Gen. Math. Notes, Vol. 3, No. 2, April 2011, pp. 123-129
ISSN 2219-7184; Copyright © ICSRS Publication, 2011
www.i-csrs.org
Available free online at http://www.geman.in
On Mellin Transform Involving the Product of a
General Class of Polynomials, Struve’s Function
and H-Function of Two Variables
Y. Pragathi Kumar1 and B. Satyanarayana2
1
Department of Science and Humanities, Tenali Engineering College
Anumarlapudi – 522213, Andhra Pradesh, India.
E-mail: pragathi.ys@gmail.com
2
Department of Applied Mathematics, Acharya Nagarjuna University Campus,
NUZVID – 521201, A.P., India.
E-mail: drbsn63@yahoo.co.in
(Received: 24-2-11/ Accepted: 25-3-11)
Abstract
The object of this paper is to establish integrals involving the product of struve’s function,
general class of polynomials and H-function of two variables. Some special cases have also
derived.
Keywords: Mellin transform, struve’s function, general class of polynomials and H-function
of two variables.
1
Introduction
Recently, The Mellin transform of struve’s function with H-function of two variables and Mellin
transform of general class of polynomials with H-function of two variables [5] are evaluated. In
the present paper we establish the Integral transform of H-function of two variables with general
class of polynomials and struve’s function.
We shall utilized the following formulae in the present investigation. The
124
Y. Pragathi Kumar et al.
H-function of two variables given by Prasad and Gupta [7].
(1.1)
H[x , y] =
=
 σ (a j ;α j , Aj )1, P : (c j ,C j )1, p;(e j , E j )1,u
M, N : m, n; g, hγ x
HP,Q : p, q;u, v 
δ (bj ; β j , Bj )1,Q : (d j , Dj )1, q;( f j , Fj )1,v
η x
1
s t
∫ ∫ φ1( s ) φ2 (t )ψ ( s, t ) x y ds dt ,
2
(2π i ) L L
1 2



i = −1
where x , y ≠ 0,
m
n
∏ Γ (d j − D j s ) ∏ Γ (1 − c j + C j s )
j =1
j =1
φ1( s ) =
q
p
Γ (1 − d j + D j s ) ∏ Γ (c j − C j s )
∏
j = m +1
j = n +1
g
h




∏ Γ  f j − F j t  ∏ Γ1 − e j + E j t 




j =1
j =1
φ 2 (t ) =
v
u




∏ Γ 1 − f j + F j t  ∏ Γ  e j − E j t 




j = g +1
j = h +1
M
N
∏ Γ b j − β j s − B j t  ∏ Γ1 − a j + α j s + A j t 
 j =1 

j =1 
ψ (s, t ) =
Q
P
Γ1 − b + β s + B t  ∏
Γ a − α s − A t 
∏
j
j
j
j
j
j 


j = M +1
j = N +1 
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 + Ejt), (j = 1, 2, . . . , h) and Γ(1 – aj + αjs + Ajt), (j = 1, 2, . . . , N) lie to the left of the
path. All poles of the integrand are simple poles.
On Mellin Transform Involving the Product of a…
125
Mellin transform of the H-function is defined as follows [12]
∞
az (c j , γ j )1, P 
s − 1H M , N 
 dx = a − sθ (− s)
∫z
P, Q 
0
 (d j , δ j )1, Q 
(1.2)
where
N
M
∏ Γ ((1 − c j ) − γ j s ) ∏ Γ (1 − (1 − d j ) + δ j s )
j =1
j =1
θ (− s ) =
Q
P
Γ (1 − (1 − c j ) + γ j s )
Γ ((1 − d j ) − δ j s)
∏
∏
j = N +1
j = M +1
Provided the corresponding conditions stated in [12]
According Erdely [1, p.307]
∞
 c + i∞

s −1 1
−s
∫x
∫ g(s) x dsdx = g(s)
 2πi c − i∞

0
(1.3)
The Struve’s function is defined as [3],
λ, k
∞ (−1)m(z / 2)v + 2m +1
H v, y,u [z] = ∑
m = 0 Γ(km+ y)Γ(v + λ m + u)
(1.4)
Re (k) > 0, Re (λ) > 0, Re (y ) > 0, Re (v + u) > 0
The class of polynomials [10]
(1.5)
S nm [x ]
=
[ n / m] ( − n ) m k
An, k x k
∑
k = 0 k!
n = 0, 1, 2, . . .
where m is an arbitrary positive integer and the coefficients An, k (n, k ≥ 0) are arbitrary
constants.
2
Main Result
γ xσ (a j ;α j , A j )1, P : (c j , C j )1, p1 ; (e j , E j )1, p2
x s −1 H λv,,yk,u [ ax h ] S nm [bx h ] H PM,Q, N: p:m,1q, n;1p; m,2q,n2 
1 1 2 2 
δ (b ; β , B )
j
j
j 1,Q : ( d j , D j )1, q1 ; ( f j , F j )1, q2
0
η x
∞
(2.1)
∫

 dx

126
=
1
δ
Y. Pragathi Kumar et al.
∞
[n / m]
∑G(l) r ∑= 0 F (r)η
−
(s + φ (l, r ) )
δ
l =0

 s + φ (l , r ) 
 A j ,α j − (σ / δ ) A j )1, N , (c j , C j )1, n ,
 σ (a j + 
−
δ
1


M + m1 + n2 , N + n1 + m2  δ
η
γ
H
P + p1 + q2 , Q + q1 + p2 
 s + φ (l , r ) 

(b j + 
B j , β j − (σ / δ ) B j )1, M , (d j , D j )1, m ,

δ
1




 s + φ (l , r ) 
 s + φ (l , r ) 
(1 − f j − 
 F j , (σ / δ ) F j )1, q , ( a j + 
 A j ,α j − (σ / δ ) A j ) N + 1, P , (c j , C j )n + 1, p 
δ
δ
2
1
1 





 s + φ (l , r ) 
 s + φ (l , r ) 

(1 − e j − 
 E j , (σ / δ ) E j )1, p , (b j + 
 B j , β j − (σ / δ ) B j ) M + 1, Q , (d j , D j ) m + 1, q 
δ
δ
2
1
1




where
G(l) =
F (r ) =
(−1)l av+2l +1
Γ(kl + y)Γ(v + λl + u)
(−n)mr
An, r br
r!
φ(l,r) = g(v+2l +1) +hr
Provided σ > 0, δ > 0,λ > 0, h > 0,
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 =
N
∑α j −
j =1
∆2 =
N
∑A
j =1
j
−
P
M
j = N +1
j =1
P
M
∑α j + ∑ β j −
∑A +∑B
j = N +1
j
j =1
j
−
Q
m1
j = M +1
j =1
Q
m2
∑β j + ∑ Dj −
∑B +∑F
j = M +1
j
j =1
j
−
q1
∑
j = m1 +1
q2
∑
j = m2 +1
n1
Dj + ∑C j −
j =1
n2
Fj + ∑ E j −
j =1
p1
∑C
j = n1 +1
j
,
p2
∑E
j = n2 +1
j
and

δ ( a j − 1) 
Re s + g v + g +
 < 0
Aj



for j = 1,2,. . . , N

δ bj
Re s + g v + g +
Bj


for j = 1,2, . . ., M

>0



δ f j σ d 
i >0
Re s + g v + g +
+
Fj
Di 




δ (e j − 1) σ (c − 1) 
i
Re s + g v + g +
>0
+
Ej
Ci




for j = 1,2,. . . , m2; for i = 1,2,. . . , m1
for j = 1,2,. . . , n2; for i = 1,2,. . . , n1
On Mellin Transform Involving the Product of a…
127
Proof.
We have
λ, k
m h
H v, y , u  ax g  Sn [bx ]


∞ (−1) m ( ax g / 2) v + 2l + 1 [n / m] ( −n) mr
An, r (bx h )
∑
r
!
Γ
(
kl
+
y
)
Γ
(
v
+
λ
l
+
u
)
l =0
r=0
= ∑
Multiply both sides with
γ xσ (a j ; α j , A j )1, P : (c j , C j )1, p ; (e j , E j )1, p
M
,
N
:
m
,
n
;
m
,
n
1
2
1 1 2 2
x s − 1H
P, Q : p1 , q1; p 2 , q 2  δ (b ; β , B )
j j j 1, Q : (d j , D j )1, q1 ; ( f j , F j )1, q 2
η x



and integrate with respect to x from 0 to ∞. By using (1.1) , represent H-function in integral form
and put δt2 = - w. Interchange the order of integration then use result (1.3) and (1.2) to get the
result. Change of order of integration is justifiable due to convergence of integrals.
3
Special Cases
Put h = 0, b = 1 in (2.1) we get Mellin transform of product of Struve’s function with H-function
of two variables
(3.1)
γ xσ (a j ;α j , A j )1, P : (c j , C j )1, p ; (e j , E j )1, p
∞
1
2
s − 1 H λ , k [ax h ]H M , N : m1, n1; m2 , n2 
∫x
v, y , u
P, Q : p1, q1; p2 , q2  δ (b ; β , B )
0
η x
j j j 1, Q : (d j , D j )1, q1 ; ( f j , F j )1, q2
=
1
δ
∞
∑G(l )η
l =0
−
( s + φ (l , 0) )
δ

dx


 s + φ (l ,0) 
 A j ,α j − (σ / δ ) A j )1, N , (c j , C j )1, n ,
 σ (a j + 
−
δ
1


m + n , N + n1 + m2  δ
η
γ
H 1 2

P + p1 + q2 , Q + q1 + p2
 s + φ (l ,0) 

(b j + 
 B j , β j − (σ / δ ) B j )1, M , (d j , D j )1, m ,

δ
1




 s + φ (l ,0) 
 s + φ (l ,0) 
(1 − f j − 
 F j , (σ / δ ) F j )1, q , (a j + 
 A j ,α j − (σ / δ ) A j ) N + 1, P , (c j , C j ) n + 1, p 
δ
δ
2
1
1 





 s + φ (l ,0) 
 s + φ (l ,0) 

(1 − e j − 
 E j , (σ / δ ) E j )1, p , (b j + 
 B j , β j − (σ / δ ) B j ) M + 1, Q , ( d j , D j )m + 1, q 
δ
δ
2
1
1




Put g = 0, a = 1 in (2.1) we get Mellin transform of product of generals class of polynomials with
H-function of two variables [5, Result 2.1, p.45]
Put M = 0, we obtain
(3.2)
γ xσ ( a j ;α j , A j )1, P : (c j , C j )1, p ; (e j , E j )1, p 
∞
1
2
s − 1 H λ , k [ax h ]S m [bx h ]H 0, N : m1, n1; m2 , n2 
x
dx
∫
v, y , u
n
P, Q : p1, q1; p2 , q2  δ (b ; β , B )
:
(
d
,
D
)
;
(
f
,
F
)
0
j j j 1, Q
j j 1, q1 j j 1, q2 
η x
128
=
1
δ
Y. Pragathi Kumar et al.
∞
[n / m]
∑G(l ) r ∑= 0 F (r )η
−
( s + φ (l , r ) )
δ
l =0

 s + φ (l , r ) 
 A j ,α j − (σ / δ ) A j )1, N , (c j , C j )1, n ,
 σ (a j + 
−
δ
1


m1 + n2 , N + n1 + m2  δ
η
γ
H
P + p1 + q2 , Q + q1 + p2 
 s + φ (l , r ) 

(b j + 
 B j , β j − (σ / δ ) B j )1, M , (d j , D j )1, m ,

δ
1




 s + φ (l , r ) 
 s + φ (l , r ) 
(1 − f j − 
 F j , (σ / δ ) F j )1, q , ( a j + 
 A j ,α j − (σ / δ ) A j ) N + 1, P , (c j , C j )n + 1, p 
δ
δ
2
1
1 





 s + φ (l , r ) 
 s + φ (l , r ) 

(1 − e j − 
E
,
(
σ
/
δ
)
E
)
,
(
b
+
B
,
β
−
(
σ
/
δ
)
B
)
,
(
d
,
D
)
 j
 j j
j 1, p2 j 
j M + 1, Q
j j m1 + 1, q1 
δ
δ





By taking M = N = P = Q = 0, we have
 σ (a j ;α j , A j )1, P : (c j , C j )1, p ; (e j , E j )1, p
1
2
0,0 : m1, n1; m2 , n2 γx
λ
s
−
1
,
k
h
m
h
x
Hv, y, u [ax ]Sn [bx ]H
0,0 : p1, q1; p2 , q2  δ (b ; β , B )
ηx
j j j 1, Q : (d j , D j )1, q1 ; ( f j , F j )1, q2
0
∞
(3.3)
∫
=
=
1
δ
∞
 γ xσ (c j , C j )1, p
∞
1
s − 1H λ , k [ ax h ]S m [bx h ]H m1, n1 
∫x
v , y ,u
n
p1, q1 
( d j , D j )1, q
0

1
[n / m]
∑ G(l ) r ∑= 0 F (r )η
−
( s + φ (l , r ) )
δ
l =0
δ (e j , E j )1, p
 m , n η x
2
H 2 2
p
q
,

2 2
( f j , F j )1, q

2

 dx


 dx


 s + φ (l , r ) 
 A j ,α j − (σ / δ ) A j )1, N , (c j , C j )1, n ,
 σ (a j + 
−
δ
1


m1 + n2 , n1 + m2  δ
η
γ
H
p1 + q2 , q1 + p2 
 s + φ (l , r ) 

(b j + 
 B j , β j − (σ / δ ) B j )1, M , (d j , D j )1, m ,

δ
1




 s + φ (l , r ) 
 s + φ (l , r ) 
(1 − f j − 
 F j , (σ / δ ) F j )1, q , ( a j + 
 A j ,α j − (σ / δ ) A j ) N + 1, P , (c j , C j )n + 1, p 
δ
δ
2
1
1 





 s + φ (l , r ) 
 s + φ (l , r ) 

(1 − e j − 
 E j , (σ / δ ) E j )1, p , (b j + 
 B j , β j − (σ / δ ) B j ) M + 1, Q , (d j , D j ) m + 1, q 
δ
δ
2
1
1




Put M = N = P = Q = 0 and (α)j = (β)j = (A)j = (B)j = (C)j = (D)j = (E)j = (F)j = 1, we obtain
(3.4)
 γxσ (c j , C j )1, p 
 δ (e j , E j )1, p 
∞
1
2
m2 , n2 ηx
s − 1 H λ , k [ax h ] S m [bx h ] G m1, n1 
x

G
 dx
∫
v, y , u
n

p1, q1 
p
,
q

2
2
(
d
,
D
)
(
f
,
F
)
0


j j 1, q1
j j 1, q2 

=
1
δ
∞
[n / m]
∑ G(l ) r ∑= 0 F (r )η
l =0
−
( s + φ (l , r ) )
δ

 s + φ (l , r ) 
, (σ / δ ))1, q , (c j ) n + 1, p 
 σ (c j )1, n , (1 − f j − 
−
δ
2
1
1


1
m + n , n + m2  δ

G 1 2 1
γ
η

p1 + q2 , q1 + p2 
(
d
)
,


j 1, m1 (1 − e −  s + φ (l , r ) , (σ / δ ))
,
(
d
)

j
1, p2
j m1 + 1, q1 
δ



On Mellin Transform Involving the Product of a…
129
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
A. Erdely, Tables of Integral Transforms, (Vol I), Mc Graw Hill Book Co., New York.
K.C. Gupta and U.C. Jain, The H-function II, Proc. Nat. Acad. Sci. India, Sect.A.,
36(1966), 594-699.
B.N. Kanth, Jour. Inst. Sci, Nepal, 3(1) (1980), 99.
P.K. Mittal and K.C. Gupta, An integral involving generalized functions of two variables,
Proc. Indian. Acad. Sci. Sect. A, 75(1972), 117-123.
Y.P. Kumar and B. Satyanarayana, Integral transform involving the product of general
class of polynomials and H-function of two variables, ABJMI, Vol.5, 2011,p.43-48.
Y.P. Kumar and B. Satyanarayana, On Laplace transform of the product of generalized
struve’s function and H-function of two variables, International Journal of Applied
Mathematical Analysis and Applications, Vol.6 No.1-2 2011.p. 193-198
Y.N. Prasad and R.K. Gupta, Vijnana Parishad Anusandhan Patrika, 19(1976), 39-45.
P.C. Srinivas, On special functions, integral transforms and integral equations, Ph.D
Thesis, Kannur University, India.
E.D. Rainville, Special Functions, Chelsea Publ. Co., Branx, New York.
R.P. Sharms, On a new theorem involving the H-function and general class of
polynomials, Kyungpook Math. J., 43(2003), 489-494.
H.M. Srivastava and R.G. Buschman, Some convolution integral equations, Indag. Math.,
36(1974), 211-216.
H.M. Srivastava, K.C. Gupta and S.P. Goyal, The H-Function of One and Two Variables
with Applications, South Asian Publishers, New Delhi, (1982).
H.M. Srivastava and R. Panda, Some bilateral generating functions for a class of
generalized hypergeometric polynomials, J. Reine Argew. Math., 283(1976), 265- 274.