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International Journal of Engineering Inventions e-ISSN: 2278-7461, p-ISSN: 2319-6491 Volume 2, Issue 5 (March 2013) PP: 25-28

Ramanujan Integral involving the Product of Generalized Zeta -function function and H Ashish Jain, A. K. Ronghe Department of Mathematics, S.S.L.Jain P.G. Collage, Vidisha (M.P.) India, 464001

Abstract: In the present paper, some Ramanujan Integrals associated with product of generalized Riemann

 -function of one variable evaluated. Zeta-function and H Importance of the results established in

this

paper

lies

in

the

fact

they

involve

H -function which is sufficiently general in nature and capable to yielding a large of results merely by specializing the parameters their in.  -function etc. Key words: Zeta-function, H Mathematics subject classification: 2000, 24A33

I.

Introduction

 -function is a generalization of Fox’s H-function [ 2 ], introduced by Inayat Hussain [5] and The H studied by Buchman and Shrivastava [1] and other, is defined and represented in the following manner ;

 H

m, n

z 

p, q

m, n 

 H

p, q

1 2

 

   

 e j , E j ;j , e j ; E j 1,n n 1,p   z   f j , Fj 1,m , Fj , f j ;  j m1,q 

    zd,

(1.1)

L

Where, m

    

 f j  Fji

j1 q

n

   1  e j  E j  j1

j m 1

 f j  Fj, ,  j  1,...., m

i

 1  f j  Fj  ,  

p

 e j  E j

j n 1

(1.2) And the contour L is a line from

j

to

c   to c   suitable indented to keep the poles of the

right

of

the

path

and

the

singularities

of

 1  e j  E j  i , j  1,...., n to the left path.   The following sufficient condition for absolute convergence of the integral defined in equation (1.1) have been recently given by Gupta, Jain and Agrawal [4, p. 169-172] as follow, (i) (ii)

1  and   0, 2 1 arg  z    and   0 2 arg  z  

Where,

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Ramanujan Integral involving the Product of Generalized Zeta-function and H-function M



Q

N

  Fj 

j1

j E j 

j1

P

  Fj j 

jM 1

II.

Ej

(1.3)

j N 1

Formula required

In the present paper we will use following formula.

n

1  x 

x 1 x 

0

2

 n    n  1         2  dx  P1 2 n  n    2  2

(2.1) Where,

n    0,

The Riemann Zeta-function given to by Goyal and Laddha [3] are used for Drive these Integrals: 

    z,s, a, g  

  g  a  g   s

g 0

zg , g!

  1, | z | 1, Re  a   0, (2.2) If   1 in (2.2) then it reduces to generalized Riemann zeta function. 

  z,s, a, g  

 a  g s

g 0

zg , g!

| z | 1, Re  a   0, (2.3) If   1 , and s = 1, in (2.2) reduce to Hypergeometric function i.e.,

1   z,1, a, g   a

 r 0

1r  a r g!

zr 1 1 . ,  . 2 F0 1, a : z  r! a q! III.

(2.4)

Integrals

 -function and Riemann zeta function. In this section we will established integrals involving product of H First Integral

x 1 x  1  x 2

0 

n

 cg  1 vg   2  g! 

  g  a  g s 

g 0

 zx 1 . x  1  x 2   z,s, a, g  .H  

 H

1 

 dx. 

m  2, n

 R1  z  p 1, q 3  R 2 

R1 and R 2 are sets of parameter giving as follows  n   1  1  R1 : 1  n; 1  ,  ;  ,  e j , E j ;j 1,n ,  e j; E j n 1,p 2   2  n     R2 :  ; 1 , 1 ,  f j , Fj  ,  Fj , f j ;  j  ,  n; 1, 1 1,m m 1,q 2  2 

Where

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Ramanujan Integral involving the Product of Generalized Zeta-function and H-function

 Fj  n  g  1    0,  fj    1 | arg  z  |  , 2

F    1  j   0,  fj 

(3.1) Where  is mention in (1.3) Second Integral:

x 1 x  1  x 2

0 

  g  a  g 

g 0

Where

s

n

 zx 1 . x  1  x 2   z,s, a, g  .H  

 H

cg  1 vg  .2 . g

1 

 dx. 

m 1, n  2

 R3  z  p 3, q  2  R 4 

R 3 and R 4 are sets of parameter giving as follows

R 3 :  n; 1,1 ,  1  n  ; 1  1,1 , e j , E j;j  n    1 1  1  R4 :  ;  , f j , Fj 2 2  

1,n , e j; E j n1,p ,  P, 1 

1,m ,  Fj,f j;  j m1,q , 1  n; ,1

Provided,

 Fj  F  n   g  1    0,   1  j   0,  fj   fj    1 | arg  z  |  , 2 Where  is mention in (1.3) Proof: To prove the first integral we first express

 m,n H p.q  z  occurring in the left hand side of (3.1) in

term of Mellin-Barnes contour integral with the help of (1.1) and changing the order of integration, which is permissible under the conditions stated, using the (2.1) and (2.2) and interpreting the expression thus obtained in term of

 -function with the help of (1.1), we arrive at the right hand side of (3.1) after a little simplification. H The proof of second integral (3.2) can be developed proceeding on similar line to that of (3.1).

IV.

Special Cases

 -function and Remain zeta function lie in their The importance of our Integrals involving H manifold generality.  -function, on specializing the various parameters, we can obtain In view of the generality of the H from our integrals, several results involving a remarkable wide Varity of useful function, Which are expressible in terms of [2] Fox H-function, G-function and Remain zeta-function they are as follows : (1) If we take j   j  1,  i  1,...., n; j  m  1,....,q  in (3.1) and (3.2), the relation is reduce with

 -function and Fox’s H-function [2]. H  m,n  (2) If we reduce H p.q  z  to generalized Wright hypergeometric function p 1  q 1 [6, pp.271, Eq. (7)]

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Ramanujan Integral involving the Product of Generalized Zeta-function and H-function

References [1] [2] [3] [4] [5] [6]

Bushman: R.G. and Shrivastava H.M. (1990): The H -function associated with a certain cases of Feymnann integrals J. Phy. A. Math. Gen.23 4707-4720. Fox, C. (1961): The G-function and H-function as a symmetrical Fourier kernal Trans. Amer. Math. Soc. 98, 395-429. Goyal S.P. and Laddha, R.K. (1997): Ganita, Sandesh II, pp. 99. Inayat Hussain, A.A. (1987): New properties of Hypergeometric series, derivable from Feynman integrals II. A generalization of the H-function. J.Phys. A. Math. Gen. 20, 4119-4128. Qureshi, M.C. and Khan, E.H. (2005): South East Asian, J. Math. And Math. Sc. (6) Sharma A.A., Gupta K.C., Jain R. and (2003) A study of unified finite integral transforms with applications, J. Rajasthan Aca. Phy. Sci. 2, [4] pp. 269-282.

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