Communications on Applied Nonlinear Analysis
ISSN: 1074-133X
Vol 32 No. 9s(2025)

Generalized q-Mittag-Leffler Matrix Function and its

Properties

Cynthia V. Rodrigues1,2 and Bharti V. Nathwani2,*

1Mukesh Patel School of Technology Management & Engineering,
SVKM’s NMIMS, Mumbai - 400056, Maharashtra, India

2Amity School of Applied Sciences,Amity University Maharashtra,
Mumbai, Panvel - 410206, Maharashtra, India

*Corresponding Author

Article History:
Received: 13-11-2024
Revised: 25-12-2024
Accepted: 09-01-2025

Abstract: Motivated essentially by the success of the appli-
cations of the Mittag-Leffler functions and Matrix theory in
Science and Engineering, we propose here a unification of cer-
tain q-extensions of generalizations of Mittag-Leffler function
together with Saxena-Nishimoto’s function, Bessel-Maitland
function, Dotsenko function, Elliptic Function, etc. We obtain
Mellin-Barnes contour integral representation, q-difference
equation and eigen function property.

Keywords: Mittag-Leffler Matrix function, q-Mittag-Leffler
function, q-difference equation, Eigen function

1. Preliminaries

Let the spectrum of a matrix in Cr×r, denoted by σ (A), be the set of all eigenvalues of
A. Recall that a matrix A ∈ Cr×r is said to be positive stable when

β (A) = min {A (z) /z ∈ σ (A)} > 0 (1.1)

For a positive stable matrix A ∈ Cr×r, the q-gamma matrix function is defined by [10,
Eq.(3.13)]

Γq (A) =

1
1−q∫
0

tA−IE−qt
q dt (1.2)

and the reciprocal q-gamma matrix function is defined as [10, Eq.(3.20)]

Γ−1
q (A) = [A]q [A+ I]q . . . . . . [A+ (n− 1) I]q Γq

−1 (A+ nI) , n ≥ 1 (1.3)

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If A and B are positive stable matrices in Cr×r then the q-beta matrix function is defined
as [10, Eq.(4.5)]

Bq (A,B) =

1∫
0

(tq; q)∞
(
tqB; q

)−1

∞ tA−I dqt (1.4)

Furthermore, if A,B and A+B are positive stable matrices in Cr×rsuch that AB = BA
then the beta matrix function is defined as [10, Eq.(4.6)]

Bq (A,B) = Γq (A) Γq (B) Γ−1
q (A+B) (1.5)

2. Introduction

A generalized structure of the Mittag-Leffler matrix function is given by [11, Eq.(2.9)]:

EA,B,C
αI,δI,µI (λz; s, r) =

∞∑
n=0

[(A)δn]
s Γ−1 (αnI +B)

[
(C)µn

]−r (λz)n

n!
(2.1)

where A,B,C are positive stable matrices in Cp×p, α, λ, z ∈ C with ℜ(α) > 0,
δ, µ > 0, r ∈ {−1, 0} ∪ N and s ∈ {0} ∪ N.

Interestingly, the proposed function ((2.13) below) also enables us to define and include
the q-analogues of

(i) Bessel-Maitland matrix function [11] :

JµI
νI (z) =

∞∑
n=0

(−1)n Γ−1(νI + nµI + I)
zn

n!
,

(ii) Dotsenko matrix function [11] :

2R1(aI, bI; cI, ωI; ν; z)

=
∞∑
n=0

Γ(aI + nI)Γ−1(aI) Γ(bI + n
ω

ν
I)Γ−1(bI) Γ(cI) Γ−1(cI + n

ω

ν
I)
zn

n!
,

(iii) Saxena and Nishimoto’s matrix function [11]:

EγI,K [(αjI, Bj)1,2; z]

=
∞∑
n=0

(γI)KnΓ
−1(α1nI +B1)Γ

−1(α2nI +B2)
zn

n!
,

where z, γ, αj, βj ∈ C,ℜ(α1 + α2) > ℜ(K)− 1,ℜ(K) > 0, and

(iv) the Elliptic matrix function [11] :

K(I; k) =
π

2
2F1

(
1
2
I, 1

2
I; k2

I;

)
.

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The following definitions and formulas will be used in this work.

For A ∈ Cr×r and q ̸= 1, qA = eA log q then the q-shifted factorial matrix function is
defined by [10]

(A; q)n =

{
I if n = 0
(I − A)(I − Aq)...(I − Aqn−1) if n ∈ N. (2.2)

For any n,
(A; q)n = (q; q)∞(Aqn; q)−1

∞ ,

where

(A; q)∞ =
∞∏
k=0

(I − Aqk) , |q| < 1.

(A; q)−1
∞ =

∞∏
k=0

(I − Aqk)−1 , ||A|| < 1, |q| < 1.

A q-binomial coefficient is (cf. [2, Ex.(1.2), p.20] with r = 1):[
n
m

]
r

=
(qr; qr)n

(qr; qr)n−m (qr; qr)m
, r ̸= 0. (2.3)

A q-gamma function is defined as [10, Eq.(3.18)]:

Γq(A) = (q; q)∞ (qA; q)
−1

∞ ; (1− q)I−A, (2.4)

q−n /∈ σ
(
qA
)
, n = 0, 1, 2, · · ·

For A ∈ Cr×r the q-analogue of Legendre’s duplication formula is of the form [10,
Eq.(3.23)]

Γq (2A) Γq2

(
1

2

)
= Γq2 (A) Γq2

(
A+

1

2
I

)
(1 + q)2A−I

(2.5)

where q−n /∈ σ
(
q2A
)
, q−2n /∈ σ

(
q2A
)
, q−2n /∈ σ

(
qA+I/2

)
, n = 1, 2, · · ·

In view of [3, Eq.(6)], we can propose the the following theorem for matrix function.

Theorem 2.1. If f(z) =
∞∑
n=0

Vnz
n is an entire function then the order ϱ(f) of f is given

by

ϱ(f) = lim
n→∞

supn log n log(∥Vn∥)−1. (2.6)

and the type of the function σ is given by [3]

eϱσ = lim
n→∞

sup
(
n ∥Vn∥ϱ/n

)
. (2.7)

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For every positive ϵ, the asymptotic estimate [3, Eq.(8)]

|f(z)| < exp ((σ + ϵ) |z|ϱ) , |z| ≥ r0 > 0 (2.8)

holds with ϱ, σ as in (2.6), (2.7) for |z| ≥ r0(ϵ), r0(ϵ) sufficiently large.

For any complex square matrix A and |q| < 1 the q-analogues of the exponential
matrix functions are defined as [10, Eq.(3.11)]

eq(A) =
∞∑
n=0

An

(q; q)n
= ((1− q)A; q)−1

∞ , (2.9)

and [10, Eq.(3.10)]

Eq(A) =
∞∑
n=0

qn(n−1)/2 An

(q; q)n
= (−(1− q)A; q)∞, (2.10)

Let A ∈ Cr×r be a positive stable matrix then the q-beta matrix function is defined as
[10, Eq.(4.5)]

Bq(A,B) =

∫ 1

0

(tq; q)∞ (tqB; q)−1
∞ tA−Idqt, (2.11)

The q-derivative of a function f(x) is defined by [2, Ex.1.12, p.22]

Dqf(A) = [f(A)− f(Aq)][A(1− q)]−1. (2.12)

In view of two q-analogues of generalized Mittag-Leffler function [3, 4], we define q-
generalized Mittag-Leffler matrix functions in the form:

Definition 1. Let A,B,C be positive stable matrices in Cr×r , α, δ, µ ∈ C with ℜ(α) >
0, δ, µ > 0, r ∈ {−1, 0} ∪ N, s ∈ N ∪ {0} then

EA,B,C
αI, δI, µI(λz; s, r|q) =

∞∑
n=0

(−1)pnI qpn(n−1)I/2 [Γq(δnI + A)]s

× [Γq(αnI +B)]−1 [Γq(µnI + C)]−r (λz)n

(q; q)n
, (2.13)

where p = α2 + rµ2 − sδ2 + 1 with ℜ(p) > 0.

Definition 2. Let A,B,C be positive stable matrices in Cr×r , α, δ, µ ∈ C with ℜ(α) >
0, δ, µ > 0, r ∈ {−1, 0} ∪ N, s ∈ N ∪ {0} and (α2 + rµ2 + 1)I = sδ2I then

eA,B,C
αI, δI, µI(λz; s, r|q) =

∞∑
n=0

[Γq(δnI + A)]s [Γq(αnI +B)]−1

×[Γq(µnI + C)]−r (λz)n

(q; q)n
. (2.14)

Alternatively in view of the definition of q-Gamma function (2.4) these q-forms can
also be put in the form:

EA,B,C
αI, δI, µI(λz; s, r|q) =

∞∑
n=0

(−1)pnI qpn(n−1)I/2 (qαnI+B; q)∞

×[(qµnI+C ; q)∞]r [(qδnI+A; q)∞]−s (λz)n

(q; q)n
, (2.15)

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and

eA,B,C
αI, δI, µI(λz; s, r|q) =

∞∑
n=0

[
(qαnI+B; q)∞

]
[(qµnI+C ; q)∞]r

×[(qδnI+A; q)∞]−s (λz)n

(q; q)n
. (2.16)

We shall refer to these functions as q-gmlm.
The objective of constructing this function is to
(i) include certain existing generalizations of Mittag-Leffler matrix function [3, 11, 4, 7,
8, 9, 5, 6],
(ii) also include the matrix functions such as Bessel Maitland function, Dotsenko function,
Bessel function, generalized Bessel Maitland function, Lommel function etc. especially
by means of parameters r, γ, λ (Table-1 below)
(iii) obtain inverse inequality relations and some other inequalities by means of the inte-
ger ′s′.
The q-analogues of the above stated Shukla and Prajapati’s function (2.1) and those
functions listed above from (i) through (iv) are all yielded by the q-gmlm (2.13) or (2.14).

They are tabulated below together with the indicated substitutions.

Table-1

q-Function of r s α B A δ C µ Particular
case of

Mittag-Leffler 0 1 α 1 1 1 - - (2.13)

Wiman 0 1 α B 1 1 - - (2.13)

Prabhakar 0 1 α B γI 1 - - (2.13)

Shukla and 0 1 α B γI q - - (2.13)
Prajapati

Bessel-Maitland 0 0 µ (ν + 1)I - - - - (2.13)

Dotsenko -1 1 ω/ν cI a I bI ω/ν (2.14)

Saxena- 1 1 α1 B1 γI K B2 α2 (2.13)
Nishimoto

Elliptic -1 1 1 I 1
2
I 1 1

2
I 1 (2.14)

The explicit forms of the functions mentioned in this table are as stated below.

• q-Mittag-Leffler function:

EαI(λz|q) =
∞∑
n=0

[
(−1)n qn(n−1)/2

]α2I

(qαnI+I ; q)∞ (λz)n.

• q-Analogue of Wiman’s function:

EB
αI(λz|q) =

∞∑
n=0

[
(−1)n qn(n−1)/2

]α2I

(qαnI+B; q)∞ (λz)n.

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• q-Analogue of Prabhakar’s generalized ML-function:

EA,B
αI (λz|q) =

∞∑
n=0

[
(−1)n qn(n−1)/2

]α2I

(qαnI+B; q)∞

×
[
(qA+nI ; q)∞

]−1 (λz)n

(q; q)n
.

• q-ML-function of Shukla and Prajapati (q is replaced by δ):

EA,B
αI,δI(λz|q) =

∞∑
n=0

[
(−1)n qn(n−1)/2

](α2− δ2+1)I

(qαnI+B; q)∞.

×(qA+δnI ; q)∞
−1 (λz)n

(q; q)n

• q-Bessel-Maitland function:

Jµ
νII(−z; q) =

∞∑
n=0

[
(−1)n qn(n−1)/2

](µ2+1)I

(qµnI+νI+I ; q)∞

(q; q)n
zn.

• q-Dotsenko function:

2R1(aI, bI; cI, ω; ν; z; q) =
∞∑
n=0

(qcI+
ω
ν
nI ; q)∞

[
(qbI+

ω
ν
nI ; q)∞

]−1

×
[
(qnI+aI ; q)∞

]−1 zn

(q; q)n
.

• q-Form of the particular case m = 2 of the function due to Saxena and Nishimoto

EγI,K [(αjI, Bj)1,2; z|q] =
∞∑
n=0

[
(−1)n qn(n−1)/2

](α2
1+α2

2−K2+1)I

×(qα1nI+B1 ; q)∞ (qα2nI+B2 ; q)∞

×
[
(qγI+Kn; q)∞

]−1 zn

(q; q)n
.

• q-Elliptic function:

K(
√
z|q) = π

2
2ϕ1

(
1
2
I, 1

2
I; z

I;

)
.

We first show the convergence of series in (2.13) and (2.14); this is followed by Mellin-
Barnes integral representation, q-difference equation and eigen function property.

3. Main Results

In this section, we prove the following results.

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3.1 Convergence

Theorem 3.1. Let A,B,C be positive stable matrices in Cp×p , α, δ, µ ∈ C with ℜ(α) >
0, ℜ(α2) + rµ2 − sδ2 + 1 > 0, δ, µ > 0, r ∈ {−1, 0} ∪ N, s ∈ N ∪ {0} and 0 < q < 1.
Then EA,B,C

αI, δI, µI(λz; s, r|q) is an entire function of order zero.

Proof. Put

Vn = (−1)pnI qpn(n−1)I/2 [Γq(δnI + A)]s Γq
−1(αnI +B)[Γq

−1(µnI + C)]r

× 1

(q; q)n
(3.1)

to get

EA,B,C
αI, δI, µI(λz; s, r|q) =

∞∑
n=0

Vn (λz)n.

Then in view of (2.4) and applying norm, we get after some simplification,

n
√

∥Vn∥ ∼

∥∥∥∥∥(−1)pI qp(n−1)I/2

×(1− q)s(I−A)−r(I−C)−(I−B)/n (1− q)(−sδ+rµ+α)I

×

[
∞∏
h=0

(
I − qδnI+A+hI

)]s [ ∞∏
m=0

(
I − qαnI+B+mI

)]

×

[
∞∏
j=0

(
I − qµnI+C+jI

)]−r
1

(q; q)n

∥∥∥∥∥
Now by applying limn → ∞, we get

1

R
= lim

n→∞
n
√
∥Vn∥ ∼ 0

when |q| < 1, ℜ(α2) + rµ2 − sδ2 + 1 > 0. Thus, the function (2.13) is an entire function.
Its order may be determined by using Theorem 2.1.
In fact, by choosing f(z) = EA,B,C

αI, δI, µI(λz; s, r|q) and un = Vn, Theorem 2.1 gets particu-
larized to

ϱ(EA,B,C
αI, δI, µI(λz; s, r|q)) = lim

n→∞
sup

n log n

log(∥Vn∥−1)
,

where

log
(
∥Vn∥−1

)
= log

(∥∥∥∥∥Γq(αnI +B) [Γq(µnI + C)]r Γq(n+ 1)

×q−n(n−1)(α2+rµ2−sδ2+1)I/2 [Γq(δnI + A)]−s

∥∥∥∥∥
)

= log ∥Γq(αnI +B)∥+ r log ∥Γq(µnI + C)∥

+ log |Γq(n+ 1)| − 1

2
n(n− 1)

[
ℜ(α2 + rµ2 − sδ2 + 1)∥I∥

]
log q

−s log ∥Γq(δnI + A)∥. (3.2)

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From the definition (2.4) of q-gamma function, one finds

log ∥Γq(αnI +B)∥ = log
∥∥∥(q; q)∞[(qαnI+B; q)∞

]−1
(1− q)I−αnI−B

∥∥∥
= log

∥∥∥(q; q)∞ [(qαnI+B; q)∞
]−1

(1− q)I−αnI−B
∥∥∥

= log |(q; q)∞|+ ∥I − αI −B∥ log |(1− q)|
− log

∥∥(qαnI+B; q)∞
∥∥ ; (3.3)

in which

log
∥∥(qαnI+B; q)∞

∥∥ = log

(
∞∏
k=0

∥∥I − qαnI+B+k
∥∥)

= log

(
lim

m→∞

m∏
k=0

∥∥I − qαnI+B+k
∥∥)

= lim
m→∞

m∑
k=0

log
∥∥I − qαnI+B+k

∥∥
=

∞∑
k=0

log
∥∥I − qαnI+B+k

∥∥ .
Here it may be noted that [1, p.207]

log
∥∥I − qαnI+B+k

∥∥ ≤ 1− ∥qαnI+B∥ |q|k

which leads us to

∞∑
k=0

log
∥∥I − qαnI+B+k

∥∥ ≤ ∥qαnI+B∥
∞∑
k=0

|q|k = ∥qαnI+B∥
1− |q|

.

This implies that

lim
n→∞

log
∥∥(qαnI+B; q)∞

∥∥
n log n

= 0.

Consequently from (3.3), it follows that

lim
n→∞

log ∥Γq(αnI +B)∥
n log n

= 0.

This last limit and the trivial limit

lim
n→∞

n− 1

log n
= ∞

when used in (3.2), yields

lim
n→∞

log (∥Vn∥)−1

n log n
= ∞.

Thus,
ϱ(EA,B,C

αI, δI, µI(λz; s, r|q)) = 0.

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Theorem 3.2. The function eA,B,C
αI, δI, µI(λz; s, r|q) represents the series which converges

absolutely for |z| <
∣∣(1− q)(sδ−α−rµ−1)

∣∣ and |q| < 1.

Proof. Take

Un = [Γq(γ + δn)]s [Γq(β + αn)]−1 [Γq(λ+ µn)]−r [Γq(n+ 1)] (3.4)

then

eγ,δα, β, λ, µ(z; s, r|q) =
∞∑
n=0

Un zn.

Now in view of (2.4), we get

n
√

∥Un∥ ∼
∥∥(1− q)(sδ−ℜ(α)−rµ)I−1

∥∥
whence

1

R
= lim

n→∞
n
√
|Un| ∼

∣∣(1− q)α+rµ−sδ+1
∣∣ .

Thus, the series in (2.14) converges absolutely if |z| < R = (1− q)sδ−ℜ(α)−rµ−1.

3.2 Contour integral

Theorem 3.3. Let A,B,C be positive stable matrices in Cp×p , α, δ, µ ∈ C with ℜ(α) >
0, ℜ(α2)+rµ2−sδ2+1 > 0, δ, µ > 0, r ∈ {−1, 0}∪N, s ∈ N∪{0} and 0 < q < 1. Then
the function EA,B,C

αI, δI, µI(λz; s, r|q) is expressible as the Mellin - Barnes q-integral given by

EA,B,C
αI, δI, µI(λz; s, r|q) =

1

2πi

∫
L

(−1)−pSq−pS(−S−1)/2 Γq(S) [Γq(A− δIS)]s

×[Γq(B − αIS)] [Γq(C − µIS)]−r (−z)−S dqS, (3.5)

where |argz| < π. The contour L of integration begins from −i∞ and proceeds towards
+i∞, and is indented to keep the poles of integrand at S = −nI to the left; and the poles
at S = (A+ nI)/δ to the right of the path for all n ∈ N ∪ {0}.

Proof. The integral on the right hand side of (3.5) may be evaluated as the sum of the
residues at the poles S = 0,−1,−2, . . . . In fact, in view of the definition of residue,

I =
1

2πi

∫
L

(−1)−pS q−pS(−S−1)/2 Γq(S) [Γq(A− δIS)]s (−z)−S

×[Γq(B − αIS)] [Γq(C − µIS)]−r dqS

=
∞∑
n=0

Res
S = −n

[
(−1)−pS q−pS(−S−1)/2 Γq(S) (−z)−S [Γq(B − αIS)]−1

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×[Γq(C − µIS)]−r [Γq(A− δIS)]s

]

=
∞∑
n=0

lim
S→−n

π(S + n)

sinπS
(−1)−pS q−pS(−S−1)/2 [Γq(A− δIS)]s (−z)−S

×[Γq(B − αIS)]−1 [Γq(C − µIS)]−r [Γq(1− S)]−1

=
∞∑
n=0

(−1)pn qpn(n−1)/2 [Γq(A+ δIn)]s [Γq(B + αIn)]−1 [Γq(C + µIn)]−r

× zn

Γq(n+ 1)

= EA,B,C
αI, δI, µI(λz; s, r|q).

Theorem 3.4. Let A,B,C be positive stable matrices in Cr×r , α, δ, µ ∈ C with ℜ(α) >
0, δ, µ > 0, r ∈ {−1, 0} ∪ N, s ∈ N ∪ {0} and (α2 + rµ2 + 1)I = sδ2I, 0 < q < 1. Then
the function eA,B,C

αI, δI, µI(λz; s, r|q) is expressible as the Mellin - Barnes q-integral given by

eA,B,C
αI, δI, µI(λz; s, r|q) =

1

2πi

∫
L

Γq(S) [Γq(A− δIS)]s (−z)−S

×[Γq(B − αIS)]−1 [Γq(C − µIS)]−rdqS, (3.6)

where |argz| < π; the contour L of integration begins from −i∞ and proceeds towards
+i∞, and is indented to keep the poles of integrand at S = −nI to the left; and the poles
at S = (A+ nI)/δ to the right of the path, for all n ∈ N ∪ {0}.

3.3 Difference equation

With the aid of the following operators, the difference equations of both q-analogues will
be derived. Put

Λqf(A) = f(A)− f(Aq−1), Θf(A) = f(A)− f(Aq), (3.7)

Dq f(A) = (1− q) Dqf(A) := (1− q) [f(A)− f(Aq)](A− Aq)−1

= [f(A)− f(Aq)](A)−1, (3.8)

{
a−1∏
u=0

a−1∏
v=0

[ΘI + c−uqI−(B+vI)/a − I]m

}{
a−1∏
u=0

a−1∏
v=0

[c−uqI−(B+vI)/a]m

}−1

= Φ(a,B,c;m)
u,v (3.9)

and {
a−1∏
u=0

a−1∏
v=0

[ΘI + c−uq(B+vI)/a − I]m

}{
a−1∏
u=0

a−1∏
v=0

[c−uq−(B+vI)/a]m

}−1

= Ψ(a,B,c;m)
u,v . (3.10)

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In these notations, the q-difference equation satisfied by (2.13) is derived in the following
theorem.

Theorem 3.5. Let α, µ, δ ∈ N, then EA,B,C
αI, δI, µI(λz; s, r|q) satisfies the equation[

Φ
(µ,C,η;r)
ℓ,k Φ

(α,B,σ;1)
h,m Θ

]
EA,B,C

αI, δI, µI(λz; s, r|q)

−
[
(−1)p z Ψ

(δ,A,ζ;s)
j,i

]
EA,B,C

αI, δI, µI(λzq
p; s, r|q) = 0 (3.11)

in which ζ is δth root of unity, η is µth root of unity, σ is αth root of unity.

Proof. In the first place, the coefficient of zn in the series representation ofEA,B,C
αI, δI, µI(λz; s, r|q)

will be expressed in q-factorial notation with the help of the set of formulas [2, Appendix
I]:

(A; q)kn = (A,Aq, . . . , Aqk−1; qk)n,

(Ak; qk)n = (A,Aωk, . . . , Aω
k−1
k ; qk)n ;ωk = e(2πi)/k,

(A; qn)νk = (A1/n; q)νk (A
1/nω; q)νk . . . (A

1/nωn−1; q)νk, ωn = 1,

and

(qA; qδ)n = (qA/δ; q)n (ϖqA/δ; q)n . . . (ϖ
δ−1qA/δ; q)n =

δ−1∏
i=0

(ϖiqA/δ; q)n, ϖδ = 1.

Then following the notation used in (3.1) for the coefficient of zn, we get

Vn = (−1)pn qpn(n−1)/2 [(qA; q)δn]
s[(qC ; q)µn]

−r [(qB; q)αn]
−1 1

(q; q)n

= (−1)pn qpn(n−1)/2 [(qA; qδ)n]
s [(qA+I ; qδ)n]

s · · · [(qA+δI−I ; qδ)n]
s

×[(qC ; qµ)n]
−r [(qC+I ; qµ)n]

−r · · · [(qC+µI−I ; qµ)n]
−r

×[(qB; qα)n]
−1 [(qB+I ; qα)n]

−1 · · · [(qB+αI−I ; qα)n]
−1 1

(q; q)n

= (−1)pn qpn(n−1)/2

{
δ−1∏
j=0

δ−1∏
i=0

[(ζj q(A+iI)/δ; q)n]
s

}

×

{
µ−1∏
ℓ=0

µ−1∏
k=0

[(ηℓ q(C+kI)/µ; q)n]
−r

}

×

{
α−1∏
h=0

α−1∏
m=0

[(σh q(B+mI)/α; q)n]
−1

}
1

(q; q)n
, (3.12)

where ζ is δth root of unity, η is µth root of unity, σ is αth root of unity.
Now take

δ−1∏
j=0

δ−1∏
i=0

[(ζj q(A+iI)/δ; q)n]
s = An,

µ−1∏
ℓ=0

µ−1∏
k=0

[(ηℓ q(C+kI)/µ; q)n]
r = Bn, (3.13)

and

α−1∏
h=0

α−1∏
m=0

(σh q(B+mI)/α; q)n = Cn, (−1)pn qpn(n−1)/2 = Dn (3.14)

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then
∞∑
n=0

Vn zn =
∞∑
n=0

An Dn B−1
n C−1

n

zn

(q; q)n
= W, say.

Since the series in (2.13) converges, we have

ΘW =
∞∑
n=0

An Dn B−1
n C−1

n

1

(q; q)n
Θzn

=
∞∑
n=0

An Dn B−1
n C−1

n

1− qn

(q; q)n
zn

=
∞∑
n=1

An Dn B−1
n C−1

n

zn

(q; q)n−1

.

Next operating by Φ
(α,B,σ;1)
h,m , we get

Φ
(α,B,σ;1)
h,m ΘI W =

∞∑
n=1

An Dn B−1
n C−1

n

1

(q; q)n−1

×

{
α−1∏
h=0

α−1∏
m=0

(ΘI + σ−hqI−(B+mI)/α − I)

}

×

{
α−1∏
h=0

α−1∏
m=0

(σ−hqI−(B+mI)/α)

}−1

zn

=
∞∑
n=1

An Dn B−1
n C−1

n

1

(q; q)n−1

×

{
α−1∏
h=0

α−1∏
m=0

(I − σhq(n−1)I+(B+mI)/α)

}

×

{
α−1∏
h=0

α−1∏
m=0

(σhq(B+mI)/α)

}−1

zn

=
∞∑
n=1

An Dn B−1
n C−1

n

1

(q; q)n−1

Cn C−1
n−1 zn

=
∞∑
n=1

An Dn B−1
n C−1

n−1

zn

(q; q)n−1

Finally,

Φ
(µ,C,η;r)
ℓ,k Φ

(α,B,σ;1)
h,m ΘI W

=
∞∑
n=1

An Dn C−1
n−1 B−1

n

1

(q; q)n−1

{
µ−1∏
l=0

µ−1∏
k=0

[(ΘI + η−lqI−(C+kI)/µ − I)]r

}

×

{
µ−1∏
l=0

µ−1∏
k=0

[(η−lqI−(C+kI)/µ)]r

}−1

zn

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=
∞∑
n=1

An Dn C−1
n−1 B−1

n

1

(q; q)n−1

{
µ−1∏
l=0

µ−1∏
k=0

[−qn + η−lqI−(C+kI)/µ)]r

}

×

{
µ−1∏
l=0

µ−1∏
k=0

[η−lqI−(C+kI)/µ]r

}−1

zn

=
∞∑
n=1

An Dn C−1
n−1 B−1

n

1

(q; q)n−1

Bn B−1
n−1 zn

=
∞∑
n=1

An Dn B−1
n−1 C−1

n−1

zn

(q; q)n−1

Thus,

Φ
(µ,C,η;r)
ℓ,k Φ

(α,B,σ;1)
h,m ΘI W =

∞∑
n=0

An+1 Dn+1 B−1
n C−1

n

zn+1

(q; q)n
(3.15)

On the other hand,

Ψ
(δ,A,ζ;s)
j,i EA,B,C

αI, δI, µI(zq
p; s, r|q)

=
∞∑
n=0

An Dn B−1
n C−1

n

1

(q; q)n

{
δ−1∏
j=0

δ−1∏
i=0

[(ΘI + ζ−jq−(A+iI)/δ − I)]s

}

×

{
δ−1∏
j=0

δ−1∏
i=0

[(ζ−jq−(A+iI)/δ)]s

}−1

zn

=
∞∑
n=0

An Dn B−1
n C−1

n

1

(q; q)n

{
δ−1∏
j=0

δ−1∏
i=0

[(I − ζjqnI+(A+iI)/δ)]s

}
zn

=
∞∑
n=0

An Dn B−1
n C−1

n

1

(q; q)n
An+1 A−1

n zn

=
∞∑
n=0

An+1 Dn B−1
n C−1

n

zn

(q; q)n

that is,

z (−1)p Ψ
(δ,A,ζ;s)
j,i EA,B,C

αI, δI, µI(zq
p; s, r|q) =

∞∑
n=0

An+1 Dn+1 B−1
n C−1

n

zn+1

(q; q)n
. (3.16)

On comparing (3.15) and (3.16), the equation (3.11) is obtained.

The q-difference equation satisfied by the function (2.14) is given in following theorem
whose proof follows line-to-line just dropping the factor qn(n−1)/2 that is, dropping Dn in
(3.14).

Theorem 3.6. Let α, µ, δ ∈ N, then Y = eγ,δα,β,λ,µ(z; s, r|q) satisfies the equation[
Φ

(µ,λ,η;r)
ℓ,k Φ

(α,β,σ;1)
h,m Θ− z Ψ

(δ,γ,ζ;s)
j,i

]
Y = 0, (3.17)

where ζ is δth root of unity, η is µth root of unity, σ is αth root of unity.

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3.4 Eigen function property

Take {
a−1∏
u=0

a−1∏
v=0

[(Λq + c−uqI−(B+vI)/a − I)]m

}{
a−1∏
u=0

a−1∏
v=0

[c−uqI−(B+vI)/a]m

}−1

= Ω(a,B,c;m)
u,v , (3.18)

and

∆q = Dq Ω
(δ,A,ζ;−s)
j,i Φ

(µ,C,η;r)
ℓ,k Φ

(α,B,σ;1)
h,m . (3.19)

Here the operators Ω
(δ,γ,ζ;−s)
j,i ,Φ

(µ,λ,η;r)
ℓ,k ,Φ

(α,β,σ;1)
h,m in (3.19) are not commutative with the

operator Dq.

This property does not hold for the function Eγ,δ
α,β,λ,µ(z; s, r|q), but for the other version:

eγ,δα,β,λ,µ(z; s, r|q), it is established in

Theorem 3.7. Let α, µ, δ ∈ N and q-difference operator Θ be defined by (3.7) then
eγ,δα,β,λ,µ(z; s, r|q) is an eigen function with respect to the operator ∆q defined by (3.19).
That is, for any non zero c,

∆q eA,B,C
αI, δI, µI(cz; s, r|q) = c eA,B,C

αI, δI, µI(z; s, r|q). (3.20)

(3.21)

Proof. With An, Bn and Cn as in (3.13) and in (3.14),

eA,B,C
αI, δI, µI(cz; s, r|q) =

∞∑
n=0

An B−1
n C−1

n

zn

(q; q)n

Now if eA,B,C
αI, δI, µI(cz; s, r|q) = Yc then in the notation (3.9),

Φ
(α,B,σ;1)
h,m Yc =

∞∑
n=0

cn An B−1
n C−1

n

1

(q; q)n

×

{
α−1∏
h=0

α−1∏
m=0

(ΘI + σ−h qI−(B+mI)/α − I)

}

×

{
α−1∏
h=0

α−1∏
m=0

(σ−hqI−(B+mI)/α)

}−1

zn

=
∞∑
n=0

cn An B−1
n C−1

n

{
α−1∏
h=0

α−1∏
m=0

(I − σhq(n−1)I+(B+mI)/α)

}

×

{
α−1∏
h=0

α−1∏
m=0

(σhq(B+mI)/α)

}−1

zn

(q; q)n

=
∞∑
n=0

cn An B−1
n C−1

n

1

(q; q)n
Cn C−1

n−1 zn

=
∞∑
n=0

cn An B−1
n C−1

n−1

zn

(q; q)n

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Next

Φ
(µ,C,η;r)
ℓ,k Φ

(α,B,σ;1)
h,m Yc =

∞∑
n=0

cn An C−1
n−1 B−1

n

1

(q; q)n

×

{
µ−1∏
l=0

µ−1∏
k=0

[(ΘI + η−lqI−(C+kI)/µ − I)]r

}

×

{
µ−1∏
l=0

µ−1∏
k=0

[(η−lqI−(C+kI)/µ)]r

}−1

zn

=
∞∑
n=0

cn An C−1
n−1 B−1

n

1

(q; q)n

×

{
µ−1∏
l=0

µ−1∏
k=0

[−qn + η−lqI−(C+kI)/µ)]r

}

×

{
µ−1∏
l=0

µ−1∏
k=0

[η−lqI−(C+kI)/µ]r

}−1

zn

=
∞∑
n=0

cn An C−1
n−1 B−1

n

1

(q; q)n
Bn B−1

n−1 zn

=
∞∑
n=0

cn An B−1
n−1 C−1

n−1

zn

(q; q)n

Further using (3.18),

Ω
(δ,A,ζ;−s)
j,i Φ

(µ,C,η;r)
ℓ,k Φ

(α,B,σ;1)
h,m Yc

=
∞∑
n=0

cn An B−1
n−1 C−1

n−1

1

(q; q)n

{
δ−1∏
j=0

δ−1∏
i=0

[(∆qI + ζ−jq−(A+iI)/δ − I)]−s

}

×

{
δ−1∏
j=0

δ−1∏
i=0

[(ζ−jq−(A+iI)/δ)]−s

}−1

zn

=
∞∑
n=0

cn An B−1
n−1 C−1

n−1

1

(q; q)n

{
δ−1∏
j=0

δ−1∏
i=0

[(I − ζjqnI+(A+iI)/δ)]s

}
zn

=
∞∑
n=0

cn An B−1
n−1 C−1

n−1

1

(q; q)n
An−1 A−1

n zn

=
∞∑
n=0

cn An−1 B−1
n−1 C−1

n−1

zn

(q; q)n

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Finally,

∆q Yc = Dq Ω
(δ,A,ζ;−s)
j,i Φ

(µ,C,η;r)
ℓ,k Φ

(α,B,σ;1)
h,m Yc

=
∞∑
n=1

cn An−1 B−1
n−1 C−1

n−1

zn−1

(q; q)n−1

=
∞∑
n=0

cn+1 An B−1
n C−1

n

zn

(q; q)n

= c eA,B,C
αI, δI, µI(cz; s, r|q).

References

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[2] G. Gasper and M. Rahman. Basic Hypergeometric Series. Cambridge University
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[3] B. V. Nathwani. Generalized q-Mittag-Leffler function and its properties. Journal
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[8] J. C. Prajapati and B. V. Nathwani. Recurrence relation of a unified generalized
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