Generating Function_Updated Paper publish


Communications on Applied Nonlinear Analysis 
ISSN: 1074-133X 
Vol 32 No. 8s (2025) 
 

951 https://internationalpubls.com  

Generating Function Involving General Function Related to Hurwitz-
Lerch Zeta Function 

B. B. Jaimini1 and M.K. Tatwal2 
1Department of Mathematics, Government College, Kota (Rajasthan), India-324001 
2Department of Mathematics, Government College, Bundi (Rajasthan), India-323001 

1Email Id: bbjaimini_67@rediffmail.com 
2Email Id: manojtatwal@gmail.com 

 
Article History: 

Received: 01-01-2025 

Revised: 22-02-2025 

Accepted: 27-02-2025 

Abstract:  
In this paper, we have studied a general function which unifies the Hurwitz-Lerch Zeta 
function and Mittage-Leffler function. The integral representation of the function and certain 
generating functions involving this general function are established. A mild extension of this 
general function is also presented in this paper. The generating functions for this extended 
function are also studied in this paper. Certain known results involving Hurwitz-Lerch Zeta 
function for several parameters are shown to be obtained here. 

Keywords: Hurwitz-Lerch Zeta function; Mittag-Leffler function; Generating functions; 
Generalized Hypergeometric function; Integral representation 

MSC 2020: Primary 11M35; 33E20 Secondary: 33E12; 33C20 

1. Introduction and Preliminaries 

A general Hurwitz-Lerch Zeta function is defined in the literature in the following manner 
[1]: 

,    …(1.1) 

when  and  when  

Where  represents the Set of complex numbers, Set of real numbers, Set of positive real 

numbers and Set of integer number respectively and ,  

The Hurwitz-Lerch Zeta function is defined in (1.1) contains, its special cases, as the 

Riemann Zeta function and Hurwitz-Lerch Zeta function defined by (see, for details,[1, 
chapter I]) 

    …(1.2) 

( )! !! " #φ

( )
( )!

"#"
!

"
!

## $
! $

φ
∞

=

=
+

∑

!
" #$!" !# −∈ ∈ !! < ( )!" #! > !"! =

! ! !! " " #+

{ }! !! !− −= ∪ { }!" #" $%%%! − = − − −

( )! !! " #φ

( )!ζ ( )!! "ζ

( )
( )

( )
!

" "# #"
" !

"
! !

"
ζ φ

∞

=

= =
+

∑ ( )( )!" #! >



Communications on Applied Nonlinear Analysis 
ISSN: 1074-133X 
Vol 32 No. 8s (2025) 
 

952 https://internationalpubls.com  

   …(1.3) 

A generalization of the Hurwitz-Lerch Zeta function was studied by Lin and Srivastava [2, 
p.727, eq.(8)] 

   …(1.4)  

 when  and  when 

and  when  

Where denotes the general pochhammer symbol which is defined by in terms of the gamma 
function by 

 …(1.5) 

The function at and yields the Hurwitz-Lerch Zeta function defined and studied by 
Goyal and Loddha [3, p.100, eq.(1.5)] 

    …(1.6) 

when and  when  

Further generalization of the functions and was considered by Garg et.al [4] in 
the following manner [4,p.313, eq.(1.7)]: 

   …(1.7) 

when and  when  

A further generalization of a family of Hurwitz-Lerch Zeta function defined and studied by 
Srivastava et al [5, p.491, eq.(1.20)] in the following form: 

( )
( )

( )
!

"# "# #!
"

! # ! #
" #

ζ φ
∞

=

= =
+

∑ ( )
!

"# $% !" # $ 
 > ∈ 
 

( )! !! " #φ

( ) ( )
( )
( ) ( )

!
!

"
! !

!
!

"
! !

## " $
! $

ρ σ ρ
µ ν

σ

µ
φ

ν

∞

=

=
+

∑

!
" # $ $ $ #!" # $%µ ν ρ σ ρ σ+

−∈ ∈ ∈ < ! "! " # ρ σ∈ = ! "∈ !! ρ σδ ρ σ ρ σ−< = =

( )!" #! µ − − > !! δ=

( )!λ

( ) ( )

( ) ( )

{ }

( )

!" !#

# $$$ # "
!

"!
!

! ! # "

λ
λ

λ
λ

λ λ λ λ λ

  = ∈  
 Γ + = = Γ  + + − = ∈



!ρ σ= = !ν =

( ) ( )
( )

!

"
# #

$

!
!

"
!

## " $
! ! $µ

µ
φ

∞

=

=
+

∑

!
" # $!! " # !$µ −∈ ∈ ∈ !! < ( )!" #! µ > !"! =

( )! !! " #φ ( )! " "! " #µφ

( ) ( ) ( )
( ) ( )! !

"
! !

#

!
! !

"
! !

## " $
! $ !λ µ ν

λ µ
φ

ν

∞

=

=
+

∑

!
" # $ # $!! " # !$λ µ ν −∈ ∈ ∈ !! < ( )!" #!  λ µ+ − − > !"! =



Communications on Applied Nonlinear Analysis 
ISSN: 1074-133X 
Vol 32 No. 8s (2025) 
 

953 https://internationalpubls.com  

   …(1.8) 

when  and when 

and  when  

The well-known Mittage-Leffler function in defined by following manner [13](See also 
[14,15]) 

,    …(1.9) 

This function further generalized by Prabhakar[16] in the following form [See also [17]]: 

    …(1.10) 

 

In view of extension of parameters, Mittage-Leffler function was also studied by Khan et al [6, p.2, 
eq.(1.9)] 

   …(1.11) 

Where and  and,

if , it takes the 
following form 

   …(1.12) 

Where and  and,

 

Wright Introduced and studied the Taylor-Maclaurin Series [7, Pg.:424]: 

  …(1.13) 

Where is a function satisfying suitable conditions Motivated by the work of Wright[7] E.W. 
Barnes [23] considered the asymptotic expansion of functions in the class which is defined as 
follows: 

( ) ( )
( ) ( )
( ) ( )

! !
! "

#
! !

$

!
" ! !

#
! "!

$$ # %
! ! %

ρ σ ρ σ
λ µ ν

λ µ
φ

ν

∞

=

=
+

∑

!
" # $ # $ # #% & $ '!! " #$ µ ν ρ σ ρ σ+

−∈ ∈ ∈ − − > − ! " #! " # $ ρ σ∈ − − = − ! "∈

! " #!" ρ σδ ρ σ− −< = !! ρ σ− − = − ( )!" #!  λ µ+ − − > !"! δ=

( )! "α

( ) ( )! "

!

!

"# "
!α α

∞

=

=
Γ +∑ ( )( )! "#$ %! ∈ >

( ) ( )
( )!

" #

!
!

!

"
# "

! !
γ
α β

γ
α β

∞

=

=
Γ +∑

( ) ( ) ( )( )! ! "#$ %!#$ %!#$ %! β γ  β γ∈ > > >

( )
( ) ( )
( ) ( ) ( )

! ! !"
! ! ! ! !

#

!
! "!

#
! ! #!

$% $
!

ρµ ρ η
α β ν σ δ

σ

µ η

α βν δ

∞

=

=
Γ +

! ! ! ! ! ! ! " ! #! " # β η δ µ ν ρ σ ∈ > ( )! "# !α≤ +

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ){ }!"# $% &$% &$% &$% &$% &$% &$% &$% ' β η δ µ ν ρ σ > !!δ = =

( )
( ) ( )

( ) ( )
! ! !"
! ! !

# $

!
! "!

! !

#$ #
!!

ρµ ρ η
α β ν σ

σ

µ η

ν α β

∞

=

=
Γ +

∑

! ! ! ! ! ! " ! #! " # β η µ ν ρ σ ∈ > ( )! "# α≤

( ) ( ) ( ) ( ) ( ) ( ) ( ){ }!"# $% &$% &$% &$% &$% &$% &$% ' β η µ ν ρ σ >

( ) ( )
( )!

"
#

!

!

! "
"

!α β

φ
φ

α β

∞

=

∈ =
Γ +∑ ( )( )! "#$ %! β ∈ >

( )!φ



Communications on Applied Nonlinear Analysis 
ISSN: 1074-133X 
Vol 32 No. 8s (2025) 
 

954 https://internationalpubls.com  

  …(1.14) 

For suitable restricted parameters  and  

Recently for suitably restricted sequence , Srivastava [8] introduced and studied a class of 
function [See also [9][10]]: 

 where   …(1.15) 

For the following function recently studied by Virendra  [11, p.13, Eq.(1.9)]: 

   …(1.16) 

Where ; ; ; ; ; and  

To give more extension in view of the parameters, we consider here the following more generalized 
function defined and represented as below. 

  …(1.17) 

Where ; ; when ; and

when and when  

The integral representation of function defined in (1.17) is also obtained here and given as the 
following theorem. 

Theorem-1.1 

For ; ; with 

integral representation of the function defined in (1.17) holds 
true 

  …(1.18) 

Where is the Mittage-Leffler function defined in (1.12) 

Proof: The function defined in (1.17) can be written straight in the following form 

 

( ) ( )
( ) ( )!

"
#$

!
"

#
!

$% $
! " !α β

α β

∞

=

=
+ Γ +

∑ ( )( )! "#$ %! β ∈ >

! !

( ){ } !!
!φ

∞

=

( ) ( )
( ) ( )!

"
# ! !

!

"
!

! #
# " $

$ ! !α β

φ
φ

α β

∞

=

∈ =
+ Γ +

∑ ( )( )! "#$ %! β ∈ >

( ) ( )
!
!!

!
µ

φ =

( ) ( )
( ) ( )! !

"
! !

#

!
!

"
!

## " $
!! $ !

µ
α β δ

µ

δ α β

∞

=

∈ =
+ Γ +

∑

( )!" #µ > ( )!" # > ( )!" #! > ( )!" #! ≥ ( )!" # > ( )!" # > !! ≤

( ) ( ) ( )
( ) ( ) ( )

! ! !
! ! !

"
! !

#

!
! !
"

! !

#
# " $

$ ! ! !
µ δ λ ν δ ν
α β σ ξ

ξ

µ λ

σ α β

∞

=

∈ =
+  +

∑

! " " " !α β µ λ∈
!

" !" #σ −∈ ! ! " #! ν ξ ξ  ν+∈ − − > − !! " #∈ !ξ δ ν− − = −

! "∈ ! "! δ ν ξρ δ ν ξ− −< = !ξ δ ν− − = − ( )!" #!  µ λ+ − − > !"! ρ=

( ) ( ){ }!"# $% &$% '(! " > ! ! ! !α β µ λ∈
!

" !" #σ −∈ ! ! " #!δ ν ξ ξ δ ν+∈ − − ≥ −

! "! δ ν ξδ δ ν ξ− −< = ( )! ! !
! ! ! ! !! " #µ δ λ ν

α β σ ξ∈

( ) ( ){ }! ! ! " ! ! !
! ! ! ! ! !

#

"! ! !" # "$ # ! % " & $% '"
#

µ δ λ ν µ δ λ ν
α β σ ξ α β σ ξ

∞
− − − =

Γ ∫

( )! ! !
! ! !! "µ δ λ ν

α β σ ξ



Communications on Applied Nonlinear Analysis 
ISSN: 1074-133X 
Vol 32 No. 8s (2025) 
 

955 https://internationalpubls.com  

 

Now using the following elementary Gamma integral 

  …(1.19) 

and then changing the order of integration and summation, we have  

 

On interpreting the inner series in view of (1.10) we at once arrive at the desired result it completes 
the proof of theorem (1.1). 

Throughout the paper  where  is the pochhammer’s notation 

2. Generating Functions 

The four generating functions involving the general function define in (1.17) are establish here. 

Theorem-2.1 

For ; ; with 

we have 

 

  …(2.1)  

 

Theorem-2.2 

For ; ; with 

we have 

  …(2.2)  

 

 

( ) ( ) ( )
( ) ( ) ( )

! ! !
! ! !

"

#! !
$
! !

"
! !

"# " $
" ! ! $ !

µ δ λ ν δ ν
α β σ ξ

ξ

µ λ
σ α β

∞

=

 Γ ∈ =  Γ Γ + +  
∑

( )
( )!

"

! " #$
$

$ # % &#
! "

∞
− +−Γ

=
+ ∫ ( ) ( ){ }( )!"#$ %& '%& !($ )! " > 

( ) ( ) ( )
( ) ( )

( )! ! ! "
! ! !

##

"! !
$

!"
#" $ ! !

! !

%&
% $ # & " '"

$ ! !
µ δ λ ν δ ν
α β σ ξ

ξ

µ λ
σ α β

−∞ ∞
− −

=

  ∈ =  Γ Γ +  
∑∫

( )!
"
!

"" !
" !
+ − 

= 
 

( )!"

( ) ( ){ }!"# $% &$% '(! " > ! ! ! !α β µ λ∈
!

" !" #σ −∈ ! ! " #!δ ν ξ ξ δ ν+∈ − − ≥ −

!! δ ν ξδ δ ν ξ− −< =

( ) ( )! ! ! ! ! !
! ! ! ! ! !

"

#
! ! ! !!

!

" !
# " ! $ % # " $ %

!
µ δ λ ν µ δ λ ν
α β σ ξ α β σ ξ

∞

=

+ − 
∈ + =∈ − 

 
∑

( ) ( ){ }!"# $% &$% '(! " > ! ! ! !α β µ λ∈
!

" !" #σ −∈ ! ! " #!δ ν ξ ξ δ ν+∈ − − ≥ −

!! δ ν ξδ δ ν ξ− −< =

( )! ! ! "
! ! !

#

" $
! " !

"
!

!

" !
# " ! $ %

!
µ δ λ ν
α β σ ξ

∞

=

+ − 
∈ + 

 
∑ ( ) ( )! ! ! ! ! !

! ! ! ! ! !
" ! ! ! !
#

! " # $ ! " # $µ δ λ ν µ δ λ ν
α β σ ξ α β σ ξ = ∈ + +∈ − 



Communications on Applied Nonlinear Analysis 
ISSN: 1074-133X 
Vol 32 No. 8s (2025) 
 

956 https://internationalpubls.com  

 

Theorem-2.3 

For ; ; with 

we have 

 …(2.3) 

 

Theorem-2.4 

For ;  

 

  …(2.4) 

Where is a generalized hypergeometric function defined by 

   …(2.5) 

Outlines Of Proofs 

Proof of Theorem 2.1: 

Let left hand side of (2.1) is denoted by i.e. 

 

On using the definition (1.17) and changing the order of summation we have 

 

( ) ( ){ }!"# $% &$% '(! " > ! ! ! !α β µ λ∈
!

" !" #σ −∈ ! ! " #!δ ν ξ ξ δ ν+∈ − − ≥ −

!! δ ν ξδ δ ν ξ− −< =

( )! ! ! " #
! ! !

$

"
! " #!

" #
!

!

" !
# " ! $ %

!
µ δ λ ν
α β σ ξ

∞
+

=

+ 
∈ + + + 

∑ ( ) ( )! ! ! ! ! !
! ! ! ! ! !

" ! ! ! !
#

! " # $ ! " # $µ δ λ ν µ δ λ ν
α β σ ξ α β σ ξ = ∈ − −∈ + 

! !
"# "# $

! "

# $
# $
 β

= =

  
> >  

   
∑ ∑ !!

"
<

( )

( )
! ! !"
! ! !

# " "

"

! !
$

!

" #! $#
"

" %$
# " %

% #
%

&' # (
#

µ δ λ ν
α β σ ξ

θ
θ β

β

∞
=

= = =

=

 
∈ − + 

 

∏
∑ ∑ ∑
∏

( ) ( )
( ) ( ) ( )

( )
( )

! !

"!
# "

! "

# $
# $

% !% %
! "

% "

& '(
% % F %

F %

δ ν

θ βξ

θµ λ
σ α β β

= =

−

 
 =

Γ + + ∑ ∑  +
∑

! "#

( )
( )

( )

( )
!

"

!

#
#
$#

!

" ##! "
! $ $

#$
% #

%

&& '( '
#F F

∞
=

=

=

 
  =
  

∏
∑
∏

( )!! <

!∆

( )! ! !
" ! ! !

#

"
! ! !

!

" !
# " ! $ %

!
µ δ λ ν
α β σ 

∞

=

+ − 
∆ = ∈ + 

 
∑

( ) ( )
( ) ( ) ( )

( )
!

" " ##

!"
" " !
#

" !"

$ # %
! & "& " " "

δ ν

ξ

µ λ

σ α 

∞ ∞

= =

   ∆ =   + + Γ +   
∑ ∑



Communications on Applied Nonlinear Analysis 
ISSN: 1074-133X 
Vol 32 No. 8s (2025) 
 

957 https://internationalpubls.com  

Now on using the binomial expansion , we have 

 

Now on interpreting the resulting series in view of (1.17) arrive at the desired result in (2.1) 

Proof of Theorem 2.2: 

Let left hand side of (2.2) is denoted by i.e 

 

On using the definition (1.17) and changing the order of summation we have 

 

Now in view of the binomial expansion 

 

It takes the following form 

 

On interpreting the resulting series in view of definition (1.17), we at once arrive at the desired result 
in (2.2) 

Proof of Theorem 2.3: 

Let left hand side of (2.3) is denoted by  i.e 

 

On using the definition (1.17) and changing the order of summation we have 

 

Now in view of the binomial expansion 

( ) ( )
!

"
#

! ""

"

!
# #

"

∞
−

=

− =∑

( ) ( )
( ) ( ) ( )!

"

!
#
! !

"
! !

! ! # $ !
δ ν

ξ

µ λ
σ α 

∞

=

∆ =
 + − +

∑

!∆

( )! ! ! "
" ! ! !

#

" $
! " !

"
!

!

" !
# " ! $ %

!
µ δ λ ν
α β σ 

∞

=

+ − 
∆ = ∈ + 

 
∑

( ) ( )
( ) ( ) ( )

( )
( )

!
!

!
" " ! ##

!"
" " !
#

" !"

$ # %
! & "& " " "

δ ν

ξ

µ λ

σ α 

∞ ∞

= =

   ∆ =   + + Γ +   
∑ ∑

( )
( ) ( ) ( )!!

"

# # #
! $ !

! !""

"

!
# # #

"

∞
− −

=

 = − + + ∑

( ) ( )
( ) ( )( )

( ) ( )
( ) ( )( )!

" "

#
! $ $

! !
! ! ! !

" "
! !! !

# #

! ! $ % ! ! ! $ % !
δ ν δ ν

ξ ξ

µ λ µ λ

σ α  σ α 

∞ ∞

= =

 
 ∆ = +
 Γ + − + Γ + + + 
∑ ∑

!∆

( )! ! ! " #
$ ! ! !

%

"
! " #!

" #
!

!

" !
# " ! $ %

!
µ δ λ ν
α β σ 

∞
+

=

+ 
∆ = ∈ + + + 

∑

( ) ( )
( ) ( )( )

( )
( )

! "
! "

#
$ $ ! " %%

!"
" " !

#
" !"

$ # %
! & "" " & "

δ ν

ξ

µ λ

σ α 

+∞ ∞
+

= =

   ∆ =   + + Γ + +   
∑ ∑



Communications on Applied Nonlinear Analysis 
ISSN: 1074-133X 
Vol 32 No. 8s (2025) 
 

958 https://internationalpubls.com  

 

It takes the following form 

 

On interpreting the inner series in view of definition (1.17), we at once arrive at the desired result in 
(2.3) 

Proof of Theorem 2.4: 

Let left hand side of (2.4) is denoted by  i.e, 

 

On using the definition (1.17) and changing the order of summation we have 

 

Now interpreting the inner series into generalized hypergeometric function in view of (2.5) at  
once arrive at the desired result in (2.4) 

3. Mild Extension And Associated Generating Functions 

A mild extension is also considered here, it is defined and represented in the following manner: 

  …(3.1) 

Where ; ; ;

 

The integral representation for the above function defined in (3.1) is given in the following theorem. 

 

  

( )
( ) ( ) ( )! "! "

#

" " "
! " $ !

! !""

"

!
# # #

"

∞
− −++

=

 = − − + +∑

( ) ( )
( ) ( )( )

( ) ( )
( ) ( )( )!

" "

#
$ % %

! !
! ! ! !

" "
! !! !

# #

! $ % ! ! ! $ % ! !
δ ν δ ν

ξ ξ

µ λ µ λ

σ α  σ α 

∞ ∞

= =

 
 ∆ = −
 Γ + − + Γ + + + 
∑ ∑

!∆

( )

( )
! ! !"

# ! ! !
$ " "

"

! !
%

!

" #! $#
"

" %$
# " %

% #
%

&' # (
#

µ δ λ ν
α β σ 

θ
θ β

β

∞
=

= = =

=

 
∆ = ∈ − + 

 

∏
∑ ∑ ∑
∏

( ) ( )

( ) ( )( )

( )

( )! !

!

" "

!

!
#

#
! "

# $
# $

!

%& # %
& & #

"
& %

$& %
$

' (
) & %

& & ) &

δ ν

θ β

ξ

θµ λ

βσ α β = =

∞ ∞
=

= =

=

 
   =   + ∑ ∑  Γ + +
  

∏
∑ ∑

∏

! "#

( ) ( )
( ) ( ) ( )

( )

( ) ( ) ( )

! "
! ! !

#

"

! !
$

!
" "

# #

$

"

! %%
!

#
&%

$ %
$

'' & (
%( % %

δµ δ

α β σ ρ

ρ

µ

σ α β

∞
=

=

=

∈ =
+ Γ +

∏

∏

!" #! " #∈ ! ! ! !!" # $µ α β ∈ ( )!"#"$$$$$$$%! =
!

" !
"# $σ −∈ ( )!"#"$$$$$$$%! = !! " #δ ρ +∈

( )!"#"$$$$$$$%& !"#"$$$$$$$' (! "= =



Communications on Applied Nonlinear Analysis 
ISSN: 1074-133X 
Vol 32 No. 8s (2025) 
 

959 https://internationalpubls.com  

Theorem 3.1: 

For ; ;  ;

 

 

   …(3.2) 

Where is the Mittage-Leffler function defined as follows: 

   …(3.3)  

Proof: Theorem 3.1 can be easily proved following the similar lines as to prove Theorem 1.1 

The four generating functions involving general function defined in (3.1) are establish here and given 
in the following theorem (3.2) to (3.5). 

Theorem 3.2: 

For ; ; ;

, we have 

=  …(3.4) 

 

Theorem 3.3: 

For ; ; ;

, we have 

= …(3.5) 

 

  

!"! " #∈ ! ! ! !!" # $µ α β ∈ ( )!"#"$$$$$$$%! =
!

" !
"# $σ −∈ ( )!"#"$$$$$$$%! = !! " #δ ρ +∈

( )!"#"$$$$$$$%& !"#"$$$$$$$'! "= =

( ) ( )
( ) ( ) ( ) ( ) ( )

( ) ( ) ( )! !"
! # ! ! # !

$

"! !! ! ! !

" " " "

#$ % $& % # ' $ E &' )$
%

µ δ µ δ

α β σ ρ α β σ ρ

∞
− − −∈ =

Γ 

( ) ( )
( ) ( ) ( )!

! " !
! !

" "
# $µ δ

α β σ ρ

( ) ( )
( ) ( ) ( )

( )

( ) ( )

! "
! # !

$

"

%
!

" "

# #

$

"

! %%
!

#
%

$ %
$

&' &
%%

δµ δ

α β σ ρ

ρ

µ

σ α β

∞
=

=

=

=
Γ +

∏
∑
∏

!"! " #∈ ! ! ! !!" # $µ α β ∈ ( )!"#"$$$$$$$%! =
!

" !
"# $σ −∈ ( )!"#"$$$$$$$%! = !! " #δ ρ +∈

( )!"#"$$$$$$$%& !"#"$$$$$$$'! "= =

( ) ( )
( ) ( ) ( )!

! " !
#

$
!% &!! !

" "

#

#

$ #
% & '

#
µ δ

α β σ ρ

∞

=

+ − 
∈ + 

 
∑ ( ) ( )

( ) ( ) ( )!

! " !
!#! $! !

" "
# $µ δ

α β σ ρ
∈ −

!"! " #∈ ! ! ! !!" # $µ α β ∈ ( )!"#"$$$$$$$%! =
!

" !
"# $σ −∈ ( )!"#"$$$$$$$%! = !! " #δ ρ +∈

( )!"#"$$$$$$$%& !"#"$$$$$$$'! "= =

( ) ( )
( ) ( ) ( )! "
! # !

$

" %
!& "'!

"
! !

" "

#

#

$ #
% & '

#
µ δ

α β σ ρ

∞

=

+ − 
∈ + 

 
∑ ( ) ( )

( ) ( ) ( ) ( ) ( )
( ) ( ) ( )! !

! " ! ! " !

# !$! % !$! %
&

! ! ! !

" " " "
# $ # $µ δ µ δ

α β σ ρ α β σ ρ
 ∈ + +∈ −  



Communications on Applied Nonlinear Analysis 
ISSN: 1074-133X 
Vol 32 No. 8s (2025) 
 

960 https://internationalpubls.com  

Theorem 3.4: 

For ; ; ;

 

= …(3.6) 

 

Theorem 3.5: 

For ; ; ;  

; ;  where in the generalized 

hypergeometric function defined in (2.5) 

 

  …(3.7) 

The proof of Theorem (3.2),(3.3),(3.4),(3.5) are developed following the similar lines as to proof of 
(2.1),(2.2),(2.3) and(2.4) respectively in view of the definition of general function defined in (3.1) 

4. Applications 

The functions studied in this paper are very general in nature and unify all the functions related to 
Hurwitz-Lerch Zeta function, therefore many known and New results can be obtained from our main 
results. As an application, some of these have been shown to derive as follows. 

(i) If in the result (1.17), we consider , then it reduces to the known result due to  
Srivastava et al [5, p. 494,eq. (2.4)] 

(ii) If in result (2.1) to (2.4), we apply , then these results reduce to the known result 
studied by Gupta [12, pp. 132-133, eq. (2.9.1) to (2.9.4)]respectively. 

(iii) If in result (3.2) we take , then it reduces to the integral representation for 
Hurwitz-Lerch Zeta function studied by Srivastava et al [5, pp.504, eq. (6.4)] 

(iv) If we take , the result (3.4) to (3.7) reduce to known results for Hurwitz-Zeta function 
due to Gupta [12, pp. 134-135, eq. (2.9.7) to (2.9.10)] 

(v) If we take  , ,  in (2.1), it reduces to the known generating function due to 
Virendra [11, p. 17, eq. (3.1)] at . 

!"! " #∈ ! ! ! !!" # $µ α β ∈ ( )!"#"$$$$$$$%! =
!

" !
"# $σ −∈ ( )!"#"$$$$$$$%! = !! " #δ ρ +∈

( )!"#"$$$$$$$%& !"#"$$$$$$$'! "= =

( ) ( )
( ) ( ) ( )! " #
! $ !

%

"
!& "' #!

" #
! !

" "

#

#

$ #
% & '

#
µ δ

α β σ ρ

∞
+

=

+ 
∈ + + + 

∑ ( ) ( )
( ) ( ) ( ) ( ) ( )

( ) ( ) ( )! !

! " ! ! " !

# !$! % !$! %
&

! ! ! !

" " " "
# $ # $µ δ µ δ

α β σ ρ α β σ ρ
 ∈ − −∈ +  

!"! " #∈ ! ! ! !!" # $µ α β ∈ ( )!"#"$$$$$$$%! =
!

" !
"# $σ −∈ ( )!"#"$$$$$$$%! = !! " #δ ρ +∈

( )!"#"$$$$$$$%& !"#"$$$$$$$'! "= =
! !

"# "# $
! "

# $
# $
 λ

= =

   
> >   

   
∑ ∑ !!

"
< ! "#

( )

( )
( ) ( )

( ) ( )!"
! # !

$ " "

"

! !
%

! !

" "

#

$ %# &%
$

$ '&
% $ '

%

() % *
%

µ δ

α β σ ρ

ν
ν

θ
θ λ

λ

∞
=

= = =

=

 ∈ − + 
 

∏
∑ ∑ ∑
∏

( )

( ) ( ) ( )

( )
( )

! !

!

"

!

#!
#$

!

" #

$ %
$ %

&

'

! ((
$!

$ %F
! %

& (
&

* +,
( - (( - (

δ

θ λ

ρ

µ θ
λσ α β = =

∞
=

−=

=

 
=  +∑ ∑  Γ + +

∏
∑
∏

!"# !α →

!"# !α →

!"# !α →

!α →

µ σ= δ ξ= !! =
!δ =



Communications on Applied Nonlinear Analysis 
ISSN: 1074-133X 
Vol 32 No. 8s (2025) 
 

961 https://internationalpubls.com  

(vi) For  , , , , the theorem 2.4 provides the known results due to 
Virendra [11, p. 17, Theorem (3.2)]. 
 

5. Conclusion 

The general function and its Mild generalization studied in this paper unify all the function related to 
Hurwitz-Lerch Zeta function, its extension as well as the Mittag-Leffler function and their extension 
scattered in the literature. Many properties involving these general functions like The derivatives, the 
fractional calculus and fractional order differential equations and more results related to finite and 
infinite integrals can be established. The present study is worthwhile as the properties of Mittage-
Leffler function and their generalizations are being study [18,19,20]. The Mittag-Leffler function is 
widely used to solve differential equation of fractional order [21,22]. So our proposed study will be 
very useful to the young researchers, who are engaged in the field of study. 

References 

[1] Erd´elyi A., Magnus W., Oberhettinger F. and Tricomi F. G., Higher Transcendental 
Functions, Vol.I, McGraw-Hill Book Company, New York, Toronto and London, (1953).  

[2] Lin S.-D.  and Srivastava H.M., Some families of the Hurwitz-Lerch Zeta functions and 
associated fractional derivative and other integral representations, Appl. Math. Comput. 154 
(2004),pp. 725–733. 

[3] Goyal S.P. and Laddha R.K., On the generalized Zeta function and the generalized Lambert 
function, Ganita Sandesh 11 (1997), pp.  99–108. 

[4] Garg, M., JainK., and  KallaS.L., A further study of general Hurwitz-Lerch zeta function, 
Algebras Groups Geom. 25 (2008), pp. 311–319. 

[5] Srivastava H. M. and Pogany T.K. Integral and computational representations of the extended 
Hurwitz-Lerch Zeta function, Integral Transforms and Special Functions Vol. 22, No. 
7,(2011) ,pp. 487–506  

[6] Khan and Ahmed, On some properties of the generalized Mittag-Leffler function., Springer 
Plus.2013, 2: 337.  

[7] Wright E. M., The asymptotic expansion of integral functions defined by Taylor series. I, 
Philos. Trans. Roy. Soc. London Ser. A Math. Phys. Sci. 238 (1940),pp.  423–451. 

[8] Srivastava H. M., Some families of hybrid -type fractional order Kinetic equations based 
upon the hilfer type and other related operators of fractional derivatives, J. of non-linear and 
Convex Anal, volume 25, (2024) ,pp. 2647- 2669.  

[9] Srivastava H.M., An introductory overview of fractional-calculus operators based upon the 
Fox- Wright and related higher transcendental functions, J. Adv. Engineering and 
computation, 5(3), (2021), pp. 135-166.  

[10] Srivastava H.M., Some parametric and argument variations of the operators of fractional      
calculus and related special functions and integral transformations, J. Nonlinear Convex 
Anal., 22(8), (2021), pp. 1501-1520 . 

[11] Kumar V., On a Generalized Mittag-Leffler Function and Fractional Integrals Fundamental 
Journal of Math.andApp.7(1) (2024),pp. 12-25.  

!! = !! = µ σ= !! = δ ξ=



Communications on Applied Nonlinear Analysis 
ISSN: 1074-133X 
Vol 32 No. 8s (2025) 
 

962 https://internationalpubls.com  

[12] Gupta J., Investigation in generalized special functions and the fractional calculus with it's 
application to univalent and multivalent functions, Ph.D.*thesis, University of Kota, Kota, 
India, 2011.  

[13] Mittag-Leffler GM., Sur la nouvelle fonction E (x).C R Acad. Sci. Paris., 137, (1903), pp. 
554-558. 

[14] Mittag-Leffler GM., Sur la repr'esentation analytiqued ;une branche uniformed' une function 
monogène: cinquième note. Acta. Math., 29, (1905), pp.101-181. 

[15] Wiman A., ÜBer den fundamental satz in der theorie der funcktionen E (x). Acta Math. 29, 
(1905),pp. 191-201. 

[16] Prabhakar T.R., A singular integral equation with a generalized Mittag-Leffler function in the 
kernel. Yokohama Math J.,19, (1971), pp.7-15. 

[17] Srivastava H.M., On an extension of the Mittag-Leffler function. Yokohama Math J., 16, 
(1968), pp. 77-88. 

[18] Srivastava H.M, TomovskiŽ., Fractional calculus with an integral operator containing a 
generalized Mittag-Leffler function in the kernel. Appl Math Comput., 211, (2009), pp.198-
210. 

[19] Shukla A.K., Prajapati J.C., On a generalization of Mittag-Leffler function and its properties. 
J Math Anal. Appl. 336, (2007), pp.797-811. 

[20] Tariq O. Salim, Ahmad W. Faraj, A generalization of mittag-leffler function and integral 
operator associated with fractional calculus, Journal of Fract. Calc. and Appl., Vol. 3. (2012), 
No. 5, pp. 1 - 13. 

[21] .Jaimini. B. B. and Gupta J., On certain fractional differential equations involving generalized  
multivariable Mittag-Leffler function, Note di Matematica, 32 (2013), pp.141-156. 

[22] Jaimini B. B., Sharma M., Suthar D. L. and Purohit S. D., On multi-index Mittag-Leffler 
function of several variables and fractional differential equations, Journal of Mathematics, 
(2021), Art. ID 5458037, 8 pp. 

[23] Barnes E.W., The asymptotic expansion of integral functions defined by Taylor’s series, 
Philos.Trans. Roy. Soc. London Ser.A Math. Phys. Sci. 206 (1906), pp.249–297.