General Letters in Mathematics Vol. 3, No. 2, Oct 2017, pp.81-90

e-ISSN 2519-9277, p-ISSN 2519-9269

Available online at http:// www.refaad.com

Operational Formulae of the Multivariable Hypergeometric

Matrix Functions and Related Matrix Polynomials

M. Abdalla ∗1, H. Abd-Elmageed 2 ,M. Abul-Ez 3 and Z. Kishka 4

1,2 Mathematics Department, Faculty of Science, South Valley University, Qena 83523, Egypt.3,4 Mathematics Department, Faculty of Science, Sohag University, Sohag 82524, Egypt.

1m.abdallah@sci.svu.edu.eg., 2halla mohamed2010@yahoo.com

Abstract. This paper presents an overview of the modern developments around the multivariable hypergeometric matrix

functions (MHMFs). We introduce operational representation of MHMFs. Also, certain classes of orthogonal matrix polyno-

mials are established.

Keywords: Operational formulae; Generalized hypergeometric matrix functions; Orthogonal matrix polynomials;Matrix functional calculus.

MSC2010 33D15, 33C05, 15A15, 34A05.

1 Introduction and Preliminaries

Hypergeometric functions in one and more variables occur naturally in a wide variety of problems [4]. Various gener-alizations of hypergeometric functions, in particular such involving multiple functions, appear in different branches ofmathematics and physics. Many of the special functions that arise in engineering, probability theory and Lie groupstheory also have representations as hypergeometric functions. In recent years, in one hand, various extensions of thehypergeometric matrix functions have been presented and investigated (see, e.g., [1, 2, 12, 14, 15, 17, 18, 20] and thereferences cited therein). On the other hand, we may mention (see, e.g., [4, 5, 6, 9, 10, 13, 16]) who have contributedto the study of the operational formulae of some special functions. Indeed, the use of operational formulae, currentlyexploited in the theory of algebraic decomposition of exponential operators, may significantly simplify the studyof hypergeometric matrix functions and the discovery of new relations, hardly achievable by conventional means.Taking account of operational formulae and owing to the techniques used in [13], we introduce in the present work,operator representations of multivariable hypergeometric matrix functions (MHMFs) and certain orthogonal matrixpolynomials. The structure of this paper is as follows. Section 2, we introduce the definitions and properties of thegeneralized hypergeometric matrix functions and matrix polynomials in terms of hypergeometric matrix function.Operator representations of multivariable hypergeometric matrix functions and their miscellaneous formulas are de-duced in section 3. The operational formulae of several orthogonal matrix polynomials are established in section 4.The results obtained are believed to be new.

Now, we will summarize basic concepts and notations that will be largely exploited in this work.

LetMd(C) denotes the complex vector space constituted of all square matrices with d rows and d columns withentries in complex space. For any matrix M ∈ Md(C), σ(M) (spectrum of M) denotes the set of all eigenvalues of

∗Corresponding author. M. Abdalla 1 m.abdallah@sci.svu.edu.eg.

82 M. Abdalla et al.

M,

µ(M) = maxRe(z) : z ∈ σ(M), µ(M) = minRe(z) : z ∈ σ(M), (1)

where µ(M) is referred to as the spectral abscissa (the largest of the real parts of its eigenvalues) and µ(−M) =−µ(M). The square matrix M is said to be positive stable if and only if µ(M) > 0. I and 0 stand for the identitymatrix and the zero matrix in Md(C), respectively.

If f1(z) and f2(z) are holomorphic functions of the complex variable z, which are defined in an open set Ω ofthe complex plane, and M is a matrix in Md(C) with σ(M) ⊂ Ω, then from the properties of the matrix functionalcalculus,(cf. [3, 7, 15]) we have

f1(M)f2(M) = f2(M)f1(M).

Hence, if N in Md(C) is a matrix for which σ(N) ⊂ Ω also, and if MN = NM , then

f1(M)f2(N) = f2(N)f1(M).

For M ∈Md(C) the matrix version of the Pochhammer symbol (the shifted factorial) is (cf. [3, 7, 15])

(M)n = M(M + I)...(M + (n− 1)I), n ≥ 1; (M)0 ≡ I.

The reciprocal scalar Gamma function denoted by Γ−1(z) = 1Γ(z) is an entire function of the complex variable

z. Thus, for any M ∈ Md(C), Riesz-Dunford functional calculus [7, 15] shows that Γ−1(M) is well defined and is,indeed, the inverse of Γ(M). Furthermore, if

M + nI is invertible for all integer n ≥ 0, (2)

then

(M)n = Γ(M + nI)Γ−1(M). (3)

From the Notation 2.2 of [3, pp. 1007], we have

(−1)k

(n− k)!I =

(−n)kn!

I =(−nI)kn!

; 0 ≤ k ≤ n.

In 1731, Euler defined the derivative formula

Dνx x

α =Γ(α+ ν)

Γ(α− ν + 1)xα−ν , Dx ≡

d

dx,

where α and ν are arbitrary complex numbers. By application of the matrix functional calculus to this definition,for any matrix M ∈Md(C), one gets (see [3, 7])

Dnt [tM+mI ] = (M + I)m[(M + I)m−n]−1 tM+(m−n)I , n = 0, 1, 2, 3, ...,

where Dt = ddt denotes the differentiation operator. Again, we observe that

tM (−Dt)n t−M = (M)n t

−n (4)

and

tI−M (D−1t )n tM−I = [(M)n]−1tn, (5)

where D−1t = 1

Dtis the inverse operator (primitive operator) of Dt.

On other hand, if M ∈Md(C), and z is any complex number, then the matrix exponential eMz is defined to be

eMz = I +Mz + ...+Mn

n!zn + ...,

dn

dzn[eMz] = Mn eMz = eMz Mn, n = 0, 1, 2, 3, ....

Operational Formulae of the Multivariable Hypergeometric Matrix... 83

2 Definitions and Miscellaneous Formulas

In this section, we give the definitions of the following generalized hypergeometric matrix function and certain matrixpolynomials in terms of hypergeometric matrix function. (see [17, 19]). Let p and q be finite positive integers, thenthe generalized hypergeometric matrix function has been given by the matrix power series as the following

F (Ai, Bj ; z) =∑n≥0

p∏i=1

(Ai)n

q∏j=1

[(Bj)n]−1 zn

n!. (6)

For commutative matrices Ai, 1 ≤ i ≤ p and Bj , 1 ≤ j ≤ q in Md(C) such that

Bj + nI are invertible for all integers n ≥ 0. (7)

The following results may be proved using the standard techniques of convergence theory:

• If p ≤ q, then the power series (2.1) is convergence for all finite z.

• If p = q + 1, then the power series (2.1) is convergent for |z| < 1 and diverges for |z| > 1.

• If p > q + 1, then the power series (2.1) diverges for z, z 6= 0.

• If p = q + 1, then the power series (2.1) is absolutely convergent on the circle |z| = 1 when

p∑i=0

µ(Ai) <

q∑j=0

µ(Bj),

where µ(Ai) and µ(Bj) as defined in (1.1).

With p = 1 and q = 0 in (2.1), one gets the following relation due to [15]

(1− z)−A =

∞∑n=0

(A)nzn

n!, |z| < 1, (8)

while at p = 1 and q = 1 in (2.1), we have

ez =

∞∑n=0

zn

n!. (9)

Also, with three matrix parameter; p = 2 and q = 1 in (2.1) we have the Gauss matrix function in the form(cf.[15])

2F1(A1, A2;B1; z) =

∞∑n=0

(A1)n(A2)n[(B1)n]−1 zn

n!, (10)

where A1, A2 and B1 are matrices inMd(C), and B1 satisfying the condition (1.2). (see [7, 21]). For an arbitrarymatrix A ∈Md(C), with A+nI invertible for every integer n ≥ 1, then the n-th Laguerre matrix polynomials LAn (z)is defined by

LAn (z) =(A+ I)n

n!1F1(−nI;A+ I; z). (11)

therefore, we can be reduced to Shively’s pseudo Laguerre matrix polynomial in the form

RAn (z) =(A)2n [(A)n]−1

n!1F1(−nI;A+ I; z). (12)

(see [11]). Let A and B be parameter commuting matrices inMd(C) satisfying the spectral condition (1.2). For

any natural number n ≥ 0, the n-th generalized Bessel matrix polynomial Y(A,B)n (z) is defined by

Y(A,B)n (z) = 2F0(−nI,A+ (n− 1)I;−; z B−1). (13)

84 M. Abdalla et al.

Also, the n-th reverse generalized Bessel matrix polynomial Θ(A,B)n (z) is defined in [3] by

Θ(A,B)n (z) = (−1)nΓ−1(−A− (2n− 2)I)Γ(−A+ (n− 2)I)

× 1F1(−nI;−A− (2n− 2)I;Bz).(14)

(see [8, 24]). Suppose that A and B are positive stable matrices in Md(C) satisfying the condition

Re(a) > −1, ∀ a ∈ σ(A) and Re(a) > −1, ∀ b ∈ σ(B). (15)

For any positive integer n, the Jacobi matrix polynomials PA,Bn (z) are defined by

PA,Bn (z) =(A+ I)n

n!2F1(−nI,A+B + (n+ 1)I;A+ I;

1− z2

). (16)

The special case of A = B of the Jacobi matrix polynomial is called ultraspherical matrix polynomial and is denotedby PA,An (z). It is thus defined as

PA,An (z) =(A+ I)n

n!2F1(−nI, 2A+ (n+ 1)I;A+ I;

1− z2

). (17)

(see [22]). The Gegenbauer matrix polynomials CAn (z) is defined in the form

CAn (z) =(2A)nn!

2F1(−nI, 2A+ nI;A+1

2I;

1− z2

), (18)

where A is a positive stable matrix in Md(C). With d = 1 and A = 12 , we define the class of orthogonal Legendre

polynomials in the form

Pn(z) = 2F1(−n, (n+ 1); ;1− z

2). (19)

(see [19]). The generalized Bateman’s matrix polynomials ZA,Bn (z) is defined in the form

ZA,Bn (z) = 2F2(−nI, 2A+ nI;A+1

2I,B + I; z), (20)

where A and B are commutative matrices in Md(C) and A + nI and B + nI are invertible for all integer n ≥ 0.(see [23]). The Rices matrix polynomials HA,B

n (z) is defined by means of the relation

HA,Bn (z) = 3F2(−nI, (n+ 1)I, A; I,B; z), (21)

for commutative matrices A and B in Md(C) and B satisfying the condition (1.2).

3 Operational Formulae of The MHMFs

In this section, our main theorem is stated as follows: Suppose that M, A1, A2, ..., Ap and B1, B2, ..., Bq areparameter commuting matrices in CN×N while B1, B2, ..., Bq satisfying the spectral condition (2.2) and let Dzµ = ∂

∂zµ

and Dwµ = ∂∂wµ

; µ = 1, 2, 3, ..., be partial differential operators; z1, z2, ..., zq and w1, w2, ..., wp ∈ C then, we have

the following operational representations

p+1Fq

(M,A1, A2, ..., Ap;B1, B2, ..., Bq;

z1, z2, ..., zqw1, w2, ..., wp

)

=

p∏r=1

wArr

q∏s=1

z−Bs+Is

(1−

(−1)p∏pr=1 Dwr∏q

s=1 Dzs

)−M p∏r=1

w−Arr

q∏s=1

zBs−Is

(22)

Operational Formulae of the Multivariable Hypergeometric Matrix... 85

and

pFq

(A1, A2, ..., Ap;B1, B2, ..., Bq;

z1, z2, ..., zqw1, w2, ..., wp

)

=

p∏r=1

wArr

q∏s=1

z−Bs+Is exp

(−1)p

∏pr=1 Dwr∏q

s=1 Dzs

p∏r=1

w−Arr

q∏s=1

zBs−Is .

(23)

Proof. Applying the result (2.3) in the right hand side of (3.1), it follows that

p∏r=1

wArr

q∏s=1

z−Bs+Is

(1−

(−1)p∏pr=1 Dwr∏q

s=1 Dzs

)−M p∏r=1

w−Arr

q∏s=1

zBs−Is

=

p∏r=1

wArr

q∏s=1

z−Bs+Is

∞∑k=0

(M)k(−1)kp∏pr=1 (Dwr )

k

k!∏qs=1 (Dzs)

k

p∏r=1

w−Arr

q∏s=1

zBs−Is

=

∞∑k=0

(M)kk!

p∏r=1

wArr (−Dwr )

k w−Arr

q∏s=1

z−Bs+Is (

1

Dzs

)k zBs−Is

.

(24)

According to the relations (1.4) and (1.5), we obtain

p∏r=1

wArr

q∏s=1

z−Bs+Is

(1−

(−1)p∏pr=1 Dwr∏q

s=1 Dzs

)−M p∏r=1

w−Arr

q∏s=1

zBs−Is

=

∞∑k=0

(M)kk!

p∏r=1

w−kr (Ar)k

q∏s=1

zks [(Bs)k]−1

.

(25)

Hence, from (2.1), we get the asserted result according to (3.1).Taking the right hand side of (3.2) with the definition (2.4), we have

p∏r=1

wArr

q∏s=1

z−Bs+Is exp

(−1)p

∏pr=1 Dwr∏q

s=1 Dzs

p∏r=1

w−Arr

q∏s=1

zBs−Is

=

p∏r=1

wArr

q∏s=1

z−Bs+Is

∞∑k=0

(−1)kp∏pr=1 (Dwr )

k

k!∏qs=1 (Dzs)

k

p∏r=1

w−Arr

q∏s=1

zBs−Is

=

∞∑k=0

1

k!

p∏r=1

wArr (−Dwr )

k w−Arr

q∏s=1

z−Bs+Is (

1

Dzs

)k zBs−Is

.

(26)

According to the relations (1.4) and (1.5), we obtain

p∏r=1

wArr

q∏s=1

z−Bs+Is exp

(−1)p

∏pr=1 Dwr∏q

s=1 Dzs

p∏r=1

w−Arr

q∏s=1

zBs−Is

=

∞∑k=0

1

k!

p∏r=1

w−kr (Ar)k

q∏s=1

zks [(Bs)k]−1

.

(27)

Again, from (2.1), we have proved the relation (3.2) in our main theorem.

Putting different values of p and q in (3.1) and (3.2), we obtain operational representations of MHMFs.If we take p = 1 and q = 0 in (3.2), we see

1F0(A;−;1

z) = zA exp(Dz) z

−A. (28)

86 M. Abdalla et al.

Similarly, we present the following results:

1F1(A;B;z

w) = wA z−B+I exp(

−Dw

Dz) w−A zB−I . (29)

1F1(A;B; z) = z−B+I (1 +1

Dz)−A zB−I . (30)

2F0(A,B;1

z) = zA (1 +Dz)

−B z−A. (31)

2F0(A,B;1

zw) = zAwB exp(Dz Dw) z−Aw−B . (32)

2F1(A1, A2;B1;z

w) = z−B1+IwA2 (1 +

Dw

Dz)−A1 zB1−Iw−A2 . (33)

2F1(A1, A2;B1;z

w) = z−B1+IwA1 (1 +

Dw

Dz)−A2 zB1−Iw−A1 . (34)

2F1(A1, A2;B1;z

w1w2) = z−B1+I wA1

1 wA22 exp

(Dw1Dw2

Dz

)zB1−I w−A1

1 w−A22 . (35)

2F2(A1, A2;B1, B2;z1z2

w) = z−B1+I

1 z−B2+I2 wA2

(1 +

Dw

Dz1Dz2

)−A1

zB1−I1 zB2−I

2 w−A2 . (36)

2F2(A1, A2;B1, B2;z1z2

w) = z−B1+I

1 z−B2+I2 wA1

(1 +

Dw

Dz1Dz2

)−A2

zB1−I1 zB2−I

2 w−A1 . (37)

2F2(A1, A2;B1, B2;z1z2

w) = z−B2+I

1 z−B1+I2 wA2

(1 +

Dw

Dz1Dz2

)−A1

zB2−I1 zB1−I

2 w−A2 . (38)

2F2(A1, A2;B1, B2;z1z2

w1w2) =z−B2+I

1 z−B1+I2 wA1

1 wA22 exp

(Dw1Dw2

Dz1Dz2

)× zB2−I

1 zB1−I2 w−A1

1 w−A22 .

(39)

2F2(A1, A2;B1, B2;z1z2

w1w2) =z−B1+I

1 z−B2+I2 wA1

1 wA22 exp

(Dw1Dw2

Dz1Dz2

)× zB1−I

1 zB2−I2 w−A1

1 w−A22 .

(40)

3F0(A1, A2, A3;−;1

w1w2) = wA2

2 wA31

(1−Dw1

Dw2

)−A1

w−A22 w−A3

1 . (41)

3F0(A1, A2, A3;−;1

w1w2) = wA1

2 wA31

(1−Dw1 Dw2

)−A2

w−A12 w−A3

1 . (42)

3F0(A1, A2, A3;−;1

w1w2) = wA1

2 wA21

(1−Dw1

Dw2

)−A3

w−A12 w−A2

1 . (43)

Operational Formulae of the Multivariable Hypergeometric Matrix... 87

3F0(A1, A2, A3;−;1

w1w2w3) = wA1

2 wA21 wA3

3 ; exp(Dw1 Dw2 Dw3

)w−A1

2 w−A21 w−A3

3 . (44)

3F1(A1, A2, A3;B1;z

w1w2) = z−B1+I wA1

1 wA22

(1− Dw1

Dw2

Dz

)−A3

zB1−I w−A11 w−A2

2 . (45)

3F1(A1, A2, A3;B1;z

w1w2) = z−B1+I wA1

1 wA32

(1− Dw1

Dw2

Dz

)−A2

zB1−I w−A11 w−A3

2 . (46)

3F1(A1, A2, A3;B1;z

w1w2) = z−B1+I wA2

1 wA32

(1− Dw1 Dw2

Dz

)−A1

zB1−I w−A21 w−A3

2 . (47)

3F1(A1, A2, A3;B1;z

w1w2w3) = z−B1+I wA2

1 wA32 wA1

3

exp(−Dw1

Dw2Dw3

Dz

)zB1−I w−A2

1 w−A32 w−A1

3

(48)

4 Some Basic Applications

In the present section, we give the operational representations of many orthogonal matrix polynomials as follows

LAn (z) =z−A

n!(Dz − 1)n zA+nI , (49)

LAn (z) =(A+ I)n

n!z−A (1− 1

Dz)n zA, (50)

LAn (z

w) =

(A+ I)nn!

z−A w−n exp(−Dw

Dz) zA wn, (51)

RAn (z) =(A)2n [(A)n]−1

n!z−A+(1−n)I (1 +

1

Dz)n zA+(n−1)I . (52)

RAn (z

w) =

(A)2n [(A)n]−1

n!z−A+(1−n)I w−n exp(

−Dw

Dz) zA+(n−1)I wn. (53)

Y(A,B)n (

1

z) = z−n (1−DzB

−1)−(A+(n−1)I) zn. (54)

Y(A,B)n (

1

z) = zA+(n−1)I (1−DzB

−1)−n z−(A+(n−1)I). (55)

PA,Bn (1− 2z

w) =

z−A w−n

n!Dnz (1 +

Dw

Dz)−(A+B+(n+1)I) zA+nI wn. (56)

88 M. Abdalla et al.

PA,Bn (1− 2z

w) =

(A+ I)nn!

z−A wA+B+(n+1)I (1 +Dw

Dz)n zA w−(A+B+(n+1)I). (57)

PA,Bn (1− 2z

w) =

(A+ I)nn!

z−A w−n (1 +Dw

Dz)−(A+B+(n+1)I) zA wn. (58)

PA,An (1− 2z

w) =

z−A w−n

n!Dnz (1 +

Dw

Dz)−(2A+(n+1)I) zA+nI wn. (59)

PA,An (1− 2z

w) =

(A+ I)nn!

z−A w2A+(n+1)I (1 +Dw

Dz)n zA w−(2A+(n+1)I). (60)

PA,An (1− 2z

w) =

(A+ I)nn!

z−A w−n (1 +Dw

Dz)−(2A+(n+1)I) zA wn. (61)

CAn (1− 2z

w) =

(2A)nn!

z−A+ 12 I w−n (1 +

Dw

Dz)−(2A+nI) zA−

12 I wn. (62)

CAn (1− 2z

w) =

(2A)nn!

z−A+ 12 I w2A+nI (1 +

Dw

Dz)n zA−

12 I w−(2A+nI). (63)

ZA,Bn (z1z2

w1) = w2A+nI

1 z−A+ 1

2 I1 z−B2 (1 +

Dw1

Dz1Dz2

)n

w−2A−nI1 z

A− 12 I

1 zB2 .

(64)

ZA,Bn (z1z2

w1) = w−n1 z−B1 z

−A+ 12 I

2 (1 +Dw1

Dz1Dz2

)−(2A+nI)

wn1 zB1 z

A− 12 I

2 .

(65)

ZA,Bn (z1z2

w1w2) = w

(2A+nI)2 w−n1 z−B1 z

−A+ 12

2 exp(Dw1

Dw1

Dz1Dz2

)× w

−(2A+nI)2 wn1 z

B1 z

A− 12 I

2 .

(66)

HA,Bn (

w

z) =

zA

n!Dnw

(w−B+(n+1)I (1 +

Dz

Dw)n z−A w−(B+I)

)(67)

HA,Bn (C,D;

z1z2

w1w2) =z−C1 z−B+I

2 w−n1 wA2

(1− Dw1

Dw1

Dz1Dz2

)−(C+D+(n+1)I)

× zC1 zB−I2 wn1 w−A2

(68)

Operational Formulae of the Multivariable Hypergeometric Matrix... 89

5 Outlook and Concluding Remarks

The paper proposes to introduce operational formulae of some interesting multivariable hypergeometric matrix func-tions (MHMFs). The induced MHMFs which have been investigated are Laguerre matrix polynomial, generalizedBessel matrix polynomial, Jacobi matrix polynomials, Gegenbauer matrix polynomial, generalized Bateman’s matrixpolynomial and Rices matrix polynomial. Further research on this topic is now under investigation and will bereported in forthcoming papers concerning: Lagrange matrix polynomial, Sister Celines matrix polynomial, Bedientsmatrix polynomial etc.

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