Volume 3, Number 2, 2020, Pages 297-315 https://doi.org/10.34198/ejms.3220.297315
Received: November 13, 2019; Accepted: March 3, 2020
2010 Mathematics Subject Classification: 33B99, 33C15, 33C45, 33C60.
Keywords and phrases: Exton’s double hypergeometric function, Hurwitz-Lerch zeta functions, Mellin- Barnes integrals, operational connections, summation formulas.
Copyright © 2020 Maged G. Bin-Saad et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium,
Hurwitz Zeta Function of Two Variables and Associated Properties
M. A. Pathan1, Maged G. Bin-Saad2 and J. A. Younis3
1 Centre for Mathematical Sciences, Peechi P.O., Kerala-680653, India e-mail: [email protected]
2 Department of Mathematics, Aden University, Kohrmakssar P.O. Box 6014, Yemen e-mail: [email protected]
3 Department of Mathematics, Aden University, Kohrmakssar P.O. Box 6014, Yemen e-mail: [email protected]
Abstract
The main objective of this work is to introduce a new generalization of Hurwitz-Lerch zeta function of two variables. Also, we investigate several interesting properties such as integral representations, operational connections and summation formulas.
1. Introduction
The Exton hypergeometric function HEA::GB;;CM;;DN of two variables [9, p.339 (13)] is defined by
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
x y
n m g e
d c b H a
N M G E
D C B A D C B A
N M G
E ,
;
;
; :
;
;
;
; :
; :
;
; :
[ ( ) ] [ ( ) ] ( ) [ ] ( ) [ ]
[ ( ) ] [ ( ) ] ( [ ) ] ( ) [ ]
,(
0 1,0 1)
,!
0 !
, 2
2 < < < <
=
∑
∞= + +
+
+ x y
j y i x n m g
e
d c b
a i j
j
i E i j G i j M i N j
D j C i j B i j A i
(1.1)
where for the sake of convenience (in the contracted notation),
( )
aA denotes the array of A parameters given by a1,a2,...,aA in the contracted notation of Slater [17, p.54]. The symbol ∆(
M;a)
represents an array of M parametersM M a M
a M
a 1
..., 1,
, + + −
[21, p.47, pp.213].
The special case A= E =0 in (1.1) reduces to the Kampé de Feriet double hypergeometric function [22, p.423 (26), see also[23, p.23 (1.2,1.3)]
( ) ( ) [ ( ) ] ( ) [ ] ( ) [ ]
[ ( ) ] ( [ ) ] ( ) [ ]
! ! , ,,
0 ,
;
; : 0
;
; : 0
;
;
;
; j
y i x n m g
d c b
y x H
y x F
j i j
i G i j M i N j
D j C i j B i D
C B
N M G D
C B
N M
G
∑
∞= +
= +
=
(
0< x <1,0 < y <1)
. (1.2) Similarly, when B=G = 0, then (1.1) reduce to the double hypergeometric function due to Exton [10, p.137 (1.2)]( ) [ ( ) ] [ ( ) ] ( ) [ ]
[ ( ) ] [ ( ) ] ( ) [ ]
! ! , ,0
, 2
; 2 :
;
: j
y i x n m e
d c a
y x X
j i j
i E i j M i N j
D j C i j A i D
C A
N M
E
∑
∞= +
= +
(
0< x <1, 0< y <1)
(1.3) which, for A= D = E = N =0 and y = 0 we shall obtain the generalized hypergeometric function CFM defined by [20, p.19 (23)]( )
( ) [ ( ) ]
( )
[ ]
∑
∞=
=
0
! ,
;
;
i
i M i C i M
C M
C i
x m x c
m
F c (1.4)
where C and M are positive integers or zero; mM ≠0, −1, −2,.... The Hurwitz-Lerch zeta function Φ
(
z, s,a)
is defined by (see [8, 19])( )
( ) ( ( ) )
∑
∞=
− ∈ < > =
+ ∈
= Φ
0
0; when 1; 1when 1.
\ ,
, ,
i
s i
z s
z s
a i a a z
s
z CZ C R
(1.5) Recently many researchers investigated and studied various generalizations of the Hurwitz-Lerch zeta function Φ
(
z,s,a)
. The latest development and properties of suchgeneralizations is found in the recent work of various researchers (see e.g., [2, 3, 4, 5, 6, 7, 14, 16, 18, 19, 24, 25]). Very recently, Pathan and Daman [16] introduced another generalization in terms of double series representation:
( ) ( ) ( ) ( ) ( )
( ) ( ) ( )
∑
∞ν = µ λ γ β
α γ ν + +
µ λ β
= α φ
0 , ,
; , :
;
, ,
! , !
, ,
j i
j i s j
i
j j i q i
p
j t i z qj pi a a
s t
z (1.6)
Where γ,ν,a ≠
{
0, −1, −2,...}
, s∈C, p,q >0, R( )
s >0 when z , t <1 and(
γ+ν+s−α−β−λ−µ)
>0R when z , t =1.
Motivated essentially by various extensions of the Hurwitz-Lerch zeta function, we introduce a new extension of the generalized Hurwitz-Lerch zeta function of two variables defined as follows:
(
z t s a)
w v r n m l k h q
p, Φα::β;;γ;;λ,,µ::ν;;ρ;;σ , , ,
[ ( ) ] [ ( ) ] ( ) [ ] ( ) [ ]
[ ( ) ] [ ( ) ] [ ( ) ] ( ) [ ]
! !( )
,0
, 2
2
s j i j
i n i j r i j v i w j
m j l i j k i j h i
qj pi a j i
t z
+ σ +
ρ ν
µ
λ γ β
=
∑
∞ α= + +
+
+ (1.7)
where α1,...,αh,β1,...,βk, γ1,..., γl,λ1,..., λm ∈C; µ1,...,µn,ν1,..., νr, ρ1,...,
v,
ρ σ1,...,σw,a∈C\Z0−; p,q > 0; s∈C, R
( )
s >0 when z , t <1; and(
µ1+...+µn +ν1+...+νr +ρ1+...+ρv +σ1+...+σw +s −α1−...−αhR −β1−
)
0...
...
...−βk −γ1− −γl −λ1− −λm > when z , t =1. Some special cases of (1.7) are given below.
In the case when h = k =n =r =0,l = m= 2 and v = w=1, then (1.7) reduces to equation (11) of [16] which is given in (1.6). In the case when k = r =0, then we get
( ) [ ( ) ] [ ] ( ) ( ) [ ]
[ ( ) ] [ ( ) ] ( ) [ ]
! !( )
, ,, ,
0
, 2
; 2 : ,
; :
; : ,
; : ,
s j i j
i n i j v i w j
m j l i j h i w
v n m l h q p
qj pi a j i
t a z
s t
z µ ρ σ + +
λ γ
= α
Φ
∑
∞= +
σ + ρ µ λ γ
α (1.8)
where α1,...,αh, γ1,..., γl, λ1,...,λm ∈C; µ1,...,µn,ρ1,..., ρv,σ1,...,σw, a∈
;
\Z0−
C p,q >0; s∈C, R
( )
s > 0 when z , t <1; and R(
µ1+...+µn +ρ1+h w
v +σ + +σ +s−α − −α ρ
+ ... ...
... 1 1 −γ1−...−γl −λ1−...−λm
)
> 0 when z , .=1
t Similarly, if h = n =0, we have
( ) [ ( ) ] ( ) [ ] ( ) [ ]
[ ( ) ] [ ( ) ] ( ) [ ]
! !( )
, ,, ,
0 ,
;
; ,
;
;
;
; ,
;
; ,
s j i j
i r i j v i w j
m j l i j k i w
v r m l k q p
qj pi a j i
t a z
s t
z ν ρ σ + +
λ γ
= β
Φ
∑
∞= +
σ + ρ ν λ γ
β (1.9)
where β1,...,βk, γ1,..., γl,λ1,...,λm ∈C; ν1,..., νr,ρ1,...,ρv,σ1,..., σw, a∈
;
\Z0−
C p, q >0; s∈C, R
( )
s > 0 when z , t <1; and R(
ν1+...+νr +ρ1+...+1 1+...+σ + −β σ
+
ρv w s −...−βk −γ1−...−γl −λ1−...−λm
)
>0 when z , t ,=1 which, for l = m= v = w=0, we obtain the following:
( ) [ ( ) ]
[ ( ) ]
! !( )
, ,, ,
0 , ,
, ,
s j i j
i r i j
j k i r
k q p
qj pi a j i
t a z
s t
z ν + +
= β
Φ
∑
∞= +
ν +
β (1.10)
where β1,...,βk ∈C; ν1,...,νr, a∈C\Z0−; p,q >0; s∈C, R
( )
s >0 when; 1 , t <
z and R
(
ν1+...+νr + s−β1−...−βk)
> 0 when z , t =1. Now, if in (1.10), we let p = q =1, then after some simplification, we have( ) [ ( ) ]
[ ( ) ] ( )
( )
,! ,
, ,
0 ,
, 1 , 1
s N
N r N
k N r
k N a N
t a z
s t
z +
+ ν
= β
Φ
∑
∞= ν
β (1.11)
where β1,...,βk ∈C; ν1,...,νr, a∈C\Z0−; s∈C, R
( )
s > 0 when z , t <1; and(
ν1+...+νr +s−β1−...−βk)
>0R when z , t =1. By making suitable
adjustments in the number of numerator and denominator parameters of (1.11), it gives the following relationships:
(
, , ,) (
, ,)
,1 1 ,
1 Φβ z t s a =φ∗β z +t s a (1.12)
(
, , ,) (
, ,)
,11 1 ,
1 Φ z t s a =Φ z +t s a (1.13)
(
,)
, ,2, , 1 2
1 1
1 1 ,
1 s a = ζ s a
Φ (1.14)
where φ∗β, Φ and ζ
(
s,a)
are the generalized Hurwitz-Lerch zeta function (see [11, p.100, (1.5)]), the Hurwitz-Lerch zeta function defined by (1.5) and the Hurwitz zeta function (see [8, p.24, (1)]), respectively.2. Integral Representations
Theorem 2.1. The following integral representation holds true for
(
, , ,)
:;
; : ,
;
; :
;
; : ,
;
; :
,q h k l m n r v w z t s a
p Φα β γ λ µ νρ σ
(
z t s a)
w v r n m l k h q
p, Φα::β;;γ;;λ,,µ::ν;;ρ;;σ , , ,
( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
: ; ; ; , ,;
;
; 1 :
0
;
; :
;
; :
1e H ze te dx
s x
qx px w
v r n
m l k h m l k h
w v r n ax
∫
∞ s− − µα νβ ργ λσ − − = Γ (2.1)
(
min{
R( ) ( )
s ,R a}
>0 when z ≤1(
z ≠1)
, t ≤1(
t ≠1)
; R( )
s >1 when z =1,)
.=1 t
Proof. Using Eulerian integral formula (see, e.g., [8]):
( )
( ) ∫∞ − −( + + ) ( { ( ) ( ) } )
− > ∈
= Γ +
+ 1 0 1 , min , 0; , 0
j N i a
s dx
e s x
j i
a s s a i j x R R (2.2)
in (1.7) and then, changing the order of the integral and the summation, and using definition (1.1), yields the proof of (2.1).
Similarly, by means of the relation (2.2), we can give the following corollary.
Corollary 2.1. The following integral representations for p,qΦhα::lγ;;mλ,,nµ::vρ;;wσ
(
z,t,s,a)
,(
z t s a)
w v r m l k q
p, Φβ;;γ;;λ,,ν;;ρ;;σ , , , and p,qΦβk,,νr
(
z,t, s,a)
in (1.8), (1.9) and (1.10) holds true:(
z t s a)
w v n m l h q
p, Φα::γ;;λ,,µ::ρ;;σ , , ,
( )
( ) ( ) ( ) ( ) ( ) ( )
∫
∞ − − µα ργ λσ − − = Γ
0
; : ;
1 : , ,
;
; :
;
; 1 :
dx te ze X
e s x
qx px w
v n
m l m h
l hv w ax n
s (2.3)
(
z t s a)
w v r m l k q
p, Φβ;;γ;;λ,,ν;;ρ;;σ , , ,
( ) ( ) ( ) ( )
( ) ( ) ( )
∫
∞ − − νβ ργ λσ − − = Γ
0
;
;;
1 ; ,
;
;
;
;
; 1 ;
dx te ze F
e s x
qx px w
v r
m l m k
l kv w ax r
s (2.4)
and
( )
( )
( )
∫
∞ − −( )
− −ν
β
+
ν β
= Γ
Φ 0
1 ,
, ,
; 1 ;
, ,
, x e F ze te dx
a s s t
z px qx
r k r ax k s r
k q
p (2.5)
(
min{
R( ) ( )
s ,R a}
>0 when z ≤1(
z ≠1)
, t ≤1(
t ≠1)
; R( )
s >1 when z =1,)
.=1 t
Theorem 2.2. Each of the following integrals for p,qΦhα::kβ;;lγ;;mλ,,nµ::rν;;vρ;;wσ
(
z,t, s, a)
holds true when h = n,k = r and l =v, respectively(
z t s a)
w v r n m l k n q
p, Φα::β;;γ;;λ,,µ::ν;;ρ;;σ , , ,
( ) ( )
( )
α Γ( ) (
α Γ µ −α)
Γ(
µ −α) ∫ ∫ α − α − ( − )µ−α − Γ
µ Γ µ
= Γ 1
0 1 0
1 1 1 1
1 1 1 1
1 x 1 xnn 1 x 1 1
n n n
n L L
L L
L
(
1) (
2 2 , 1 , ,)
1 ,1
;
; ,
;
;
;
; ,
;
; ,
1 1 k l m r v w n n n
q
p x x z x x t s a dx dx
x n n L L L
L − µ −α − Φβ γ λ ν ρσ
×
( ) ( ) ( )
(
R αu > 0,R µu −αu >0, u =1, 2,..., n)
, (2.6)(
z t s a)
w v k n m l k h q
p, Φα::β;;γ;;λ,,µ::ν;;ρ;;σ , , ,
( ) ( )
( )
β Γ( ) (
β Γν −β)
Γ(
ν −β) ∫π ∫π( )β − ( )ν −β − Γ
ν Γ ν
= Γ 2
0 2
0 2
1 2 1
2 1 2 1
1 1 1
1 sin 1 cos 1 1
2 x x
k k k
k k
L L L
L
(
sin) (
cos)
: ; , : ;(
sin2 1 sin2 ,; : ,
; : 2 ,
2 1 2 2 1
z x x
x
xk k k k k pq h l m n v w L k
L β − ν −β − Φα γ λ µ ρ σ
×
)
,, , sin
sin2x1L 2xkt s a dx1Ldxk
( ) ( ) ( )
(
R βu > 0, R νu −βu > 0, u =1,2,..., k)
, (2.7)(
z t s a)
w l r n m l k h q
p, Φα::β;;γ;;λ,,µ::ν;;ρ;;σ , , ,
( ) ( )
( ) ( ) ( ) ( ) ∫∞ ∫∞( ) ( )ρ
− γ ρ
− γ
+ γ +
− ρ Γ γ
− ρ Γ γ Γ γ Γ
ρ Γ ρ
= Γ
0 0
1 1
1 1 1
1 1
1
1
1 1
1
l l
l l l
l l
l
x x x
x L
L L L
L
(
1 1) (
1)
, , , 1 ,; 1 : ,
; :
; : ,
; :
, l
l l w
r n m k h q
p z t s a dx dx
x x x
x L L
+ Φ +
× α β λ µ ν σ
( ) ( ) ( )
(
R γu >0,R ρu −γu >0, u =1, 2,...,l)
. (2.8) Proof. To prove each of the integral representations from (2.6) to (2.8), it is enough to substitute the expression of the generalized Hurwitz-Lerch zeta function in each integrand and then to change the order of the integral and the summation, and finally taking into account the following Eulerian integrals (see, e.g., [8, p. 9-11], [19, Section 1.1]:(
a,b)
=∫
01x −1(
1−x)
−1dx,B a b (2.9)
(
a,b)
=2∫
0π2(
sinx)
2 −1(
cosx)
2 −1dx,B a b (2.10)
( )
( )
∫
∞ + − += 0
1
, 1
, dx
x b x
a
B a b
a
(2.11)
( ) ( )
(
R a > 0, R b >0)
.Theorem 2.3. The following integral representations holds true:
(
z t s a)
w v r n m l k n q
p, Φα::β;;γ;;λ,,µ::ν;;ρ;;σ , , ,
( ) ( )
( )
α Γ( ) (
α Γµ −α)
Γ(
µ −α) ( )
Γ∫
∞∫
∞ −(α + +α + )Γ
µ Γ µ
= Γ
0 0
1 1 1
1 1x1 x ay
n n n
n e n n
s L L
L L
L
(
1− − 1)
µ1 1 1(
1− −)
µ −α −1 −1× e x −α − L e xn n n ys
( ) ( ) ( )
( ) ( ) ( )
; ; ; ( ), ( ) ,;
;
;
2 1 2
;
;;
; ze 1 te 1 dx dx dy
F x x py x x qy n
w v r
m l m k
l kv w
r L n L n L
σ ρ ν
λ γ
× β − + + + − + + +
( ) ( )
{ } ( ) ( ) ( )
(
min R s ,R a >0;R αu > 0, R µu −αu >0, u =1, 2,..., n)
, (2.12)(
z t s a)
w v k n m l k h q
p, Φα::β;;γ;;λ,,µ::ν;;ρ;;σ , , ,
( ) ( )
( ) ( ) ( ) ( ) ( ) ∫∞ ∫∞ ( )ν ( )ν
−
− −
− β β
+ Γ +
β
− ν Γ β
− ν Γ β Γ β Γ
ν Γ ν
= Γ
0 0
1 1 1 1
1 1
1 1
1
1
1 1
1
k k
k ay s k k
k k
k
x x
e y x x
s L
L L L
L
L
( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( )
+ +
+ σ +
ρ µ
λ γ
× α ν − ν ν − ν t
x x
e x z x
x x
e x X x
k
k k
k qy k
k py w
v n
m l m h
l hv w
n 1 1
, 1
; 1
; :
;
; :
1
1 1
1 1
; 1 : ;
: L
L L
L
1 dx dy, dx L k
×
( ) ( )
{ } ( ) ( ) ( )
(
min R s ,R a >0; R βu >0,R νu −βu >0, u =1, 2,..., k)
. (2.13) Proof. Using (2.2) and the Eulerian Beta function formula [8, p.11 (24)](
,) ( ) (
1)
,( ( )
0,( )
0)
,0
1 > >
−
=
∫
∞ e− e− − dx a bb a
B x a x b R R
which implies that
(
z t s a)
w v r n m l k n q
p, Φα::β;;γ;;λ,,µ::ν;;ρ;;σ , , ,
( ) ( )
( )
α Γ( ) (
α Γµ −α)
Γ(
µ −α) ( )
Γ∫
∞∫
∞(
− −)
µ −α − Γµ Γ µ
= Γ
0 0
1 1
1 1
1 1 x1 1 1
n n n
n e
s L
L L
L
(
−e−xn)
µn−αn− e−α x e−αnxnys− e−ay×L1 1 11L 1
[ ( ) ] ( ) [ ] ( ) [ ]
[ ( ) ] [ ( ) ] ( ) [ ] (
( )) (
( ))
∑
∞=
+ + +
− +
+ +
− +
+
σ ρ ν
λ γ
× β
0 ,
2 2
!
!
1 1
j i
j qy x x i py x x w j
v i j r i
m j l i j k i
j i
te
ze L n L n
1 dx dy. dx L n
×
in view of definition (1.2), immediately yields the first equality (2.12). By employing formulas (2.2) and (2.11), and exploiting the same procedure leading to (2.12) one can derive the second equality (2.13).
Now, by using the contour integral [8, p.14 (4)]
( )
= −( )
π∫
∞( )+( )
− − −( )
− ≤ πΓ 0 1 , arg ,
sin 2
1 x e dx x
s
s i s x (2.14)
we can give the following theorem without proof.
Theorem 2.4. Let R
( )
a >0 and arg( )
−x ≤π. Then(
z t s a)
w v r n m l k h q
p, Φα::β;;γ;;λ,,µ::ν;;ρ;;σ , , ,
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
∫
∞( )+ − − − µα νβ ργ σλ − − π− Γ
= − 0 1 :: ;;;; , .
;
;
; :
;
;
; : 2
1 x e H ze te dx
i
s px qx
w v r n
m l k m h
l k hr v w ax n
s
(2.15) Theorem 2.5. The following Mellin-Barnes contour integral representation of (1.7) holds true:
(
z t s a)
w v r n m l k h q
p, Φα::β;;γ;;λ,,µ::ν;;ρ;;σ , , ,
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ∫ ∫
∏ ∏ ∏ ∏
∏ ∏ ∏ ∏
= = = =
= = = =
λ Γ γ
Γ β Γ α Γ π
σ Γ ρ Γ ν
Γ µ
Γ
−
=
z t
C C
h u
k u
l u
m u
u u
u u
n u
r u
v u
w u
u u
u u
1 1 1 1
2
1 1 1 1
4
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
∏ ∏ ∏ ∏
∏ ∏ ∏ ∏
= = = =
= = = =
+ σ Γ +
ρ Γ +
+ ν Γ +
+ µ Γ
+ λ Γ +
γ Γ +
+ β Γ +
+ α Γ
− Γ
− Γ
× n
u
r u
v u
w u
u u
u u
h u
k u
l u
m u
u u
u u
y x
y x y
x
y x
y x y
x y
x
1 1 1 1
1 1 1 1
2 2
( )
[ ] ( ) ( )
( )
[
a px qy 1]
dxdy, tz qy px a
s y x s
+ + + Γ
−
− +
+
× Γ (2.16)
( ) ( )
(
arg− z < π, arg−t <π)
.Proof. If we evaluate the integral as a sum of the residues by calculus of residues at the simple poles of Γ
( )
−x at the points x = −i(
i∈N0)
and Γ( )
− y at the points(
∈N0)
,−
= j j
y we immediately find the following series expansion:
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
∏ ∏ ∏ ∏
∏ ∏ ∏ ∏
= = = =
= = = =
λ Γ γ
Γ β Γ α
Γ
σ Γ ρ
Γ ν
Γ µ
Γ
h u
k u
l u
m u
u u
u u
n u
r u
v u
w u
u u
u u
1 1 1 1
1 1 1 1
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
∑ ∏ ∏ ∏ ∏
∏ ∏ ∏ ∏
∞
=
= = = =
= = = =
+ σ Γ +
ρ Γ +
+ ν Γ +
+ µ Γ
+ λ Γ + γ Γ +
+ β Γ +
+ α Γ
×
0 ,
1 1 1 1
1 1 1 1
2 2
j i
n u
r u
v u
w u
u u
u u
h u
k u
l u
m u
u u
u u
j i
j i j
i
j i
j i j
i
( )
,!
! s
j i
qj pi a j i
t z
+
× +
which is just the p,qΦhα::kβ;;l
