EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 4, Article Number 6194
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

On Some Applications of Generalized
Meijer G - Functions

Syed Ali Haider Shah1, Imran Siddique1,2,∗, Ilyas Khan3,4,5,
Ashit Kumar Dutta6,7, Mujeeb Ahmed Shaikh8, Aten Aldawood9

1 Department of Mathematics, University of Sargodha, Sargodha 40100, Pakistan
2 Mathematics in Applied Sciences and Engineering Research Group, Scientific Research
Center, Al-Ayen University, Nasiriyah, 64001, Iraq
3 Department of Mathematical Sciences, Saveetha School of Engineering, SIMATS, Chennai,
Tamil Nadu, India
4 Hourani Center for Applied scientific Research, Al-Ahliyya Amman University, Amman,
Jordan
5 Department of Mathematics, College of Science Al-Zulfi, Majmaah University, Al-Majmaah
11952, Saudi Arabia
6 Department of Computer Science and Information Systems, College of Applied Sciences,
AlMaarefa University, Dariyah, 13713, Saudi Arabia
7 Research Center, Deanship of Scientific Research and Post-Graduate Studies, AlMaarefa
University, Dariyah, 13713, Saudi Arabia
8 Department of Basic Medical Science, College of Medicine, AlMaarefa University, Diriyah,
13713, Riyadh, Saudi Arabia.
9 Department of Computer Science and Information Systems, College of Applied Sciences,
AlMaarefa University, Dariyah, 13713, Saudi Arabia

Abstract. In recent few years, the researchers are investigating many generalization and extension
of special functions because of their applicability in various fields of study especially in mathemat-
ical modeling. Keeping in view, some summations involving generalized Meijer G-functions in
k-form for different combination of parameters have been investigated in this paper. By using con-
tour integral approach and generalized hypergeometric k-functions we derived some new relations
and identities. By taking k = 1, we can obtain the classical form of the derived results. Some im-
portant applications of generalized Meijer G-functions to the generalized hypergeometric functions
have also been considered. The main objective of this paper is to give some useful computational
techniques to tackle with the applications related to generalized Meijer G-functions in different
mathematical and physical problems, also connect them with some generalized special functions.
2020 Mathematics Subject Classifications: 33C60, 33C20, 33C05, 33C15
Key Words and Phrases: Meijer G-function; Generalized Meijer G - function; Gauss hyperge-
ometric function; Confluent hypergeometric function.

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6194

Email addresses: ali.bukhari78699@gmail.com, imransmsrazi@gmail.com, i.said@mu.edu.sa.

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



S. A. Haider Shah et a. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6194 2 of 12

1. Introduction and Preliminaries

Special functions are widely applicable in different areas of study like pure and applied
mathematics, physics, engineering, and mathematical modeling. In the field of special
functions the main aim is to derive and investigate the results, which are helpful in solv-
ing the problems of different fields. With this aim many researchers are working through
different directions. In the last two decades the researchers like Diaz, Pariguan, Mubeen,
Habibullah, Kokologiannaki, Krasniqi, Mansour etc. worked on the specific k-symbol and
proved number of properties and applications.

Diaz and Pariguan [1], presented the extension of beta and gamma functions respec-
tively, in the form of k > 0 as

Γk(o) = lim
n→∞

n!kn(nk)
o
k
−1

(o)n,k
where k > 0, o ∈ C\kZ− (1)

and
βk(o, ω) =

1
k

∫ 1

0
t
o
k
−1(1 − t)

ω
k
−1dt, ℜ(o) > 0,ℜ(ω) > 0 (2)

on the basis of extended Pochhammer’s symbol

(o)n,k = (o)(o+ k)(o+ 2k)...(o+ (n− 1)k), n ≥ 1, k > 0.

They also discussed the integral form of extended gamma function

Γk(o) =

∫ ∞

0
t
o
k
−1e−tk

k dt, ℜ(o) > 0, k > 0

and the hypergeometric function k-function [1] for all ϵ, ℘, ω ∈ C and ω ̸= 0,−1,−2,−3, · · · , |α| <
1, as

2F1,k((ω, k), (℘, k); (ϵ, k);α) =
∞∑

m=0

(ω)m,k(℘)m,k
(ϵ)m,k

αm

m!
, k > 0.

After that the researchers [2–11] worked on k-functions, proved many properties and re-
sults in k-form. Mubeen et al. [12] introduced generalized hypergeometric differential
equation also proved twenty four solutions of that equation. Some other functions like
Mittag-Leffler function, Fox H-function, Meijer G-function etc. are the functions upon
which many researchers [13–25] have worked and gave different properties.
In the last few years, the researchers [26–31] started to see the applicability of the special
functions and fractional operators in various fields like Magneto hydrodynamics (MHD)
and hybrid nanofluids, entropy analysis in special fluid flows, heat and mass transfer via
special functions. Khan et.al [26] concentrated on many generalizations of MHD fluids
through different fractional operators. They also suggested that many complicated prac-
tical life phenomena’s can be dealt easily with the help special functions and fractional
operators. The Caputo–Fabrizio time-fractional derivatives have been used by [31] to gen-
eralize the idea of dusty tetra hybrid nanofluid, also discussed several applications like in



S. A. Haider Shah et a. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6194 3 of 12

signal processing, diffusion, image processing, damping, and bioengineering etc. of gener-
alized fractional operators.

Among several special functions, Meijer G-function as a particular instance of Fox
H-function is an interesting function due to its generality because many other classical
functions are the special cases of this function. Meijer G-function is an essential math-
ematical tool in studying different applied mathematical models, integral equations, and
differential equations. Milgram [32] gave some techniques to tackle the summations in-
volving Meijer G-functions and proved some useful results.

In our current investigations, we prove the generalization of some summations involving
Meijer G-function in k-form, k > 0, and give some techniques to tackle with applications
related to generalized Meijer G-functions. All the obtained results will take the classical
forms while choosing k = 1.

Generalization of Meijer G-function (i.e. Meijer (G, k)-function) [33, 34], for k > 0
can be defined as:

Gm,n
k,p,q

[
ep
fq

∣∣∣∣z] =
1

2πi

∮
L

Γk(f1,m − s)Γk(k − e1,n + s)

Γk(en+1,p − s)Γk(k − fm+1,q + s)
zs/kds, (3)

where

Γk(on,p + s) =

p∏
j=n

Γk(oj + s)

is the gamma k-function.

Luke [21] considered the path of contour integral of the form given above for various
ranges of z. Here, we consider |z| ≤ 1

k , k > 0 and p ≤ q, q ≥ 1. We are keen in the
summations involving generalized Meijer G-function as:

∑
l

xlgl
l!

Gm,n
k,p,q

[
ep(l)
fq(l)

∣∣∣∣z] , (4)

where ep(l) and fq(l) are of the form ω ± lk and ρ± lk respectively.

The basic idea, is to observe that if gl, ep(l) and fq(l) are such that after replacing the
summation and integral in (4), the l dependence take the form of Gauss hypergeometric
function 2F1,k and then different known results and transformations can be applied on
2F1,k. After solving 2F1,k, the order of summation and integration can be interchanged
again and we will obtain generalized G-function.



S. A. Haider Shah et a. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6194 4 of 12

2. Main Result

In this section, we investigate a result which implements the technique discussed
above.

Theorem 1. For ℜ(h − g − k + ω) > min(ℜ(fj)), j = 1, ...,m, k > 0, the following
equation holds:

Sk(
1

k
, z) =

Γk(g)

Γk(h− g)
Gm+1,n+1

k,p+2,q+1

[
ω, ep, ω + h− k
ω + h− g − k, fq

∣∣∣∣z] .
Proof. Consider the summation

Sk(x, z) =
∑
l

Γk(g + lk)xl

l!Γk(h+ lk)
Gm,n+1

k,p+1,q

[
ω − lk, ep

fq

∣∣∣∣z] . (5)

By using the definitions of generalization of Meijer G-function and hypergeometric k-
function, we get

Sk(x, z) =
Γk(g)

Γk(h)

1

2πi

∮
L

∏m
j=1 Γk(fj − s)

∏n
j=1 Γk(k − ej + s)∏p

j=n+1 Γk(ej − s)
∏q

j=m+1 Γk(k − fj + s)
Γk(k−ω+s)2F1,k

[
g, k − ω + s

h

∣∣∣∣x] zs/kds
from equation (3) for |x| < 1

k .

To evaluate the summation at x = 1
k , we use the relation

lim
x→( 1

k
)−

2F1,k

[
ω, η
ρ

∣∣∣∣x] =
Γk(ρ)Γk(ρ− ω − η)

Γk(ρ− ω)Γk(ρ− η)
, ℜ(ρ− ω − η) > 0.

Thus, we obtain

Sk(
1

k
, z) =

Γk(g)

Γk(h− g)
Gm+1,n+1

k,p+2,q+1

[
ω, ep, ω + h− k
ω + h− g − k, fq

∣∣∣∣z] (6)

for ℜ(h − g − k + ω) > min(ℜ(fj)), j = 1, ...,m, in other case, we obtain residues of
Γk(h− g − k + ω − s) for the poles which lies outside the contour L.

Corollary 1. For |x| < 1
k , the transformation [13]

2F1,k(g, k − ω + s;h;x) = (1− kx)
h−g−k+ω−s

k 2F1,k

[
h− g, h− k + ω − s

h

∣∣∣∣x] (7)

can be used in (5) to get

Sk(x, z) =
Γk(g)(1− kx)

h−g−k+ω
k

Γk(h− g)

∑
l

Γk(h− g + lk)xl

Γk(h+ lk)l!
Gm+1,n+1

k,p+2,q+1

[
ω, ep, ω + h− k
ω + h− k + lk, fq

∣∣∣∣ z

1− kx

]
.

(8)



S. A. Haider Shah et a. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6194 5 of 12

Remark:
Similar and even more difficult results can also be obtained by using p+1Fq,k for known
arguments.

3. some variation formulas involving generalized Meijer G-functions

In this section, we consider some summations involving generalized Meijer
G-functions for x = ± 1

k and use some transformations to obtain the some useful results.

Theorem 2. For 0 ≤ n < p + 1, p + 2 ≤ q, 0 ≤ m ≤ q and (η−g−k+ω)
2 which is enclosed

by contour for fj , j = 1, ...,m, the following holds:

1

Γk(g)

∑
l

Γk(g + lk)( 1k )
l

l!
Gm,n+1

k,p+2,q

[
ω − lk, ep, η + lk

fq

∣∣∣∣z]

=
1

2
g
k

Gm+2,n+1
k,p+4,q+2

[
ω, ep,

(ω+η−k)
2 , (η+ω)

2 , η − g
(η+ω−g−k)

2 , (η+ω−g)
2 , fq

∣∣∣∣z
]
. (9)

Proof. Consider the left hand side as

1

Γk(g)

∑
l

Γk(g + lk)( 1k )
l

l!
Gm,n+1

k,p+2,q

[
ω − lk, ep, η + lk

fq

∣∣∣∣z]

=
1

Γk(g)

∑
l

Γk(g + lk)( 1k )
l

l!

1

2πι

∮
L

∏m
j=1 Γk(fj − s)

∏n
j=1 Γk(k − ej + s)∏p

j=n+1 Γk(ej − s)
∏q

j=m+1 Γk(k − fj + s)
z

s
k ds

=
1

2πι

∮
L

∏m
j=1 Γk(fj − s)

∏n
j=1 Γk(k − ej + s)∏p

j=n+1 Γk(ej − s)
∏q

j=m+1 Γk(k − fj + s)

∑
l

(k − ω + s)l,kΓk(k − ω + s)(g)l,k(
1
k )

l

(η − s)l,kΓk(η − s)l!
z

s
k ds

=
1

2πι

∮
L

∏m
j=1 Γk(fj − s)

∏n
j=1 Γk(k − ej + s)Γk(k − ω + s)∏p

j=n+1 Γk(ej − s)
∏q

j=m+1 Γk(k − fj + s)Γk(η − s)

× Γk(η − s)Γk(η − s− k − g + ω − s)

Γk(η − s− g)Γk(η − s− k − s+ ω)
z

s
k ds.

After doing simple mathematical calculations we finally obtain

1

Γk(g)

∑
l

Γk(g + lk)( 1k )
l

l!
Gm,n+1

k,p+2,q

[
ω − lk, ep, η + lk

fq

∣∣∣∣z]

=
1

2
g
k

Gm+2,n+1
k,p+4,q+2

[
ω, ep,

(ω+η−k)
2 , (η+ω)

2 , η − g
(η+ω−g−k)

2 , (η+ω−g)
2 , fq

∣∣∣∣z
]
.



S. A. Haider Shah et a. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6194 6 of 12

Corollary 2. For η = g + ω, the quadratic transformation x → −4x
(1−kx)2

of the function
2F1,k in equation (9) can be used, as for x = −1

k , we get

1

Γk(g)

∑
l

Γk(g + lk)(−1
k )l

l!
Gm,n+1

k,p+2,q

[
ω − lk, ep, g + ω + lk

fq

∣∣∣∣z] =
Γk(

k
2 )

2
g
kΓk

(g+k)
2

Gm,n+1
k,p+2,q

[
ω, ep,

g
2 + ω

fq

∣∣∣∣z] .
(10)

Theorem 3. For 0 ≤ n < p + 1, p + 2 ≤ q, 0 ≤ m ≤ q and (η−g−k+ω)
2 which is enclosed

by contour for fj , j = 1, ...,m, the following holds:

1

Γk(g)

∑
l

Γk(g + lk)( 1k )
l

l!
Gm,n+1

k,p+2,q

[
ω + lk, ep, η − lk

fq

∣∣∣∣z]

=
1

2
g
k

Gm+3,n+1
k,p+5,q+3

[
ω, ep,

(ω+η−k)
2 , (ω+η)

2 , η, ω − g

ω, (ω+η−g−k)
2 , (ω+η−g)

2 , fq

∣∣∣∣z
]
. (11)

Proof. Considering the left hand side as

1

Γk(g)

∑
l

Γk(g + lk)( 1k )
l

l!
Gm,n+1

k,p+2,q

[
ω + lk, ep, η − lk

fq

∣∣∣∣z]

=
∑
l

(g)l,k(
1
k )

l

l!

1

2πι

∮
L

∏m
j=1 Γk(fj − s)Γk(k − ω − lk + s)

∏n+1
j=2 Γk(k − ej + s)∏p+1

j=n+2 Γk(ej − s)Γk(η − lk − s)
∏q

j=m+1 Γk(k − fj + s)
z

s
k ds.

Now, by interchanging the order of summation and integration and then applying

(k − ω)−n,k =
(−1)n

(ω)n,k
,

we obtain

1

Γk(g)

∑
l

Γk(g + lk)(−1
k )l

l!
Gm,n+1

k,p+2,q

[
ω − lk, ep, g + ω + lk

fq

∣∣∣∣z]

=
1

2πι

∮
L

∏m
j=1 Γk(fj − s)

∏n
j=1 Γk(k − ej + s)∏p

j=n+1 Γk(ej − s)
∏q

j=m+1 Γk(k − fj + s)
×
∑
l

(g)l,kΓk(k − ω − lk + s)( 1k )
l

Γk(η − lk − s)l!
z

s
k ds.

Applying Legendre Duplication k-formula for z = ω−g−k+η−2s
2 and z = ω+η

2 − s, we get

1

Γk(g)

∑
l

Γk(g + lk)(−1
k )l

l!
Gm,n+1

k,p+2,q

[
ω − lk, ep, g + ω + lk

fq

∣∣∣∣z]

=
1

2
g
k 2πι

∮
L

∏m
j=1 Γk(fj − s)

∏n+1
j=2 Γk(k − ej + s)∏p+1

j=n+2 Γk(ej − s)
∏q

j=m+1 Γk(k − fj + s)



S. A. Haider Shah et a. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6194 7 of 12

×
Γk(ω − s)Γk(

ω−g+η
2 − s)Γk(k − ω − s)Γk(

ω−g−k+η
2 − s)

Γk(η − s)Γk(ω − s− g)Γk(
ω+η
2 − s)Γk(

ω+η−k
2 − s)

z
s
k ds

=
1

2
g
k

Gm+3,n+1
k,p+5,q+3

[
ω, ep,

ω−k+η
2 , ω+η

2 , η, ω − g

ω, ω−k+η−g
2 , ω+η−g

2 , fq

∣∣∣∣z] .
Corollary 3. If we choose ω = η + g in Theorem (3), then

1

Γk(g)

∑
l

Γk(g + lk)(−1
k )l

l!
Gm,n+1

k,p+2,q

[
η + g + lk, ep, η − lk

fq

∣∣∣∣z]

=
Γk(

k
2 )

2
g
kΓk

(g+k)
2

Gm+1,n+1
k,p+3,q+1

[
η + g, ep, η + g

2 , η
η + g, fq

∣∣∣∣z] ,
where m,n, p and q are identical to classical Meijer G-function.

By keeping in view the above variants, we consider another variant as

Theorem 4. For 0 ≤ m ≤ q, 0 ≤ n ≤ p ≤ q, ℜ(h− η+ω) > 1, k > 0 the following holds:

∑
l

( 1k )
l

Γk(h+ lk)l!
Gm+1,n+1

k,p+1,q+1

[
ω − lk, ep
η + lk, fq

∣∣∣∣z]
= Γk(h− η + ω − k)Gm+1,n+1

k,p+2,q+2

[
ω, ep, ω + h− k
η, fq, η + h+K

∣∣∣∣z] .
Proof. Consider the left side as

∑
l

( 1k )
l

Γk(h+ lk)l!
Gm+1,n+1

k,p+1,q+1

[
ω − lk, ep
η + lk, fq

∣∣∣∣z]

=
∑
l

( 1k )
l

Γk(h+ lk)l!

1

2πι

∮
L

∏m+1
j=1 Γk(fj − s)

∏n+1
j=1 Γk(k − ej + s)∏p+1

j=n+2 Γk(ej − s)
∏q+1

j=m+2 Γk(k − fj + s)
z

s
k ds.

Now, interchanging the order of summation and integral also doing some calculations, we
have

∑
l

( 1k )
l

Γk(h+ lk)l!
Gm+1,n+1

k,p+1,q+1

[
ω − lk, ep
η + lk, fq

∣∣∣∣z]

=
Γk(h+ ω − η − k)

2πι

∮
L

∏m+1
j=1 Γk(fj − s)

∏n+1
j=1 Γk(k − ej + s)∏p+2

j=n+2 Γk(k − ej − s)
∏q+1

j=m+2 Γk(k − fj + s)



S. A. Haider Shah et a. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6194 8 of 12

× Γk(η − s)Γk(k − ω + s)

Γk(k − k + h− η + s)Γk(h− η + s)Γk(h− k + ω − s)
z

s
k ds.

Hence, we finally obtain

∑
l

( 1k )
l

Γ(h+ lk)l!
Gm+1,n+1

k,p+1,q+1

[
ω − lk, ep
η + lk, fq

∣∣∣∣z]
= Γk(h− η + ω − k)Gm+1,n+1

k,p+2,q+2

[
ω, ep, h+ ω − k
η, fq, h+ η + k

∣∣∣∣z] .

4. Special Cases and Applications

In this section, some results are given for special functions which can be
identified as generalized Meijer G-functions. The results that are derived are useful in
numerical computations. Here, we consider generalized hypergeometric functions in sum-
mation and then after some mathematical calculations, we derive the results in k-form as
follows:

Theorem 5. For ℜ(fq − ω − lk − ep) > 0, k > 0 the following holds:

∑
l

Γk(g + lk)( 1k )
l

l!
p+1Fq,k

[
ω + lk, ep

fq

∣∣∣∣y]

=
(−y)

−g
k Γk(fq)Γk(g)Γk(ep − g)

Γk(ep)Γk(fq − g)
p+1Fq,k

[
ω − g, ep − g

fq − g

∣∣∣∣y] . (12)

Proof. As

G1,p+1
k,p+1,q+1

[
k − ω − lk, k − ep

0, k − fq

∣∣∣∣− y

]
=

Γk(ω + lk)Γk(ep)

Γk(fq)
p+1Fq,k

[
ω + lk, ep

fq

∣∣∣∣y] .
So∑

l

Γk(g + lk)( 1k )
l

l!
p+1Fq,k

[
ω + lk, ep

fq

∣∣∣∣y]

= Γk(g)
∑
l

(g)l,kΓk(fq)(
1
k )

l

Γk(ep)Γk(ω + lk)l!
G1,p+1

k,p+1,q+1

[
k − ω − lk, k − ep

0, k − fq

∣∣∣∣− y

]
=

Γk(fq)Γk(g)

Γk(ep)
G1,p+1

k,p+1,q+1

[
k − ω, k − ep
−g, k − fq

∣∣∣∣− y

]
.

Also



S. A. Haider Shah et a. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6194 9 of 12

G1,p+1
k,p+1,q+1

[
k − ω, k − ep
−g, k − fq

∣∣∣∣− y

]
=

Γk(ω)Γk(ep − g)

Γk(fq − g)
(−y)

−g
k p+1Fq,k

[
ω − g, ep − g

fq − g

∣∣∣∣y] .
So∑
l

Γk(g + lk)( 1k )
l

l!
p+1Fq,k

[
ω + lk, ep

fq

∣∣∣∣y] =
Γk(fq)Γk(g)Γk(ep − g)

Γk(ep)Γk(fq − g)
(−y)

−g
k p+1Fq,k

[
ω − g, ep − g

fq − g

∣∣∣∣y] .
Remark:

For q = 0 and p = 0 or p = 1, the above equation can be expressed in the k-form of result
[35].

We also derive some results by using generalization of Meijer G-function given as

∑
l

Γk(g + lk)( 1k )
l

l!
pFq+1,k

[
ep

fq, η − lk

∣∣∣∣y]

=
y
∏p

j=1 ejΓk(g)Γk(k − η)Γk(k − g)

kηΓk(k − η − g)
∏q

j=1 fq
p+1Fq+2,k

[
k + ep, k − g

k + fq, k + η, 2k

∣∣∣∣y] (13)

and∑
l

Γk(g + lk)Γk(ep + lk)yl

l!Γk(h+ lk)Γk(fq + lk)
pFq,k

[
ep + lk
fq + lk

∣∣∣∣y] =
Γk(g)Γk(ep)

Γk(h)Γk(fq)
p+1Fq+1,k

[
ep, h− g
fq, h

∣∣∣∣y] . (14)

Many other similar results can also be derived for some special functions.
If we take ϕk(e; g; y) as confluent k-hypergeometric function of second kind, then in term
of generalization of Meijer G-function we can write it as

ϕk

[
e
g

∣∣∣∣y] =
1

Γk(e)Γk(k + e− g)
G2,1

k,1,2

[
k − e
0, k − g

∣∣∣∣y] . (15)

After doing some calculations on equation (15), we obtain∑
l

Γk(e+ lk)Γk(f + lk)

l!
ϕk

[
e+ lk

g

∣∣∣∣y] = Γk(e)Γk(f)ϕk

[
e

f + g

∣∣∣∣y] . (16)

5. Conclusions

In this paper, we look into the summations involving generalization of Meijer G-
functions, hypergeometric functions and confluent hypergeometric functions. By applying
different techniques and transformations on summations we obtain the variations of said
functions. Throughout this paper if we choose k = 1 then we will get the classical Mei-
jer’s G-function, Hypergeometric function and Confluent Hypergeometric functions and
the results obtained will take the classical form.



S. A. Haider Shah et a. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6194 10 of 12

Funding statement

Ashit Kumar Dutta would like to thanks AlMaarefa University for supporting this
research under project number MHIRSP2025017.

Acknowledgements

Dr. Mujeeb Ahmed Shaikh would like to thank AlMaarefa University for supporting
this research under project number MHIRSP2025010.

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	Introduction and Preliminaries
	Main Result
	some variation formulas involving generalized Meijer G-functions
	Special Cases and Applications
	Conclusions