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ISSN 2229 – 5046
INTEGRAL TRANSFORMS OF H-FUNCTION
OF TWO VARIABLES INVOLVING GENERALIZED M-SERIES
B. SATYANARAYANA
1, ALEM MEBRAHTU
2, Y. PRAGATHI KUMAR*
31
Department of Mathematics,
Acharya Nagarjuna University, Nagarjuna Nagar-522 510, India.
2
Department of Physics, Adigrat University, Adigrat, Ethiopia
3
Department of Mathematics, Adigrat university, Adigrat, Ethiopia.
(Received On: 24-09-15; Revised & Accepted On: 05-11-15)
ABSTRACT
T
he object of this paper is to establish some integral transforms involving product of generalized M-series and H-function of two variables. Some special cases have also been derived.Key words: H-function of two variables, Mellin transform, Laplace transform and Generalized M-series.
1. INTRODUCTION
Recently, the Mellin-Barnes type contour integral of H-function of two variables evaluated by P.C.Srinivas [9]. In the present paper we establish the Mellin transform and Laplace transform of H-function of two variables with generalized M-series.
We shall utilized following formulae in present investigation. The H-function of one variable given by Charles Fox [2]
[ ]
(
,
)
1,
,
,
[ ]
,
,
(
,
)
1,
a
A
x
j
j
p
m n
m n
H x
H
x
H
p q
p q
b
B
j
j
q
=
=
s
L
1
F(s)x
,
i
-1, x
0
2 i
π
ds
=
∫
=
≠
(1.1)Where
(
)
(1
)
1
1
F(s)
(1
)
(
)
1
1
j j j j
j j j j
m
n
b
B s
a
A s
j
j
q
p
b
B s
a
A s
j
m
j
n
Γ
−
Γ −
+
∏
∏
=
=
=
Γ − +
Γ
−
∏
∏
= +
= +
An empty product is interpreted as unity; m,n,p and q are integers satistying 0 ≤ m ≤ q, 0 ≤ n ≤ p; Aj(j = 1,…..,p),
Bj(j = 1,…..,q) are positive numbers and aj(j = 1,..,p), bj(j = 1,…..,q) are complex numbers such that no poles of
Γ(bj-Bjs), j = 1,… ..,m coincide with any pole of Γ(1-aj+Ajs), j = 1,…..,n i.e Aj(bk+N) ≠ Bk(aj-M-1), where
k= 1,…..,m; j = 1,…..,n; M = 0, 1, 2, …...
Corresponding Author: Y. Pragathi Kumar*
3The contour L runs from σ-i∞ to σ+i∞, σ be a positive constant such that the points h
,
1,..., ;
1,..., ;
hb
N
s
h
m N
n
B
+
=
=
=
which are the poles of Γ(bj-Bjs), j = 1,…..,m lie to the right and the points , 1,..., ; 0,1,...
1
= =
− −
= j mn M
a M a s
j j
which are the poles of Γ(1-aj+Ajs), j = 1,…..,n lie to the left of L.
The H-function of two variables given by Prasad and Gupta[6]
(
;
,
)
: (
,
)
; (
,
)
1,
1,
1,
,
: ,
; ,
[ , ]
,
: ,
; ,
(
;
,
)
: (
,
)
; (
,
)
1,
1,
1,
j
a
A
c
C
e
E
x
j
j
P
j
j
p
j
j
u
M N m n g h
H x y
H
P Q p q u v
b
B
d
D
f
F
x
j
j
j
Q
j
j
q
j
j
v
α
σ
γ
δ
β
η
=
1
( )
( ) ( , )
,
1
1
2
2
(2
)
1 2
s t
s
t
s t x y ds dt
i
i
L L
ϕ
ϕ
ψ
π
=
∫ ∫
= −
(1.2) where x , y ≠ 0,
∏ +
= Γ − + =∏+ Γ − ∏
= Γ − ∏= Γ − + =
q
m j
p
n j
s j C j c s
j D j d m
j
n
j
s j C j c s
j D j d
s
1 1
) (
) 1
(
1 1
) 1
( ) (
) ( 1
φ
( )
∏
+
=
=
∏
+
−
Γ
−
+
Γ
∏
=
−
+
Γ
∏
=
−
Γ
=
v
g
j
u
h
j
t
j
E
j
e
t
j
F
j
f
h
j
t
j
E
j
e
g
j
t
j
F
j
f
t
1
1
1
1
1
1
2
φ
( )
∏
+
=
=
∏
+
−
−
Γ
−
+
+
Γ
∏
=
−
+
+
Γ
∏
=
−
−
Γ
=
Q
M
j
P
N
j
t
j
A
s
j
j
a
t
j
B
s
j
j
b
N
j
t
j
A
s
j
j
a
M
j
t
j
B
s
j
j
b
t
s
1
1
1
1
1
1
,
α
β
α
β
ψ
,
where M, N, P, Q, m, n, p, q, g, h, u, v are all non negative integers such that 0 ≤ N ≤ P, Q ≥ 0, 0 ≤ m ≤ q, 0 ≤ n ≤ q, 0
≤ g ≤ v, 0 ≤ h ≤ u and αj, βj, Aj, Bj, Cj, Dj, Ej, Fj are all positive. The sequence of parameters (aP), (bQ), (cp), (dq), (eu)
and (fv) are so restricted that none of the poles of the integrand coincide.
The contour L1 lies in the complex s-plane and runs from -i∞ to +i∞ with loops, if necessary, to ensure that the poles
of Γ(dj - Djs), (j = 1, 2, ….. , m), lie to the right of the path; and those of Γ(1 – cj + Cjs), (j = 1, 2,….., n) and
Γ(1 – aj + αjs + Ajt), (j = 1, 2,….., N) lie to the left of the path.
Also the contour L2 lies in the complex t-plane running from -i∞ to +i∞ with loops, if necessary, to ensure that the
poles of Γ(fj - Fjt), (j = 1, 2,….., g), lie to the right of the path; and those of Γ(1 – ej + Ej t), (j = 1, 2,….., h) and
Γ(1 – aj + αj s + Aj t), (j = 1, 2,….., N) lie to the left of the path. All poles of the integrand are simple poles.
The Mellin transform of the function f(x) is defined as
{
( );
}
1
( )
, Re( )
0
0
s
M f x s
x
f x d x
s
∞
−
=
∫
>
(1.3)If Laplce transform of f(t) is F(p) and G(s) is Mellin transform of f(t), then
(
)
( )
(
1)
!
0
s
p
F p
G s
s
s
∞ −
=
∑
+
According Erdelyi [1, p.307]
1
1
( )
( )
2
0
c i
s
s
x
g s x
ds dx
g s
i
c i
π
∞
−
+ ∞
−
=
∫
∫
− ∞
(1.5)
The Generalized M-Series is defined by Sharma and Renu[8] as
, ,
1 1
( ,....,
p; ,....,
q; )
p q p q
M
M a
a b
b z
α β α β
=
1
0 1
( ) ....(
)
( ) ....( )
(
)
k
k p k
k k q k
a
a
z
b
b
α
k
β
∞
=
=
Γ
+
∑
z,α,β∈ , ℜ(α) > 0 (1.6)Series is convergent for all z if q ≥ p, it is convergent for |z| < 1 if p = q+1 and divergent if p > q+1. Where p = q+1 and |z| = 1, the series convergent in some case. Let
∑
∑
= =
−
= q
j j p
j
j b
a
1 1
β
It can be shown that when p = q+1 the series is absolutely convergent for |z| = 1 if ℜ(β) < 0. Conditionally convergent for z = -1 if 0 ≤ ℜ(β) ≤ 1 and divergent for |z|=1 if 1 ≤ ℜ(β).
2. MAIN RESULTS
Theorem 1: Prove that
,
1 1
(
;
,
)
: (
,
)
; (
,
)
1,
1,
1,
,
:
,
;
,
1
2
1 1
2
2
( ,...., ; ,....,
;
)
;
,
:
,
;
,
(
;
,
)
: (
,
)
; (
,
)
1 1
2
2
1,
1,
1,
1
2
l m
l m
a
A
c
C
e
E
j
j
j
P
j
j
p
j
j
p
M N m n m
n
x
M M a
a b
b ax H
s
P Q p q p
q
b
B
d
D
f
F
x
j
j
j
Q
j
j
q
j
j
q
α β
λ
α
σ
γ
δ
β
η
( )
1 1
0
1 1
( ) ( ( ) , ( / ) ) ,
, 1,
1 1 2 1 2
,
( ) 1 2 1 2
( ) ( ( ) , ( / ) ) ,
1,
m l
j j k s k
j j
l m
k
j j
j j
s k
b a k a A A
M m n N n m j j j j P
a
H
P p q Q q p s k
k
a b k b B B
j j j j Q
λ δ
λ
σ
α
σ δ
δ
δ
η
η
γ
λ
δ
α
β
β
σ δ
δ
+
∞ −
= =
=
= =
+
+ + + + + − + −
= + + + +
+
Γ +
+ + −
∏
∏
∑
∏
∏
2
2
(1
(
)
, ( / )
)
, (
,
)
, (1
(
)
, ( / )
)
, (
,
)
1,
1,
1,
1,
2
1
2
1
1
(1
(
)
, ( / )
)
, (
,
)
, (1
(
)
, ( / )
)
, (
,
)
1,
1,
1,
1,
2
1
2
1
1
s
k
s
k
f
F
F
c
C
f
F
F
c
C
j
j
j
n
j
j
n
j
j
j n
p
j
j n
p
s
k
s
k
e
E
E
d
D
e
E
E
d
D
j
j
j
m
j
j
m
j
j
j m
q
j
j m
q
λ
σ δ
λ
σ δ
δ
δ
λ
σ δ
λ
σ δ
δ
δ
+
+
−
−
−
−
+
+
+
+
−
−
−
−
+
+
Provided σ > 0, δ > 0, λ is complex number αj - (σ/δ) Aj > 0 for j = 1, 2, - - -, P
βj - (σ/δ) Bj > 0 for j = 1, 2, - - -, Q
|arg γ| < (1/2) π∆1, |arg η| < (1/2) π∆2
where ∆1 =
∑
∑
∑
∑
∑
∑
∑
∑
= = + = = +
+ = = + = =
− + −
+ −
+
− 1 1
1
1 1
1
1 1 1 1
1 1
1 1
m
j
q
m j
n
j
p
n j
j j
j j
Q
M j
j M
j j P
N j
j N
j
j α β β D D C C
α
and
∆
2 =∑
∑
∑
∑
∑
∑
∑
∑
= = + = = +
+ = =
+ = =
−
+
−
+
−
+
−
2 22
2 2
2
1 1 1 1
1 1
1 1
m
j
q
m j
n
j
p
n j
j j
j j
Q
M j
j M
j j P
N j
j N
j
j
A
B
B
F
F
E
E
A
and
Re
(
1
)
<
0
−
+
j j
A
a
Re >0 + j j B b
s δ for j = 1, 2,….., M
Re >0
+ + i i j j D d B b
s δ σ for i = 1, 2,….., m1 ; for j = 1, 2,….., m2
Re
(
1
)
(
1
)
>
0
−
+
−
+
i i j jC
c
E
e
s
δ
σ
for i = 1, 2,….., n1 ; for j = 1, 2,….., n2Proof: To prove this theorem, take f(x) as
2 , 1 ) , ( ; 1 , 1 ) , ( : , 1 ) , ; ( 2 , 1 ) , ( ; 1 , 1 ) , ( : , 1 ) , ; ( 2 , 2 ; 1 , 1 : , 2 , 2 ; 1 , 1 : , ) ; ,...., ; ,....,
( 1 1
, q j F j f q j D j d Q j B j j b p j E j e p j C j c P j A j j a x x n m n m N M q p q p Q P H ax b b a a
M l m
m l β α δ η σ γ λ β α in (1.3),
then expression becomes
,
1 1
(
;
,
)
: (
,
)
; (
,
)
1,
1,
1,
,
:
,
;
,
1
2
1 1
2
2
( ,...., ; ,....,
;
)
,
:
,
;
,
(
;
,
)
: (
,
)
; (
,
)
1 1
2
2
1,
1,
1,
1
2
l m
l m
a
A
c
C
e
E
j
j
j
P
j
j
p
j
j
p
M N m n m
n
x
M M a
a b
b ax
H
P Q p q p
q
b
B
d
D
f
F
x
j
j
j
Q
j
j
q
j
j
q
α β λ
α
σ
γ
δ
β
η
(2.2) , 1 1 1 0(
;
,
)
: (
,
)
; (
,
)
1,
1,
1,
,
:
,
;
,
1
2
1 1
2
2
( ,...., ; ,....,
;
)
,
:
,
;
,
(
;
,
)
: (
,
)
; (
,
)
1 1
2
2
1,
1,
1,
1
2
s
l m
l m
a
A
c
C
e
E
j
j
j
P
j
j
p
j
j
p
M N m n m
n
x
x
M M a
a b
b ax
H
dx
P Q p q p
q
b
B
d
D
f
F
x
j
j
j
Q
j
j
q
j
j
q
α β λ
α
σ
γ
δ
β
η
∞ −
=
∫
By using (1.2) and (1.6) represent H-function in integral form and M-series as series form. Put δt2 = -u, we get
( ) ( )
( ) ( )
∑
∞=∫
∫ − + + − − − − + Γ 0 2 1 1 1 2 1 1 1 1 , 1 2 ) 1 ( 1 ) 2 ( 1 ) ( ... ... k L k k m k k l k L dx dt du s k t x u x u t u t u t i k a b b a a δ λ σ δ η γ δ ψ δ φ φ π β αInterchange the order of integration, we get
( ) ( )
( ) ( )
∑
∞∫
∫
∫
= − − ∞ − + + − − + Γ = 0 1 1 2 0 1 1 1 1 1 2 1 1 1 , 2 1 ) ( 2 1 1 ) ( ... ... k L u u s k t L t k k m k k lk u t u x du dx dt
i x t i k a b b a
a σ λ δ
η δ ψ δ φ π γ φ π δ β α
Use result (1.5) and (1.1) to get the result. Change of order of integration is justifiable due to convergence of integrals.
Theorem 2:
,
1 1
(
;
,
)
: (
,
)
; (
,
)
1,
1,
1,
,
:
,
;
,
1
2
1 1
2
2
( ,...., ; ,....,
;
)
;
,
:
,
;
,
(
;
,
)
: (
,
)
; (
,
)
1 1
2
2
1,
1,
1,
1
2
l m
l m
a
A
c
C
e
E
j
j
j
P
j
j
p
j
j
p
M N m n m
n
x
L M a
a b
b ax
H
s
P Q p q p
q
b
B
d
D
f
F
x
j
j
j
Q
j
j
q
j
j
q
α β λ
α
σ
γ
δ
β
η
− + + + − + + + − + + + + + + + + + Γ + + − =∑
∏
∏
∑
∏
∏
∞ = + + − = = ∞ = = = , , 1 ) ) / ( , ) 1 ( ( , , 1 ) ) / ( , ) 1 ( ( 2 1 , 2 1 2 1 , 2 1 ) ( ) ( ) ( ! ) ( 1 0 ) 1 ( 1 1 0 1 1 Q j B j j B k s j b P j A j j A k s j a m n N n m M p q Q q p P H k a k b k a s p a b k k s k m j j l j j s s l j j m j j δ σ β δ λ δ σ α δ λ γ δ σ η η β α δ δ λ + + + + − − + + − − + + + + − − + + − − 1 , 1 1 ) , ( , 2 , 1 ) ) / ( , ) 1 ( 1 ( , 1 , 1 ) , ( , 2 , 1 ) ) / ( , ) 1 ( 1 ( 1 , 1 1 ) , ( , 2 , 1 ) ) / ( , ) 1 ( 1 ( , 1 , 1 ) , ( , 2 , 1 ) ) / ( , ) 1 ( 1 ( 2 2 q m j D j d q m j E j E k s j e m j D j d m j E j E k s j e p n j C j c p n j F j F k s j f n j C j c n j F j F k s j f δ σ δ λ δ σ δ λ δ σ δ λ δ σ δ λ3. SPECIAL CASES (i) Take
( ) ( )
αP = AP =( ) ( )
BQ = bQ =( ) ( ) ( ) ( )
Cp1 = Dq1 = Ep2 = Fq2 =1We get Mellin and Laplace transforms of G-function of two variables with Generalized M-series
(ii) Put M = N = P = Q = 0 in (2.1) and (2.3), we get
,
1 2
1 1 2 2
1 1
1 1 2 2
1 2
(
,
)
(
,
)
1,
1,
,
,
( ,...., ; ,....,
;
)
;
,
(
,
)
,
(
,
)
1,
1,
l m l mc
C
e
E
j
j
p
j
j
p
x
x
m n
m n
M M a
a b
b ax
H
H
s
p q
d
D
p q
f
F
j
j
q
j
j
q
σ δ
α β
λ
γ
η
(3.1),
1 2
1 1 2 2
1 1
1 1 2 2
1 2
(
,
)
(
,
)
1,
1,
,
,
( ,...., ; ,....,
;
)
;
,
(
,
)
,
(
,
)
1,
1,
l m l mc
C
e
E
j
j
p
j
j
p
x
x
m n
m n
M M a
a b
b ax
H
H
s
p q
d
D
p q
f
F
j
j
q
j
j
q
σ δ
α β
λ
γ
η
+ − − + − − − + + + + + Γ + + =∑
∏
∏
∏
∏
∞ = + − = = = = , 1 , 1 ) , ( , 2 , 1 ) ) / ( , ) ( 1 ( , 1 , 1 ) , ( , 2 , 1 ) ) / ( , ) ( 1 ( 2 1 , 2 1 2 1 , 2 1 ) ( ) ( ) ( 1 0 ) ( 1 1 1 1 m j D j d q j E j E k s j e n j C j c p j F j F k s j f m n n m p q q p H k a k b k a a b k k s k m j j l j j l j j m j j δ σ δ λ δ σ δ λ γ δ σ η η β α δ δ λ + + 1 , 1 1 ) , ( 1 , 1 1 ) , ( q m j D j d p n j C j c , 1 21 1 2 2
1 1
1 1 2 2
1 2
(
,
)
(
,
)
1,
1,
,
,
( ,...., ; ,....,
;
)
,
(
,
)
,
(
,
)
1,
1,
l m l mc
C
e
E
j
j
p
j
j
p
x
x
m n
m n
L M a
a b
b ax
H
H
p q
d
D
p q
f
F
j
j
q
j
j
q
σ δ
α β
λ
γ
η
(3.2) + + − − + + − − − + + + + + Γ + + − =
∑
∏
∏
∑
∏
∏
∞ = + + − = = ∞ = = = , 2 , 1 ) ) / ( , ) 1 ( 1 ( , 2 , 1 ) ) / ( , ) 1 ( 1 ( 2 1 , 2 1 2 1 , 2 1 ) ( ) ( ) ( ! ) ( 1 0 ) 1 ( 1 1 0 1 1 q j E j E k s j e p j F j F k s j f m n n m p q q p H k a k b k a s p a b k k s k m j j l j j s s l j j m j j δ σ δ λ δ σ δ λ γ δ σ η η β α δ δ λ + + 1 , 1 1 ) , ( , 1 , 1 ) , ( 1 , 1 1 ) , ( , 1 , 1 ) , ( q m j D j d m j D j d p n j C j c n j C j c(iii) For β = 1 in (2.1) and (2.3), we get Mellin and Laplace transform of H-function of two variables with generalized M-series by M. Sharma [7]
(iv) For l = m = 0 in (2.1) and (2.3), we get Mellin and Laplace transform of H-function of two variables involving Mittag-Leffler function , ( )
λ β α ax E
[4]
(v) Take α = β = 1 in (2.1) and (2.3), we get Mellin the Laplace transform of H-function of two variables involving generalized hypergeometric function [3]
(vi) Put l = 0, m = 1 and b1 = 1 in (2.1) and (2.3), we get Mellin and Laplace transform of H-function of two variables
involving Wright function W(axλ
;α,β) [5].
CONCLUSION
REFERENCES
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Source of support: Nil, Conflict of interest: None Declared
