′, if 〈w, BnBm〉 = 0, n �= m and〈w, B2

n〉 �= 0, n ≥ 0. In this case,w is said to be quasi-definite (regular) [5]. Note thatw = (w)0w0,

with (w)0 �= 0. Referring to [10], quasi-definite linear functionals are weakly regular. Note that,in general, the converse is not true.

Proposition 1.2 ([12]) An MPS {Bn}n≥0 with dual sequence {wn}n≥0 is orthogonal with respectto w0 if and only if one of the following statements hold.

(i) wn = 〈w0, B2n〉−1Bnw0, n ≥ 0.

(ii) {Bn}n≥0 satisfies a three-term recurrence relation (TTRR).

B0(x) = 1, B1(x) = x − β0,

Bn+2(x) = (x − βn+1)Bn+1(x) − γn+1Bn(x), n ≥ 0, (1.7)

where βn = 〈w0, xB2n〉/〈w0, B

2n〉 ∈ C and γn+1 = 〈w0, B

2n+1〉/〈w0, B

2n〉 ∈ C

∗, n ≥ 0.

Recall that w ∈ P′ is quasi-definite if and only if for each non-negative integer n the Hankel

determinant �n(w) = det((w)i+j )0≤i,j≤n �= 0, n ≥ 0 [5].For an MOPS {Bn}n≥0, we can define the sequence of associated polynomials of the first kind

{B(1)n }n≥0 as follows [5]: B(1)

n (x) = w0θ0Bn+1(x), n ≥ 0. The sequence {B(1)n }n≥0 is also a MOPS

Dow

nloa

ded

by [

Mos

kow

Sta

te U

niv

Bib

liote

] at

10:

03 3

0 O

ctob

er 2

013

Integral Transforms and Special Functions 881

and satisfies the shifted TTRR

B(1)0 (x) = 1, B

(1)1 (x) = x − β1,

B(1)n+2(x) = (x − βn+2)B

(1)n+1(x) − γn+2B

(1)n (x), n ≥ 0.

An MOPS {Bn}n≥0 with respect to w0 is said to be of D-Laguerre–Hahn if w0 satisfies thefollowing distributional equation (see [3,4,8]):

(�w0)′ + (1w0) + B(x−1w2

0) = 0, (1.8)

where (�, 1, B) ∈ P3 and � is a monic polynomial. In this case, w0 is said to be a D-Laguerre–

Hahn linear functional. We can associate with (�, 1, B) the non-negative integer number:max(deg(�) − 2, deg(B) − 2, deg(1) − 1). Obviously, a D-Laguerre–Hahn linear functionalw0 satisfies an infinite number of equations like (1.8). Indeed, it is enough to multiply by any χ ∈ P

on both sides of (1.8) in such a way that (χ�w0)′ + (χ1 − χ ′�)w0 + (χB)(x−1w2

0) = 0. So,we can associate with w0 a non-negative integer s, the so-called class of w0, which is definedby s := min(max(deg(�) − 2, deg(B) − 2, deg(1) − 1)), where the minimum is taken overall possible choices of polynomials (�, 1, B) satisfying (1.8). It is well known (see [1]) that aD-Laguerre–Hahn linear functional w0 satisfying (1.8) is of class s, if and only if for any zero c

of �, we have |1(c) + �′(c)| + |B(c)| + |〈w0, θc1 + θ2c � + w0(θ0θcB)〉| > 0.

When B vanishes, {Bn}n≥0 or w0 is said to be D-semiclassical. In particular, a D-semiclassicalpolynomial sequence (respectively, a linear functional) of class zero is said to be D-classical. Bymeans of a suitable affine transformation, there exist four canonical classical cases, the well-knownfamilies of Hermite, Laguerre, Bessel and Jacobi MOPS, respectively (see [2,3,10]).

If B �= 0, then {Bn}n≥0 or w0 is said to be of strict D-Laguerre–Hahn. In the same way, by meansof a suitable affine transformation, there exist four families of D-Laguerre–Hahn polynomials ofclass zero. Each of them includes two subfamilies: a singular family and a non-singular one. Formore details, see [4].

Proposition 1.3 ([7]) Let {Bn}n≥0 be an MOPS with respect to w0, satisfying (1.7). Thefollowing statements are equivalent.

(i) w0 is a D-Laguerre–Hahn linear functional of class s satisfying (1.8).

(ii) {Bn}n≥0 satisfies a first structure relation (FSR)

�(x)B ′n+1(x) − B(x)B(1)

n (x) = Cn+1(x) − C0(x)

2Bn+1(x)

− γn+1Dn+1(x)Bn(x), n ≥ 0, (1.9)

where Cn and Dn are polynomials with coefficients depending on n, such that deg Cn ≤ s +1, deg Dn ≤ s, and given by

Cn+1(x) = −Cn(x) + 2(x − βn)Dn(x),

γn+1Dn+1(x) = −�(x) + γnDn−1(x) − (x − βn)Cn(x) + (x − βn)2Dn(x),

for every n ≥ 0, and the initial conditions

C0(x) = −�′(x) − 1(x),

D0(x) = −(w0θ0�)′(x) − w0θ0(1)(x) − (w20θ

20 B)(x), D−1(x) = 0.

Dow

nloa

ded

by [

Mos

kow

Sta

te U

niv

Bib

liote

] at

10:

03 3

0 O

ctob

er 2

013

882 F. Marcellán and R. Sfaxi

Using the orthogonality of {Bn}n≥0, (1.9) can be written as follows:

�(x)B ′n+1(x) − B(x)B(1)

n (x) =n+d∑

ν=n−s

λn,νBν(x), n ≥ s, (1.10)

where d = max(deg �, deg B) and s is the class of the linear functional w0.

In this paper, we are dealing with the non-singular D-Laguerre–Hahn MOPS of class zeroanalogous to the Hermite MOPS that we denote by {Hn(.; ξ)}n≥0, where ξ = (τ, λ, ρ) ∈ C

3,

with τ �= −n, n ≥ 1, and ρ �= 0. Such a polynomial sequence is orthogonal with respect to aunique monic linear functional H(ξ). Recall that {Hn(.; ξ)}n≥0 appears in the description of theLaguerre–Hahn MOPS of class zero [4]. In the sequel, the following properties of {Hn(.; ξ)}n≥0

will be needed.

(TTRR)

{H0(x; ξ) = 1, H1(x; ξ) = x − λ,

Hn+2(x; ξ) = xHn+1(x; ξ) − γn+1Hn(x; ξ), n ≥ 0,(1.11)

where γ1 = (ρ/2)(τ + 1), γn+1 = ( 12 )(n + τ + 1), n ≥ 1.

(FSR) H ′n+1(x; ξ) − B(x)H (1)

n (x; ξ) = −2ρ−1[(ρ − 1)x + λ]Hn+1(x; ξ),

(n + τ + 1)Hn(x; ξ) + (τ + 1)(ρ − 1)δn,0, n ≥ 0, (1.12)

where B(x) = 2ρ−1(ρ − 1)x2 + 2ρ−1λ(2 − ρ)x + 1 − ρ(τ + 1) − 2ρ−1λ2.

Note that {Hn(. ; ξ)}n≥0 is a generalization of the Hermite MOPS {Hn}n≥0, which appears whenξ = (0, 0, 1). It is well known that {Hn}n≥0 is an Appell sequence with respect to the usual deriva-tive operator D = (d/dx) or, simply, a D-AMPS, since D(Hn+1) = (n + 1)Hn, n ≥ 0 [9]. Let usshow that {Hn(.; ξ)}n≥0 satisfies a similar property with respect to a new operator. Indeed, from theexpression of B(x) already quoted and (1.11), we obtain (BH(ξ))θ0Hn+1(x; ξ) = −2ρ−1((ρ −1)x + λ)Hn+1(x; ξ) + B(x)H (1)

n (x; ξ) + (τ + 1)(ρ − 1)δn,0, n ≥ 0.Accordingly, H ′n+1(x; ξ) −

(BH(ξ))θ0Hn+1(x; ξ) = (n + τ + 1)Hn(x; ξ), n ≥ 0, by (1.12). If we set U(ξ) := −BH(ξ) ∈P

′, then (U(ξ))0 = −τ �= −n, n ≥ 1, and we can write H ′n+1(x; ξ) + U(ξ)θ0Hn+1(x; ξ) =

(n + τ + 1) Hn(x; ξ), n ≥ 0. Hence, by introducing the operator DU(ξ)(p)(x) := p′(x) +U(ξ)θ0p(x), for all p ∈ P, it follows that

DU(ξ)Hn+1(x; ξ) = (n + τ + 1)Hn(x; ξ), n ≥ 0. (1.13)

Thus, {Hn(., ξ)}n≥0 is a DU(ξ)-AMPS. Besides, from (1.11), it follows that

D2U(ξ)Hn+1(x; ξ) − 2xDU(ξ)Hn+1(x; ξ) = −2(n + τ + 1)Hn+1(x; ξ), n ≥ 2.

From a more general point of view, for a given u ∈ P′M, let Du : P −→ P be the linear operator

defined by

Du(p)(x) = p′(x) +⟨uy,

p(x) − p(y)

x − y

⟩, p ∈ P, (D0 = D).

The aim of this work is to introduce the concept of Du-semiclassical MOPS and to show that theybelong to the D-Laguerre–Hahn class.A necessary and sufficient condition for a D-Laguerre–HahnMOPS to be a Du-semiclassical MOPS will be given.

Dow

nloa

ded

by [

Mos

kow

Sta

te U

niv

Bib

liote

] at

10:

03 3

0 O

ctob

er 2

013

Integral Transforms and Special Functions 883

The structure of the paper is as follows. Section 2 contains some properties of the operator Du aswell as the definition of the Du-semiclassical polynomials. In Section 3, we give a characterizationof a Du-semiclassical linear functional by a linear distributional equation. In Section 4, first wecharacterize a Du-semiclassical MOPS by means of an FSR, i.e. a finite-type relation betweenthe polynomials and the monic polynomials of its first Du-derivative [11]. Second, we will provethat every Du-semiclassical linear functional belongs to the family of D-Laguerre–Hahn linearfunctionals. Finally, an example of Du-semiclassical MOPS is presented.

2. The Du-semiclassical polynomials

2.1. Some properties of the operator Du

Among the properties of the linear operator Du, we emphasize the following one related to thelowering character of such operator. Indeed,we have

Du(1) = 0, Du(x) = (u)0 + 1, Du(xn) = (n + (u)0)x

n−1 +n−2∑ν=0

(u)n−ν−1xν, n ≥ 2.

The assumption u ∈ P′M is related to the fact that deg(Du(p)) = deg(p) − 1, for every polynomial

p.The transpose tDu of Du is defined as follows. For all (p, w) ∈ P × P

′, 〈tDu(w), p〉 =〈w, Du(p)〉 = 〈w, p′ + uθ0p〉 = 〈−w′ + x−1(uw), p〉. Thus, tDu(w) = −w′ + x−1(uw), w ∈P

′. For the sake of simplicity, we will denote Du := −tDu, and then we have: 〈Du(w), p〉 =−〈w, Du(p)〉, (p, w) ∈ P × P

′.The linear operator Du is one-to-one. Indeed, if Du(w) = 0, w ∈ P

′, then 〈w, Du(xn+1)〉 =

0, n ≥ 0. Hence, w = 0, since we have deg(Du(xn+1)) = n.

The proof of the following operational rules is a straightforward exercise.

Proposition 2.1 For v ∈ P′ and (f, g) ∈ P

2, we have

Du(fg) = Du(f )g + f Du(g) + uθ0(fg) − (uθ0f )g − f (uθ0g), (2.1)

Du(f v) = f Du(v) + Du(f )v + (vθ0f )u − (uθ0f )v. (2.2)

For an MPS {Bn}n≥0 and u ∈ P′M, we can define

B [1]n (x; u) := (n + (u)0 + 1)−1(DuBn+1)(x), n ≥ 0. (2.3)

Clearly, {B[1]n (.; u)}n≥0 is an MPS, deg B [1]

n (.; u) = n, n ≥ 0. If {w[1]n (u)}n≥0 denotes the dual

sequence of {B[1]n (.; u)}n≥0, then we have the following.

Lemma 2.2

Du(w[1]n (u)) = −(n + (u)0 + 1)wn+1, n ≥ 0. (2.4)

Proof Let n ≥ 0 be a fixed integer. Since 〈w[1]n (u), B [1]

m (.; u)〉 = δn,m, then 〈Du(w[1]n (u)),

Bm+1〉 = −(m + (u)0 + 1)δn,m = −(n + (u)0 + 1)δn,m, m ≥ 0. In addition, 〈Du(w[1]n (u)), 1〉 =

0. Hence, Du(w[1]n (u)) = −(n + (u)0 + 1)wn+1. �

Dow

nloa

ded

by [

Mos

kow

Sta

te U

niv

Bib

liote

] at

10:

03 3

0 O

ctob

er 2

013

884 F. Marcellán and R. Sfaxi

2.2. The Du-semiclassical MOPS

We start by recalling the notion of the quasi-orthogonality and some of its useful results(see [5,13]).

Definition 2.3 Let v ∈ P′ and σ is a non-negative integer. A MPS {Bn}n≥0 is said to be quasi-

orthogonal of order σ with respect to v, if

〈v, xmBn(x)〉 = 0, 0 ≤ m ≤ n − σ − 1, n ≥ σ + 1, (2.5)

∃r ≥ σ, 〈v, xr−σBr(x)〉 �= 0. (2.6)

The MPS {Bn}n≥0 is said to be strictly quasi-orthogonal of order σ with respect to v, if it satisfies(2.5), and

〈v, xn−σBn〉 �= 0, n ≥ σ. (2.7)

Remark 2.4

(i) Here, v is not assumed to be quasi-definite. In general, a quasi-orthogonal sequence can notbe strictly quasi-orthogonal.

(ii) In particular, a strictly quasi-orthogonal sequence of order zero with respect to v is anorthogonal sequence with respect to v.

In order to give a suitable definition of a Du−semiclassical MOPS, we need the following results.

Lemma 2.5 Let {Bn}n≥0 be a MOPS with respect to w0. Then, for any integer σ ≥ 0 there existsAσ+1 ∈ P, deg Aσ+1 = σ + 1, such that

〈w0, Aσ+1〉 = 0, (2.8)

〈w0, xAσ+1〉 �= −n, n ≥ 0. (2.9)

Proof Assume that {Bn}n≥0 is a MOPS with respect to w0. So, w0 is quasi-definite and hence�1(w0) �= 0, i.e. (w0)2 − (w0)

21 �= 0.

If σ = 0, we can take A1(x) = λ(x − (w0)1), with λ �= −n((w0)2 − (w0)21)

−1, n ≥ 0. We get〈w0, A1〉 = λ((w0)1 − (w0)1) = 0 and 〈w0, xA1〉 = λ((w0)2 − (w0)

21) �= −n, n ≥ 0.

If σ ≥ 1, we can take Aσ+1(x) = xσ+1 + θ(x − (w0)1) − (w0)σ+1, with θ �= (−n −(w0)σ+2 + (w0)σ+1(w0)1)((w0)2 − (w0)

21)

−1, n ≥ 0. Then 〈w0, Aσ+1〉 = 0, and 〈w0, xAσ+1〉 =(w0)σ+2 + θ((w0)2 − (w0)

21) − (w0)σ+1(w0)1 �= −n, n ≥ 0. �

Proposition 2.6 For any MOPS {Bn}n≥0 with respect to w0 and any integer σ ≥ 0, there existsAσ+1 ∈ P, deg Aσ+1 = σ + 1, such that for any c ∈ C, the MPS {B [1]

n (.; u)}n≥0, where u =−δc + (x − c)Aσ+1w0, is quasi-orthogonal of order σ with respect to δc.

Proof Assume that {Bn}n≥0 is a MOPS with respect to w0 and σ a non-negative integer. FromLemma 2.5, there exists Aσ+1 ∈ P, deg Aσ+1 = σ + 1 such that (2.8) and (2.9) hold. Given c ∈C and let u = −δc + (x − c)Aσ+1w0 ∈ P

′. Since, (u)0 = −1 + 〈w0, xAσ+1〉 − c〈w0, Aσ+1〉 =−1 + 〈w0, xAσ+1〉 �= −n, n ≥ 1, then {B [1]

n (.; u)}n≥0 is an MPS, deg B [1]n (.; u) = n, n ≥ 0.

Next, let us show that B[1]n (c; u) = 0, n ≥ σ + 1, and B [1]

σ (c; u) �= 0, i.e. 〈δc, xmB [1]

n (x; u)〉 =0, 0 ≤ m ≤ n − σ − 1, n ≥ σ + 1, and 〈δc, B

[1]σ (x; u)〉 �= 0, and then {B [1]

n (.; u)}n≥0 is quasi-orthogonal of order σ with respect to δc.

Indeed, by (2.3), B [1]n (c; u) = (n + (u)0 + 1)−1〈δ′

c − x−1(uδc), Bn+1〉. Since δ2c = −xδ′

c and((x − c)Aσ+1w0)δc = xAσ+1w0, then uδc = x(δ′

c + Aσ+1w0), and hence x−1(uδc) = δ′c +

Dow

nloa

ded

by [

Mos

kow

Sta

te U

niv

Bib

liote

] at

10:

03 3

0 O

ctob

er 2

013

Integral Transforms and Special Functions 885

Aσ+1w0. So, B [1]n (c; u) = −(n + (u)0 + 1)−1〈w0, Aσ+1Bn+1〉, n ≥ 0. Using the orthogonality of

{Bn}n≥0, we get B [1]n (c; u) = 0, n ≥ σ + 1, and B [1]

σ (c; u) = −K〈w0, B2σ+1〉/(σ + (u)0 + 1) �=

0, where K is the leading coefficient of Aσ+1. �

To avoid analogous situations to those found in the above proposition, we give the followingdefinition of a Du-semiclassical MOPS.

Definition 2.7 Let u ∈ P′M . An MPS {Bn}n≥0 is said to be Du-semiclassical, when it is orthog-

onal and such that the MPS {B [1]n (.; u)}n≥0 given by (2.3) is quasi-orthogonal of order σ with

respect to a weak-regular linear functional v.

The quasi-definite linear functional w0 corresponding to {Bn}n≥0 is also said to be Du-semiclassical.

As a straightforward consequence, the non-singular D-Laguerre–Hahn MOPS of class zeroanalogous to the Hermite polynomial sequence, is DU(ξ)-semiclassical, according to (1.13) andRemark 2.4(ii).

3. Characterization of a Du-semiclassical linear functional by means of a distributionalequation

To give a characterization of a Du-semiclassical linear functional by a distributional equation, wewill need some information concerning u and v.

Lemma 3.1 Under the assumption of the previous definition, we have the following.

(i) There exists a non-zero φ ∈ P, deg φ = t ≤ σ + 2, such that v = φw0.

(ii) If (v)0 �= 0, then there exists B ∈ P, deg B ≤ σ + 2, such that u = Bw0.

Proof From the assumption, there exist a weak-regular linear functional v and a non-negativeinteger r ≥ σ, such that

〈v, xmB [1]n (x; u)〉 = 0, 0 ≤ m ≤ n − σ − 1, n ≥ σ + 1, (3.1)

〈v, xr−σB [1]r (x; u)〉 �= 0. (3.2)

Applying Du on both sides of (1.7) and using (2.3) and (2.1), we obtain

Bn+1 + 〈u, Bn+1〉 = (n + (u)0 + 2)B[1]n+1(x; u) + (n + (u)0 + 1)βn+1B

[1]n (x; u)

+ (n + (u)0)γn+1B[1]n−1(x; u) − (n + (u)0 + 1)xB [1]

n (x; u), n ≥ 0.

(3.3)

Then, 〈v, xmBn+1〉 + 〈u, Bn+1〉(v)m = 0, n ≥ m + σ + 2, m ≥ 0, due to (3.1) and (3.3). So,〈xmv + (v)mu, Bn〉 = 0, n ≥ σ + 3 + m, m ≥ 0. Using the orthogonality of {Bn}n≥0, thereexists a polynomial sequence {�m+σ+2(x)}n≥0, deg �m+σ+2 ≤ m + σ + 2, such that

xmv + (v)mu = �m+σ+2w0, m ≥ 0, (3.4)

where �m+σ+2(x) = ∑m+σ+2ν=0 〈xmv + (v)mu, Bν〉〈w0, B

2ν 〉−1Bν(x), m ≥ 0.

Dow

nloa

ded

by [

Mos

kow

Sta

te U

niv

Bib

liote

] at

10:

03 3

0 O

ctob

er 2

013

886 F. Marcellán and R. Sfaxi

For m = 0, (3.4) gives

v + (v)0u = �σ+2w0. (3.5)

Substituting (3.5) into (3.4), we obtain

((v)m − (v)0xm)u = (�m+σ+2 − xm�σ+2)w0, m ≥ 0. (3.6)

So, we need to discuss two cases.(A) (v)0 = 0. Directly, (i) holds, with φ = �σ+2, due to (3.5) with (v)0 = 0, and the weak-

regularity of v to obtain deg φ ≥ 0. In this case, (3.6) becomes

(v)mu = (�m+σ+2 − xm�σ+2)w0, m ≥ 0. (3.7)

From the weak-regularity of v, there exists an integer k ≥ 1 such that

k = min{m ≥ 1 | (v)m �= 0}. (3.8)

Thus, for m = k, (3.7) yields

u = Bw0, (3.9)

where B(x) = (v)−1k (�k+σ+2(x) − xk�σ+2(x)).

Besides, by (3.9), (3.7) and the quasi-definiteness of w0, we obtain

�m+σ+2(x) = xm�σ+2(x) + (v)mB(x), m ≥ 0. (3.10)

Hence, if (v)0 = 0 then (ii) holds.(B) (v)0 �= 0. There exists an integer l ≥ 2 such that (v)l �= ((v)0)

1−l(v)l1. Otherwise, if weset c = (v)1(v)−1

0 , then (v)m = (v)0cm = 0, m ≥ 0, i.e. v = (v)0δc. This contradicts the weak-

regularity of v.

By taking successively m = 1 and m = l in (3.6), we obtain

((v)1 − (v)0x)u = (�σ+3 − x�σ+2)w0, (3.11)

((v)l − (v)0xl)u = (�l+σ+2 − xl�σ+2)w0. (3.12)

Since, (v)l �= (v)1−l0 (v)l1, then (v)l − (v)0x

l and (v)1 − (v)0x are coprimes. From Bézout’sidentity, there exist two polynomials E(x) and F(x) such that

((v)l − (v)0xl)E(x) + ((v)1 − (v)0x)F (x) = 1. (3.13)

By (3.11)–(3.13), we can easily deduce that

u = Bw0, (3.14)

where B(x) = E(x)(�σ+3(x) − x�σ+2(x)) + F(x)(�l+σ+2(x) − xl�σ+2(x)).

Hence, if (v)0 �= 0, then (ii) holds.By inserting (3.14) in (3.6) and using the quasi-definiteness of w0, we get ((v)m −

(v)0xm)B(x) = �m+σ+2(x) − xm�σ+2(x),m ≥ 0.The consideration of the degrees on both sides

of the previous formula yields deg B ≤ σ + 2. Substituting (3.14) into (3.5), we get v = φw0 withφ(x) = �σ+2(x) − (v)0B(x), and 0 ≤ deg φ ≤ σ + 2. Hence, if (v)0 �= 0, then (i) holds. �

Next we will give a characterization of a Du-semiclassical MOPS.

Dow

nloa

ded

by [

Mos

kow

Sta

te U

niv

Bib

liote

] at

10:

03 3

0 O

ctob

er 2

013

Integral Transforms and Special Functions 887

Theorem 3.2 Let u ∈ P′M and let {Bn}n≥0 be an MOPS with respect to w0. The following

statements are equivalent.

(i) {Bn}n≥0 is Du-semiclassical.(ii) There exists (�, , B) ∈ P

3, � monic, deg � = t, deg = p ≥ 1, and deg B ≤ k + σ +2, where k = min{m ≥ 0 | (�w0)m �= 0}, and σ = max(t − 2, p − 1), such that w0 is asolution of the distributional equation:

Du(�w0) + w0 = 0 where u = Bw0. (3.15)

Proof (i) ⇒ (ii). From Lemma 3.1(i), let us write φ(x) = λ�(x), where λ is a normalizationconstant and � is a monic polynomial with deg � = t ≤ σ + 2. From (2.3), we get

〈Du(�w0), Bn+1〉 = −λ−1(n + (u)0 + 1)〈v, B [1]n (x; u)〉, n ≥ 0. (3.16)

But, we have 〈v, B[1]n (x; u)〉 = 0, n ≥ σ + 1, according to (3.1), with m = 0. As a conse-

quence, since v �= 0, there exists an integer p, 1 ≤ p ≤ σ + 1, such that 〈v, B[1]p−1(x; u)〉 �=

0, 〈v, B[1]n (x; u)〉 = 0, n ≥ p. Using (3.16), it follows that 〈Du(�w0), Bp〉 �= 0 and

〈Du(�w0), Bn〉 = 0, n ≥ p + 1. So, from the orthogonality of {Bn}n≥0, there exists a poly-nomial (x) = ∑p

ν=1(ν + (u)0)λ−1〈v, B

[1]ν−1(.; u)〉〈w0, B

2ν 〉−1Bν(x), deg = p ≥ 1, such that

Du(�w0) + w0 = 0. Hence, (ii) holds.Before showing that (ii) ⇒ (i), we can determine the nature of the quasi-orthogonality, i.e. if

it is strict or not.For every integer n ≥ σ, let us consider,

(n + (u)0 + 1)〈v, xn−σB [1]n (x; u)〉 = −λ〈Du(x

n−σ�w0), Bn+1〉. (3.17)

But, from (2.2), we can write Du(xn−σ�w0) = xn−σ Du(�w0) + �Du(x

n−σ )w0 +[((�w0)θ0x

n−σ )B − ((Bw0)θ0xn−σ )�]w0. So, in view of (3.15), we obtain

Du(xn−σφw0) = �n+1w0, n ≥ σ (3.18)

where

�n+1(x) = −xn−σ(x) + �(x)Du(xn−σ )

+ ((�w0)θ0xn−σ )(x)B(x) − ((Bw0)θ0x

n−σ )(x)�(x). (3.19)

Note that deg �n+1 ≤ n + 1, n ≥ σ. Indeed, we will analyse the two previous situations.(A) (v)0 = 0. By Lemma 3.1, deg � = t ≤ σ + 2, deg = p ≤ σ + 1 and deg B = r ≤ k +

σ + 2, where k is given by (3.8).Note that deg((�w0)θ0x

n−σ ) = n − σ − k − 1, because we have v = λ�w0, (v)0 = · · · =(v)k−1 = 0 and (v)k �= 0. Since deg Du(x

n−σ ) = n − σ − 1, and deg((Bw0)θ0xn−σ ) ≤ n − σ −

1, then deg �n+1 ≤ n + 1, n ≥ σ.

(B) (v)0 �= 0. By Lemma 3.1, deg � = t ≤ σ + 2, deg = p ≤ σ + 1 and deg B = r ≤σ + 2. Besides, deg((�w0)θ0x

n−σ ) = n − σ − 1, because we have (�w0)0 = λ−1(v)0 �= 0.

Therefore, since deg Du(xn−σ ) = n − σ − 1 and deg(Bw0)θ0x

n−σ ≤ n − σ − 1, we obtaindeg �n+1 ≤ n + 1, n ≥ σ.

Dow

nloa

ded

by [

Mos

kow

Sta

te U

niv

Bib

liote

] at

10:

03 3

0 O

ctob

er 2

013

888 F. Marcellán and R. Sfaxi

By inserting (3.18) in (3.17), we obtain

(n + (u)0 + 1)〈v, xn−σB [1]n (x; u)〉 = −λ〈w0, �n+1Bn+1(x)〉, n ≥ σ. (3.20)

Denoting by ρn the coefficient of the monomial xn+1 in �n+1(x), n ≥ σ, and using (3.19)

we obtain

ρn = (n − σ)�(σ+2)(0)

(σ + 2)! − (σ+1)(0)

(σ + 1)! + (v)0B(σ+2)(0)

(σ + 2)!λ, n ≥ σ. (3.21)

Hence, (3.20) can be written as

(n + (u)0 + 1)〈v, xn−σB [1]n (x; u)〉 = −λρn〈w0, B

2n+1〉, n ≥ σ. (3.22)

For n = r in (3.22), where r is given by (3.2), (r ≥ σ), since we have 〈v, xr−σB [1]r (x; u)〉 �= 0,

then ρr �= 0 and so that deg �r+1 = r + 1. Thus,

σ = max(t − 2, p − 1). (3.23)

Indeed, first note that we have σ ≥ max(t − 2, p − 1). If we assume that σ > max(t − 2, p −1), then the analysis of the degrees of the polynomial �r+1, yields deg �r+1 ≤ r. This is acontradiction. Hence, (3.23) holds.

As a consequence, we can analyse two situations.When t = σ + 2, then

ρn = n − σ − (σ+1)(0)

(σ + 1)! + (v)0B(σ+2)(0)

(σ + 2)!λ, n ≥ σ. (3.24)

The MPS {B [1]n (x; u)}n≥0 is strictly quasi-orthogonal of order σ if and only if

−(σ+1)(0)/(σ + 1)! + (v)0(B(σ+2)(0))/ ((σ + 2)!λ) �= −n, n ≥ 0.

When t < σ + 2, then

ρn = −(σ+1)(0)

(σ + 1)! + (v)0B(σ+2)(0)

(σ + 2)!λ = ρr �= 0, n ≥ σ. (3.25)

Thus, the MPS {B[1]n (x; u)}n≥0 is strictly quasi-orthogonal of order σ.

(ii) ⇒ (i). We have (n + (u)0 + 1)〈�w0, xmB [1]

n (x; u)〉 = −〈Du(xm�w0), Bn+1〉, for all non-

negative integers m and n. From (2.2) and (3.15) as above, we have Du(xm�w0) = �m+σ+1w0,

with �m+σ+1(x) = −xm(x) + �(x)Du,τ (xm) + ((�w0)θ0x

m)B(x) − ((Bw0)θ0xm)�(x), and

where deg �m+σ+1 ≤ m + σ + 1, m ≥ 0. Thus,

(n + (u)0 + 1)〈�w0, xmB [1]

n (x; u)〉 = −〈w0, �m+σ+1Bn+1〉. (3.26)

Using the orthogonality of {Bn}n≥0, we obtain

〈�w0, xmB [1]

n (x; u)〉 = 0, 0 ≤ m ≤ n − σ − 1, n ≥ σ + 1. (3.27)

For m = n − σ, in (3.26) we get

(n + (u)0 + 1)〈�w0, xn−σB [1]

n (x; u)〉 = −ρn〈w0, B2n+1〉, n ≥ σ, (3.28)

where ρn = (n − σ)(�(σ+2)(0))/((σ + 2)!) − ((σ+1)(0))/((σ + 1)!) + (�w0)0(B(σ+2)(0))/

((σ + 2)!), n ≥ σ.

Dow

nloa

ded

by [

Mos

kow

Sta

te U

niv

Bib

liote

] at

10:

03 3

0 O

ctob

er 2

013

Integral Transforms and Special Functions 889

Note that there exists an integer r ≥ σ such that 〈�w0, xr−σB [1]

r (x; u)〉 �= 0. Other-wise, 〈�w0, x

n−σB [1]n (x; u)〉 = 0, n ≥ σ. Then, according to (3.28) we get ρn = 0, n ≥

σ. This requires that t ≤ σ + 1 = p and −(σ + 2)(σ+1)(0) + (�w0)0B(σ+2)(0) = 0. So,

〈�w0, xmB [1]

n (x; u)〉 = 0, 0 ≤ m ≤ n − σ, n ≥ σ. In this way, there exists (l, r) ∈ N2, with1 ≤ l ≤ σ and r ≥ σ − l, such that

〈�w0, xmB [1]

n (x; u)〉 = 0, 0 ≤ m ≤ n − σ + l − 1, n ≥ σ − l + 1, (3.29)

〈�w0, xr−σ+lB [1]

r (x; u)〉 �= 0. (3.30)

Otherwise, 〈�w0, xmB [1]

n (x; u)〉 = 0, 0 ≤ m ≤ n. In particular, for m = 0, 〈�w0, B[1]n (x; u)〉 =

0, n ≥ 0, i.e. �w0 = 0, where � �= 0. This contradicts the quasi-definiteness of the linearfunctional w0.

Hence, from (3.29) and (3.30), {B [1]n (x; u)}n≥0 will be quasi-orthogonal of order σ ′ = σ − l =

p − 1 − l with respect to v = �w0.

In addition, (p + (u)0)〈�w0, B[1]p−1(.; u)〉 = 〈Du(�w0), Bp〉 = −〈w0, Bp〉 �= 0 and 〈�w0,

B[1]n (x; u)〉 = 0, n ≥ σ − l + 1, due to (3.29), where m = 0. Since 〈�w0, B

[1]p−1(x; u)〉 �= 0, then

σ − l + 1 ≥ p. But σ = p − 1 and, as a consequence, p − l ≥ p, i.e. l ≤ 0. This contradicts thefact that l ≥ 1.

Finally, there exists an integer r ≥ σ such that 〈�w0, xmB [1]

n (x; u)〉 = 0, 0 ≤ m ≤ n − σ − 1and 〈�w0, x

r−σB [1]r (x; u)〉 �= 0. Hence, (i) holds. �

4. Another characterization of a Du-semiclassical MOPS

The following result allows us to characterize the Du-semiclassical monic orthogonal polynomialsequence by a first structure relation.

Theorem 4.1 Let u ∈ P′M and {Bn}n≥0 be an MOPS with respect to w0. The following statements

are equivalent.

(i) {Bn}n≥0 is Du-semiclassical.(ii) There exists (r, σ ) ∈ N

2, with 0 ≤ σ ≤ r, such that

�(x)B [1]n (x; u) =

n+t∑ν=n−σ

λn,νBν(x), n ≥ σ, (4.1)

λr,r−σ �= 0. (4.2)

Proof (i) ⇒ (ii). First, we always have �B [1]n (.; u) = ∑n+t

ν=0 λn,νBν, n ≥ 0, with t = deg �

and λn,ν = 〈w0, Bν�B [1]n (.; u)〉〈w0, B

2ν 〉−1, 0 ≤ ν ≤ n + t, n ≥ 0. But, {B [1]

n (x; u)}n≥0 is quasi-orthogonal of order σ = max(t − 2, p − 1) with respect to �w0, according to Definition 2.7and Theorem 3.2. Therefore, there exists an integer r ≥ σ such that 〈�w0, BνB

[1]n (x; u)〉 =

0, 0 ≤ ν ≤ n − σ − 1, n ≥ σ + 1 and 〈�w0, Br−σB [1]r (x; u)〉 �= 0. Thus, we have λn,ν = 0,

0 ≤ ν ≤ n − σ − 1, n ≥ σ + 1, and λr,r−σ �= 0. Hence, (4.1) and (4.2) hold, with λr,r−σ =〈w0, B

2r−σ 〉−1〈�w0, Br−σB [1]

r (x; u)〉 �= 0.

(ii) ⇒ (i). From the assumption and the orthogonality of {Bn}n≥0, we get 〈�w0, BmB [1]n

(.; u)〉 = ∑n+tν=n−σ λn,ν〈w0, BνBm〉 = ∑n+t

ν=n−σ λn,ν〈w0, B2ν 〉δm,ν, n ≥ σ. Then, 〈�w0, BmB [1]

n

(.; u)〉 = 0, 0 ≤ m ≤ n − σ − 1, n ≥ σ + 1 and 〈�w0, Br−σB [1]r (.; u)〉 = λr,r−σ 〈w0, B

2r−σ 〉 �=

0. Thus, {B[1]n (x; u)}n≥0 is quasi-orthogonal of order σ with respect to �w0. �

Dow

nloa

ded

by [

Mos

kow

Sta

te U

niv

Bib

liote

] at

10:

03 3

0 O

ctob

er 2

013

890 F. Marcellán and R. Sfaxi

The following result shows that any Du-semiclassical MOPS is a D-Laguerre–Hahn MOPS.

Theorem 4.2 Let u ∈ P′M and {Bn}n≥0 be an MOPS with respect to w0. The following statements

are equivalent.

(i) {Bn}n≥0 is a Du-semiclassical MOPS.(ii) There exists (�, , B) ∈ P

3, � monic, deg � = t, deg = p ≥ 1, deg B ≤ k + σ + 2,with k = min{m ≥ 0 : (�w0)m �= 0}, and σ = max(t − 2, p − 1), such that Bw0 ∈ P

′M, and

where w0 satisfies

(�w0)′ + 1w0 − (�B)(x−1w2

0) = 0, (4.3)

where

1(x) = (x) + �(x)(w0θ0B)(x) + B(x)(wθ0�)(x). (4.4)

Proof (i) ⇒ (ii). Assuming (i). By Theorem 3.2(ii), there exists (�, , B) ∈ P, � monic,deg � = t, deg = p ≥ 1, deg B ≤ k + σ + 2, with σ = max(t − 2, p − 1) and k = min{m ≥0 | (�w0)m �= 0}, such that Du(�w0) + w0 = 0, with u = Bw0. But, Du(�w0) = (�w0)

′ −x−1u(�w0), then

(�w0)′ − x−1((Bw0)(�w0)) + w0 = 0. (4.5)

Using (1.4) we can write

(Bw0)(�w0) = �Bw20 − xFw0, (4.6)

where

F(x) := �(x)(w0θ0B)(x) + B(x)(w0θ0�)(x). (4.7)

From (4.6), (1.3) and (1.6), we get

x−1((Bw0)(�w0)) = (�B)x−1w20 − Fw. (4.8)

Then, (4.5) becomes (�w0)′ + (F + )w0 − (�B)(x−1w2

0) = 0. Hence, (4.3) holds, with1(x) = F(x) + (x).

(ii) ⇒ (i). Assuming (ii), let F(x) = �(x)(w0θ0B)(x) + B(x)(w0θ0�)(x). Setting u =Bw0, (x) = 1(x) − F(x) and using (4.3), (4.4) and (4.8), we get Du(�w0) + w0 =(�B)(x−1w2

0) − x−1((Bw0)(�w0)) − Fw0 = 0. Hence, by Theorem 3.2(i) holds. �

Corollary 4.3 A D-Laguerre–Hahn linear functional w0 satisfying (�w0)′ + (1w0) +

B(x−1w20) = 0, with (�, 1, B) ∈ P

3 and � monic, is Du-semiclassical if and only if the followingconditions hold.

(i) �(x) divides B(x). We will write B(x) = −�(x)B(x).

(ii) deg B ≤ k + σ + 2, where k = min{m ≥ 0 | (�w)m �= 0}, σ = max(deg(�) − 2,

deg() − 1), and = 1 − �(w0θ0B) − B(w0θ0�).

(iii) (Bw0)0 �= −n, n ≥ 1.

In this case, u = Bw0.

As an example, let us consider the singular D-Laguerre–Hahn MOPS of class zero analogousto the Laguerre’s one which we denote by {ln(.; ξ)}n≥0 with ξ = (α, λ, ρ) (see [1,4]). We willdenote by �(ξ) the corresponding normalized linear functional. In Table 1, we summarize thecharacteristic elements of the MOPS {ln(.; ξ)}n≥0.

Dow

nloa

ded

by [

Mos

kow

Sta

te U

niv

Bib

liote

] at

10:

03 3

0 O

ctob

er 2

013

Integral Transforms and Special Functions 891

Table 1. The singular Laguerre–Hahn sequences of class zero analogousto Laguerre’s ones with parameter ξ = (α, λ, ρ).

�(x) = x, 1(x) = −x + α − 1B(x) = x2 + (2(1 − α) − λ)x + α(α + λ − 1) − ρ

β0 = α + λ − 1, βn+1 = 2n + α + 1, n ≥ 0γ1 = ρ, γn+1 = n(n + α), n ≥ 1C0(x) = x − α, Cn+1(x) = −x + 2n + α, n ≥ 0D0(x) = 0, Dn+1(x) = −1, n ≥ 0λ ∈ C, ρ �= 0, α �= −n, n ≥ 1

Table 2. The singular Laguerre–Hahn sequences of class zero analogousto the Laguerre’s ones with parameter ξ∗ = (α, λ, α(α + λ − 1)).

�(x) = x, (x) = x − (α + λ − 1), B(x) = −x + 2(α − 1) + λ

β0 = α + λ − 1, βn+1 = 2n + α + 1, n ≥ 0γ1 = α(α + λ − 1), γn+1 = n(n + α), n ≥ 1C0(x) = x − α, Cn+1(x) = −x + 2n + α, n ≥ 0D0(x) = 0, Dn+1(x) = −1, n ≥ 0λ �= 1 − α, α �= −n, n ≥ 0

Using Corollary 4.3, the condition (i) is equivalent to ρ = α(α + λ − 1). Then, B(x) =−x + 2(α − 1) + λ, (x) = x − (α − 1 + λ), and σ = 0. Besides, (xw0)0 = α + λ − 1 �= 0,

ρ = α(α + λ − 1) �= 0, then k = 0. Therefore, (ii) holds because deg B = 1 ≤ k + σ + 2 = 2.

Finally, since (B�(ξ))0 = α − 1, the condition (iii) is equivalent to α �= −n, n ≥ 0. Thus,the MOPS {ln(.; ξ ∗)}n≥0, where ξ ∗ = (α, λ, α(α + λ − 1)), with α �= −n, n ≥ 0, and α +λ − 1 �= 0 is DU(ξ∗)-semiclassical, with U(ξ ∗) = B�(ξ ∗) and B(x) = −x + 2(α − 1) + λ. Thelinear functional �(ξ ∗) satisfies: DU(ξ∗)(x�(ξ ∗)) + (x − α − λ + 1)�(ξ ∗) = 0. Also, �(ξ ∗) isa D-Laguerre–Hahn linear functional of class zero, since it satisfies: (x�(ξ ∗))′ + (−x + α −1)�(ξ ∗) − x(−x + 2(α − 1) + λ)(x−1(�(ξ ∗))2) = 0.

In summary, we have the following.Since the D-Laguerre–Hahn MOPS {ln(.; ξ ∗)}n≥0 satisfies an FSR, then from Table 2 and (1.9),

we get

xl′n+1(x; ξ ∗) + xB(x)l(1)n (x; ξ ∗) = (−x + n + α)ln+1(x; ξ ∗)

+ n(n + α)ln(x; ξ ∗), n ≥ 1. (4.9)

But, from the orthogonality of {ln(.; ξ ∗)}n≥0 with respect to �(ξ ∗), we get

(U(ξ ∗)θ0ln+1(.; ξ ∗))(x) = ln+1(x; ξ ∗) + B(x)(�(ξ ∗)θ0ln+1(.; ξ ∗))(x),

= ln+1(x; ξ ∗) + B(x)l(1)n (x; ξ ∗), n ≥ 0.

Then, (4.9) becomes xl[1]n (x; ξ ∗) = ln+1(x; ξ ∗) + n ln(x; ξ ∗), n ≥ 1, where {l[1]

n (.; ξ ∗)}n≥0 is theDU(ξ∗)-derivative sequence of {ln(.; ξ ∗)}n≥0, defined by

(n + α)l[1]n (x; ξ ∗) = l′n+1(x; ξ ∗) + (U(ξ ∗)θ0ln+1(.; ξ ∗))(x), n ≥ 0.

In addition, it is easy to see that xl[1]0 (x; ξ ∗) = l1(x; ξ ∗) + (α + λ − 1) l0(x; ξ ∗). Hence, the FSR

becomes

xl[1]n (x; ξ ∗) = ln+1(x; ξ ∗) + ϑnln(x; ξ ∗), n ≥ 0, (4.10)

where ϑn = n, n ≥ 1 and ϑ0 = α + λ − 1.

Dow

nloa

ded

by [

Mos

kow

Sta

te U

niv

Bib

liote

] at

10:

03 3

0 O

ctob

er 2

013

892 F. Marcellán and R. Sfaxi

By (4.10) and the orthogonality of {ln(.; ξ ∗)}n≥0 with respect to �(ξ ∗), we get

〈x�(ξ ∗), xml[1]n (x; ξ ∗)〉 = 0, 0 ≤ m ≤ n − 1, n ≥ 1,

〈x�(ξ ∗), xnl[1]n (x; ξ ∗)〉 = ϑn〈�(ξ ∗), l2

n(x; ξ ∗)〉 �= 0, n ≥ 0.

Equivalently, 〈(α + λ − 1)−1x �(ξ ∗), l[1]m (.; ξ ∗)l[1]

n (.; ξ ∗)〉 = r [1]n (ξ ∗)δn,m, n, m ≥ 0, with

r [1]n (ξ ∗) = (α + λ − 1)−1ϑn〈�(ξ ∗), l2

n(x; ξ ∗)〉 = n!�(α)−1�(n + α) �= 0, n ≥ 0.

Thus, {l[1]n (.; ξ ∗)}n≥0 is orthogonal with respect to �

[1]0 (ξ ∗) = (α + λ − 1)−1x�(ξ ∗). The MOPS

{l[1]n (.; ξ ∗)}n≥0 satisfies the following TTRR:

l[1]0 (x; ξ ∗) = 1, l

[1]1 (x; ξ ∗) = x − β

[1]0 ,

l[1]n+2(x; ξ ∗) = (x − β

[1]n+1)l

[1]n+1(x; ξ ∗) − γ

[1]n+1l

[1]n (x; ξ ∗), n ≥ 0, (4.11)

where γ[1]n+1 = r

[1]n+1(ξ

∗)(r [1]n (ξ ∗))−1 = (n + 1)(n + α) �= 0, n ≥ 0, and from (4.10) and the

orthogonality of {ln(.; ξ ∗)}n≥0 with respect to �(ξ ∗), we get

β [1]n = 〈�(ξ ∗), [ln+1(.; ξ ∗)]2〉 + ϑ2

n 〈�(ξ ∗), [ln(.; ξ ∗)]2〉ϑn〈�(ξ ∗), (ln(.; ξ ∗))2〉 = γn+1

ϑn

+ ϑn, n ≥ 0.

Thus, β[1]0 = 2α + λ − 1, β

[1]n+1 = 2n + α + 2, n ≥ 0.

Note that {l[1]n (.; ξ ∗)}n≥0 is a D-Laguerre–Hahn MOPS of class zero. More precisely, it is

a non-singular D-Laguerre–Hahn MOPS analogous of the Laguerre’s one [1,4]. Recall that thenon-singular D-Laguerre–Hahn MOPS analogous to the Laguerre’s polynomial sequence denotedby {Ln(.; �)}n≥0 with � = (λ, ρ, τ, α), satisfies the following TTRR:

L0(x; �) = 1, L1(x; �) = x − β0,

Ln+2(x; �) = (x − βn+1)Ln+1(x; �) − γn+1Ln(x; �), n ≥ 0,

where

β0 = 2τ + α + λ + 1, βn+1 = 2(n + τ + 1) + α + 1, n ≥ 0,

γ1 = ρ(τ + 1)(τ + α + 1), γn+1 = (n + τ + 1)(n + τ + α + 1), n ≥ 1,

with τ + α �= −(n + 1) and τ �= −(n + 1), n ≥ 0.

As a straightforward consequence, we get l[1]n (x; ξ ∗) = Ln(x; � ∗), n ≥ 0, where � ∗ = (α +

λ − 1, 1, 0, α − 1).

We conclude our work stating two finite-type relations linking {ln(.; ξ ∗)}n≥0 to {Ln(.; � ∗)}n≥0,

which are known in the literature as second structure relations (see [10,12]). By inserting (4.10) in(3.3), where Bn(x) = ln(x; ξ ∗), n ≥ 0, and using the fact that 〈U(ξ ∗), ln+1(x; ξ ∗)〉 = −δn,0, n ≥0, we obtain

ln+1(x; ξ ∗) + υnln(x; ξ ∗) = Ln+1(x; � ∗) + (n + 1)(2n + α + 1)

n + α + 1Ln(x; � ∗)

+ (n + α − 1)

n + α + 1Ln−1(x; � ∗) + (α + 1)−1δn,0, n ≥ 0,

where υn = n(n + α)(n + α + 1)−1.

Finally, substituting (4.11) into (3.3) where Bn(x) = ln(x; ξ ∗), n ≥ 0, and using the fact that〈U(ξ ∗), ln+1(x; ξ ∗)〉 = −δn,0, n ≥ 0, we obtain

ln(x; ξ ∗) = Ln(x; � ∗) + (n + α − 1)Ln−1(x; � ∗), n ≥ 0, L−1(x; � ∗) = 0.

Dow

nloa

ded

by [

Mos

kow

Sta

te U

niv

Bib

liote

] at

10:

03 3

0 O

ctob

er 2

013

Integral Transforms and Special Functions 893

Acknowledgements

The first author (F.M.) was supported by Dirección General de Investigación, Ministerio de Ciencia e Innovación of Spain,under grant MTM 2009-12740-C03-01. The work of the second author (R.S.) was supported by Faculté des Sciences deGabès and Faculté des Sciences de Sfax, Tunisia.

References

[1] J. Alaya and P. Maroni, Symmetric Laguerre–Hahn forms of class s = 1, Integral Transforms Spec. Funct. 4 (1996),pp. 301–320.

[2] W.A. Al-Salam, Characterization theorems for orthogonal polynomials, in Orthogonal Polynomials: Theory andPractice, NATO ASI Series Vol. 294, P. Nevai ed., Kluwer Academic Publishers, Dordrecht, 1990, pp. 1–24.

[3] P. Maroni, Le calcul des formes linéaires et les polynômes orthogonaux semi-classiques, in Orthogonal Polynomialsand Their Applications, Lecture Notes in Mathematics, Vol. 1329, M. Alfaro, J.S. Dehesa, F.J. Marcellan, J.L. Rubiode Francia, and J. Vinuesa, eds., Springer-Verlag, Berlin, 1988, pp. 279–290.

[4] A. Branquinho, F. Marcellán, and J.C. Petronilho, Classical orthogonal polynomials: A functional approach, ActaAppl. Math. 34 (1994), pp. 283–303.

[5] T.S. Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach, New York, 1978.[6] J. Dini, Sur les formes linéaires et les polynômes orthogonaux de Laguerre–Hahn. Thèse de Troisième Cycle

Université Pierre et Marie Curie, Paris, 1988 (in French).[7] J. Dzoumba, Sur les polynômes de Laguerre–Hahn. Thèse de Troisième Cycle, Université Pierre et Marie Curie,

Paris, 1985 (in French).[8] F. Marcellán and E. Prianes, Perturbations of Laguerre–Hahn linear functionals, J. Comput. Appl. Math. 105 (1999),

pp. 109–128.[9] F. Marcellán and R. Sfaxi, Second structure relation for semiclassical orthogonal polynomials, J. Comput. Appl.

Math. 200 (2007), pp. 537–554.[10] F. Marcellán and R. Sfaxi, A characterization of weakly regular linear functionals, Rev. Acad. Colombiana Cienc.

Exact. Fis. Natur. 31(119) (2007), pp. 285–295.[11] F. Marcellán and R. Sfaxi, Inverse finite-type relations between sequences of polynomials, Rev. Acad. Colombiana

Cienc. Exact. Fis. Natur. 32(123) (2008), pp. 245–255.[12] H. Bouakkaz and P. Maroni, Description des polynômes de Laguerre–Hahn de classe zéro, In Orthogonal Polynomials

and Their Applications, IMACS [Ann. Comput. Appl. Math. 9]. C. Brezinski, L. Gori, and A.A. Ronveaux, eds.,J.C. Baltzer AG, Basel, 1991, pp. 189–194.

[13] A. Ronveaux, Polynômes orthogonaux dont les polynômes dérivés sont quasi-orthogonaux, C.R. Acad. Sci. Paris.Sér. A 289 (1979), pp. 433–436.

Dow

nloa

ded

by [

Mos

kow

Sta

te U

niv

Bib

liote

] at

10:

03 3

0 O

ctob

er 2

013