“A STUDY OF EULERIAN INTEGRALS OF MULTIPLE HYPERGEOMETRIC FUNCTION WITH GENERAL ARGUMENT”

“A STUDY OF EULERIAN INTEGRALS OF MULTIPLE

HYPERGEOMETRIC FUNCTION WITH GENERAL ARGUMENT”

1_{Ms. Samta Gulia ,}2 _{Dr. R.S. Verma and} 3_{Prof. (Dr.) Harish Singh}

1_{Research Scholar, NIMS University, Jaipur, Rajasthan, India}

2_{Head of Department, Department of Mathematics}

NIMS University, Jaipur, Rajasthan, India

3 _{Professor, Maharaja Surajmal Institute, Affiliated to Guru Gobind Singh Indraprastha}

University, New Delhi, India

INTRODUCTION

In this research paper, we evaluate three unified Eulerian integrals. The first of these integrals

involve the product of a general class polynomials SU_V x , the generalized sequence of function

 x

S_n,,O and the multivariable Hypergeometric function with general argument which is

giving by the multiple series by Srivastava and Daoust (1969). The second integral involve the

product of a general class of multivariable polynomials 1 r

r 1

,...U U

,...V V

S [x_1,...,x_r] and the ABSTRACT

Our aim to present this paper is to investigate the most general nature of the functions and

polynomials occurring in the various integrals, we evaluate three unified Eulerian integrals.

Our results provide interesting unifications and extensions of a large number of new and

known results. Our integrals involve simpler polynomials such as, Jacobi polynomials,

multivariable Hypergeometric function where as the last integral contain the product of

another class of multivariable polynomials U1,...U r

V

S [x1,...,xr] which was introduce by

Srivastava and Garg (1987) and the multivariable Hypergeometric function in their integrals.

1.1 Theorem:1 Eulerian First Integral:

Prove that



1

0

xm(a-1) _(1-xm₎b-1 U

V

S



yx (1 xm)v



μ m

 , ,0

n

S 









) x (1

hx m  m

n ;...q 1 p:q

n ;...m 1 m 1;

F 

 

 

 

 

λ λ

) n (

q q

p

) n (

m m

1

x n z ,...., x 1 z ); (b ...; );... (b

: ) a (

); ( β ; ... . ); ( β : ) (

n 1

n 1

 dx

= m

1) -m

1 Γ(

) Γ( 1) -m

1 Γ(

  

 

 



 ) U V (

0 k

ɸ1 (k,v, u, e, p)

n ;...q 1 q : λ p

n ;...m 1 m λ; 1

F

 

     

 

     

 

  

   

 

n z ,...., 1 z ); (b ; );... (b ; λ

2) -m

1 (

..., ,... λ

1) -m

1 (

: ) a (

); ( β ...; . ); β ( ; λ

2) -m

1 (

,..., λ

1) -m

1 (

: ) (

) n ( n q 1

1 q p

) n ( n m 1

1 m 1

  

   

 

(1.1)

Then the p.d.f form of (1.1) is as below

F(x) =

] [ p)F

e, u, v, (k, 1)

-1 Γ(

) Γ( 1) -1 Γ(

] [ F

] ) x (1 hx [ S ] ) x (1 yx [ S ) x (1 x

2 ;...q q : λ p

;...m m λ; 1 1

) U V (

0 k

1 ;...q q : p

;...m m 1; m m

,0 , n m u

m U V 1 b m 1)

(a m

n 1

n 1

n 1

n 1

 

 

 



 

 

  

 



  

 



m m m

= 0 elsewhere (1.2)

Where

1 =

   

   

λ λ

) n (

q q

p

) n (

m m

1

x n z ,...., x 1 z ); (b ...; );... (b

: ) a (

); ( β ; ... . ); ( β : ) (

n 1

n 1



and

     

 

     

 

  

   

 

n z ,...., 1 z ); (b ...; );... (b

: λ

2) -m

1 (

., ,... λ

1 -m

1 : ) a (

); ( β ...; . ); ( β ; λ

2) -m

1 (

,..., λ

1) -m

1 (

: ) (

) n ( q q

p

) n ( m m

1 2

n 1

n 1

  

   

  

Where U

V

S [x] be the general class of polynomials, S_n,,0[x] be the generalized sequence of function. These are very general in nature and extends a number of classical polynomials introduced and studied by various research work such as Gould and Hopper (1962), Krall and Frink (1949), Singh(1981), Singh and Srivastava (1964), Dhilion(1989), etc and the

multivariable Hypergeometric function which we occurring in(1.1). Here we take xm instead of

x in the above definitions and we find such new results.

Where

1 (k, v, u, e, p) =

K!

p)h e, u, (v, K) A(V, V)

( _UK ₁ R

yk (1.3)

α = a + k + R & β = b +  + R (1.4)

Also the following conditions are assumed to be satisfied.

(ii) Min (, v, , , , , p, q1) > 0 (j = 1, 2, ...,n) (not all zero simultaneously)

(iii) (a)  > 0, v > 0, 1 > 0, 2 > 0

(b)  < 0, v < 0, 1 + [V/U] > 0, 2 + v [V/U] > 0

(c)  > 0, v 0, 1 > 0, 2 + v [V/U] > 0

(d)  < 0, v > 0, 1 +  [V/U] > 0, 2 > 0

Where

1 = Re(a) +  {(1(m+n) + p + 



n

1 j

j

q

2 = Re(a) +  {(1(m+n) + p + 



n

1 j

j

q

Proof:- To prove the first integral (1.1), We first express the general class of polynomials U

V

S

[x] and the generalized sequence of function , ,0

n

S  [x] occurring in its left hand side in their

respective series forms.

The generalized sequence of function is

0 , , n

S [x] = 

p e, u, v,

1

θ (v, u, e, p) xR ... (1.5)

The general class of polynomials is introduced as

U V

S [x] =





) U V (

0

K K!

K) A(V, V)

( _UK

xK _{N= 0,1,2... ... (1.6)}

Here we use the above formula by taking xm _{instead of x. While (1.6) is certainly useful}

a less general nature are of more immediate value. To this end, we consider the generalized Kamp’ ede Fe’riet function first given by Karlson (1973) which we define later.



1

0

xm(a-1)(1-xm)b-1 U

V

S



yx (1 xm)v



μ m

 , ,0

n

S 









) x (1

hx m  m

n 1

n 1

;...q q : p

;...m 1;m

F

 

 

 

 

λ λ

) n (

q q

p

) n (

m m

1

x n z ,...., x 1 z ); (b ...; );... (b

: ) a (

); ( β ; ... . ); ( β : ) (

n 1

n 1

 dx (1.7)



1

0

xm(a-1) _(1-xm₎b-1



 ) U V (

0

k K!

K) A(V, V)

( _UK

yk_xmk _(1-xm₎vk



 ) U V (

0 k

θ1 (k, v, u, e, p)hRxmR

×(1-xm)ζR 1 n n 1

;...q q : p

;...m 1;m

F dx

 

 

 

 

λ λ

) n ( n q 1

q p

) n ( n m 1

m 1

x n z ,...., x 1 z ); (b ...; );... (b : ) a (

); ( β ; ... . ); ( β : ) (

(1.8)

Now Using the Multivariable Hypergeometric Function which is giving by Srivastava and Daoust (1969) have giving the multiple series.

By using multivariable Hypergeometric function as:

(n) ;...q 1 :q p

(n) ;...m 1 m 1;

F

    

 

    

 

n

z z

: :

1

 1 (n)

(n) 1

;...q q : p

;...m m 1;

F

 

 

 

 

n z ,...., 1 z ); (b ...; );... (b

: ) a (

); ( β ; ... . ); ( β : ) (

) n (

q q

p

) n (

m m

1

n 1

n 1



=





0 s ,...,

s_{1 n}

A(s1,...,sn)

! s z

1 s 1

1

... ! s z

n s n

n

(1.9)

A(s1,...,sn)













          n n 1 1 n 1 (n) n 1 1 n 1 m 1 j s (n) j m 1 j ... s j 1 1 j s ... s j q 1 j s (n) j q 1 j ... s j p 1 j s ... s j ) ( β ) ( β ) ( ) (b ) (b ) (a 

Putting value p:q1;...q n n ;...m 1 m 1;

F



λ xλ



n z ,...., x 1

z with the help of (1.9), the equation (1.8) becomes



1

0

xm(a-1) _(1-xm₎b-1



 ) U V ( 0 k K! K) A(V, V)

( _UK

yk_xmk _(1-xm₎vk



 ) U V ( 0 k

1 (v, u, e, p)hRxmR

×(1-xm)ζR





0 m ,...,

m1 1

(m1, ..., mn)

! m x z 1 λ m 1 1 m 1 ... ! m x z 1 λ m n 1 m n dx Where

(m1, ..., mn)=













            (n) (n) j n j 1 (n) j n j 1 (n) (n) j 1 j 1 (n) j n j 1 D 1 j m (n) j D 1 j ... m j C 1 j ψ m ... ψ m j B 1 j m (n) j B 1 j ... m j A 1 j θ m ... θ m j ) (d ) (d ) (c ) (b ) (b ) (a     =



 ) U V ( 0 k K! K) A(V, V)

( _UK

yk



 ) U V ( 0 k

1 (v, u, e, p)hR



1

0

xm(a-1)(1-xm)b-1xmuk(1-xm)vkxmR

_×(1-xm₎ζR













             (n) (n) j n j 1 (n) j n j 1 (n) (n) j 1 j 1 (n) j n j 1 D 1 j m (n) j D 1 j ... m j C 1 j ψ m ... ψ m j B 1 j m (n) j B 1 j ... m j A 1 j θ m ... θ m j ) (d ) (d ) (c ) (b ) (b ) (a     ! m x z 1 λ m 1 1 m 1 ... ! m x z 1 λ m n 1 m n dx =



 ) U V ( 0 k K! K) A(V, V)

( _UK

yk



 ) U V ( 0 k

1 (v, u, e, p)hR



1

0













             (n) (n) j n j 1 (n) j n j 1 (n) (n) j n j 1 (n) j n j 1 D 1 j m (n) j D 1 j ... m j C 1 j ψ m ... ψ m j B 1 j m (n) j B 1 j ... m j A 1 j θ m ... θ m j ) (d ) (d ) (c ) (b ) (b ) (a     ! m x z 1 λ m 1 1 m 1 ... ! m x z 1 λ m n 1 m n dx =



 ) U V ( 0 k K! K) A(V, V)

( _UK

yk



 ) U V ( 0 k

1 (v, u, e, p)hR



1

0

xm(α-1) (1-xm)β-1













             (n) (n) j n j 1 (n) j n j 1 (n) (n) j 1 j 1 (n) j n j 1 D 1 j m (n) j D 1 j ... m j C 1 j ψ m ... ψ m j B 1 j m (n) j B 1 j ... m j A 1 j θ m ... θ m j ) (d ) (d ) (c ) (b ) (b ) (a     ! m x z 1 λ m 1 1 m 1 ... ! m x z 1 λ m n 1 m n dx (1.10)

Where α = a + k + R and β = b + k + R

=



 ) U V ( 0 k K! K) A(V, V)

( _UK

yk



 ) U V ( 0 k

1 (v, u, e, p)hR













             (n) (n) j n j 1 (n) j n j 1 (n) (n) j 1 j 1 (n) j n j 1 D 1 j m (n) j D 1 j . ... m j C 1 j ψ m ... ... ψ m j B 1 j m (n) j B 1 j ... .... m j A 1 j θ m ... ... θ m j ) (d ) (d ) (c ) (b ) (b ) (a    



 1 0 1 m ... ... m λ 1) 1 m( 1 x     n m  

(1-xm₎β-1

! m z 1 m 1 1 ... ! m z n m n n

dx (1.11)

Where x x dx x x dx x m xm dx

m m m m m m 1 1 0 1 ) 1 1 ( 1 1 0 ) 1 1 ( 1 1 0 ) 1 ( ) 1 ( ) 1 ( ) 1 (               







     



1

0

xa-1_(1-x)b-1_{f(x) dx =}

) Γ(a

) ( ) Γ(

b b a

 





n n

n n

) (

C ) (

b a





Above formula can be transformed into the following as

x m xm dx

m

1 1

0

1 ) 1 1 (

) 1

( 

  



   = m

1) -m

1 Γ(

) 1) Γ) -m

1 Γ(

  

 

 





n

n n n

1) -m

1 (

C 1) -m

1 (

 



 



(1.12)

Where f(x) =





n

Cnxn , putting the formula (1.12) in (1.11), we get

=



 ) U V (

0

k K!

K) A(V, V)

( _UK

yk



 ) U V (

0 k

1 (v, u, e, p)hR













 

 

 

 

 

 

 (n)

(n) j n j

1 (n)

j n j 1

(n)

(n) j n j

1 (n)

j n j 1

D

1 j

m (n) j D

1 j

... m j C

1 j

ψ m ... ψ m j

B

1 j

m (n)

j B

1 j

... m j A

1 j

θ m ... θ m j

) (d )

(d )

(c

) (b )

(b )

(a

 

 

× m

1) -m

1 Γ(

) Γ( 1) -m

1 Γ(

  

 

 

n 1

n 1

n 1

n 1

λm ... λm λm

... λm

λm ... λm λm ... λm

) m

1 (

) 1 -m

1 (

) ( )

1 -m

1 (

  



  



  

 

  



 

= m

1) -m

1 Γ(

) 1) Γ) -m

1 Γ(

  

 

 



 ) U V (

0 k

n 1

n 1

;...q q : λ p

;...m m λ; 1

F _



     

 

     

 

 



 

 

 n

z ,...., 1 z ); (b ...; );... (b

: λ

1 -λ 1) -m

1 ( ,..., λ

1 -m

1 : ) a (

); ( β ; ... . ); ( β λ

1 -λ 1) -m

1 (

,..., λ

1) -m

1 (

: ) (

) n (

q q

p

) n (

m m

1

n 1

n 1

 

  

 

Where



 ) U V (

0 k

1 (k,v, u, e, p) =





) U V (

0

k K!

K) A(V, V)

( _UK

yk



 ) U V (

0 k

1 (v, u, e, p)hR

= m

1) -m

1 Γ(

) 1) Γ) -m

1 Γ(

  

 

 



 ) U V (

0 k

ɸ1 (k,v, u, e, p)×

n 1

n 1

;...q q : λ p

;...m m λ; 1

F _



     

 

     

 

  

   

 n

z ,...., 1 z );

(b ...; );... (b

: λ

2) -m

1 (

,..., λ

1 -m

1 : ) a (

); ( β ; ... . ); ( β ; λ

2) -m

1 (

,..., λ

1) -m

1 (

: ) (

) n (

q q

p

) n (

m m

1

n 1

n 1

  

   

 

(1.13)

F(x) = ] [ p)F e, u, v, (k, 1) -1 Γ( ) Γ( 1) -1 Γ( ] [ F ] ) x (1 hx [ S ] ) x (1 yx [ S ) x (1 x 2 ;...q q : λ p ;...m m λ; 1 1 ) U V ( 0 k 1 ;...q q : p ;...m m 1; m m ,0 , n m u m U V 1 b m 1) (a m n 1 n 1 n 1 n 1                



      m m m v

= 0 elsewhere (1.14)

Where

1 =

        λ λ ) n ( q q p ) n ( m m 1 x n z ,...., x 1 z ); (b ...; );... (b : ) a ( ); ( β ; ... . ); ( β : ) ( n 1 n 1  and                          n z ,...., 1 z ); (b ...; );... (b : λ 2) -m 1 ( ., ,... λ 1 -m 1 : ) a ( ); ( β ...; . ); ( β ; λ 2) -m 1 ( ,..., λ 1) -m 1 ( : ) ( ) n ( q q p ) n ( m m 1 2 n 1 n 1          

Hence First integral is proved.

2.1 Theorem:2 Eulerian second Integral Involving 1 r

r 1 ;...U U ;...V V

S [x₁....x_r]

Prove that



1

0

xm(a-1)(1-xm)b-1 1 r

r 1 ;...U U ;...V V

S



v vr



) x (1 x y ,..., ) x (1 x

y r m m

= m 1) -m 1 Γ( ) Γ( 1) -m 1 Γ( 1 1 1 1       



 ) U V ( 0 k 1 1 1 ...



 ) U V ( 0 k r r r r r 1

1K ... r UK

U

t) ( v )

v

(  A(V₁, k₁ ; ...V_r, k_r)

 r 1 i ₁ i k i ! k y 

 _ 1 n

n 1 ;...q q : λ p ;...m m λ; 1 F                          n z ,...., 1 z ); (b ; );... (b : λ 2) -m 1 ( ,..., λ 1 -m 1 : ) a ( ); ( β ...; . ); β ( λ 2) -m 1 ( ,..., λ 1) -m 1 ( : ) ( ) n ( n q 1 q 1 1 p ) n ( n m 1 m 1 1 1 1 1          (2.1)

Then p.d.f form of (2.1) is giving below:

F(x) = ] [ F ! k k y k , ...v . ; k , A(v ) v ( ) v ( ... 1) -m 1 Γ( ) Γ( 1) -m 1 Γ( ] [ F ] ) x (1 x y ... ) x (1 x y [ S ) x (1 x 4 n ;...q 1 q : λ p n ;...m 1 m λ; 1 1 i i r 1 i r r 1 1 r k r v r ... 1 k 1 u r ) r U r V ( 0 r k ) 1 U 1 V ( 0 1 k 1 1 1 1 3 n ;...q 1 q : p n ;...m 1 m 1; m r u m r 1 m u m 1 r ;...v 1 u r ;...v 1 v 1 b m 1 a m                          m r v v

= 0 elsewhere (2.2)

Where

3 =

         

 z1x (1-x ) ,...., zn x (1 )

     

 

     

 

  

   

 

n z ,...., 1 z );

(b );...; (b : λ

2) -m

1 (

,..., λ

1 -m

1

: ) a (

); ( β ...; . ); ( β ; λ

2) -m

1 (

,..., λ

1) -m

1 (

: ) (

) n ( n q 1

q 1

1 p

) n ( n m 1

m 1

1 1

1 1 4

  

   

  

Where 1 r

r 1

;...U U

;...V V

S [x₁....x_r] be the multivariable polynomial which was formed by generalized the

general class of polynomial as defined below

r 1

r 1

U ... U

V ... V

S [x1, ..., xr] =





) U V (

0 k

1 1

1





) U V (

0 k

r r

r

) ....k k A(V, )

V ( ) V

( _{1 U k ... r U k ...U k 1 r}

1 r r 1

1

 

 _

! k x ... ! k x

r k r

1 k 1

r 1

(2.3)

Where Vi = 0, 1,2 ... (i = 1, ..., r), U1,...Ur are arbitrary positive integers, The coefficient

A(V1, k1; ...; Vr, kr) being arbitrary constants, real or complex.

So by using (2.3) and by using the multivariable Hypergeometric function defined in (1.9), we occurred (2.1) then make its pdf form as (2.2).

Where

α1 = a + 



n

1 j

jkj and β1 = b + 



n

1 j

vjkj

Now assumed the following condition to be satisfied.

(vii) Re (a, b, r) > 0, Re (α1, β1) > 0

(ix) (a) 1 > 0, 2 > 0

(b) 1 + [V/U] > 0, 2 + [V/U] > 0

(c) 1 > 0, 2 + [V/U] > 0

(d) 1 + [V/U] > 0, 2 > 0

Where

1 = Re (a) + p + 



n

1 j

j

q and ₂ = Re (a) + p + 



n

1 j

j

q

3.1 Theorem:3 Eulerian third Integral Involving

r ..U ;... 1 U

V

S [x₁…….x_r] :

Prove that:



1

0

xm(a-1) (1-xm)b-1 U1;...U r

V

S



m v vr



) x (1 x y ,..., ) x (1 x

y m

μ m r m

1

r 1

1

 



× 1 n

n 1

;...q q : p

;...m 1;m

F

 

 

 



 _λ

m λ

m )

n (

q q

p

) n (

m m

1

x n z ,...., x 1 z ); (b ...; );... (b

: ) a (

); ( β ; ... . ); ( β : ) (

n 1

n 1

 dx

= m

1) -m

1 Γ(

) 1) Γ -m

1 Γ(

2 2

2 2

  

 



 (



 

 v k U ... k U

0 k ,..., k

r r 1 1

r 1

r r 1 1K ...U K U

v)

n 1

n 1

...q ;... q : λ p

...m ;... m λ; 1

F _

       

 

       

 

  

   



 z1,..,zn

); (b );...; (b

: λ

2) -m

1 (

,..., λ

1 -m

1 :

) a (

); ( β ...; . ); ( β λ

2) -m

1 (

,..., λ

1) -m

1 (

: ) (

) n (

q q

2 2

p

) n (

m m

2 2 2

2

1

n 1

n 1

  

   

 

(3.1)

Where α2 = a + 



n

1 j

j kj and β2 = b + 



n

1 j

vj kj

Then p.d.f form of (3.1)

F(x)

] [ F

) k , ...v . ; k , A(v )

v ( 1)

-m

1 Γ(

) ( 1) -m

1 Γ(

] [ F

] ) x (1 x y ... ) x (1 x y [ S

) x (1 x

6 n ;...q 1 :q λ p

2 ;...m 1 m λ; 1 r r 1 1 r k r v ... 1 k 1 u r v r k r U ... 1 k 1 U

0 r k ,..., 1 k

2 2

2 2

5 n ;...q 1 p:q

n ;...m 1 m 1; m r

u m

r 1 m u

m

1 r ;...v 1 u

v 1 b m 1)

(a m

 



 



  



 



 



 

 

 



m

r v v

= 0 elsewhere (3.2)

Where

5 =

   

   

λ m λ

m )

n (

q q

p

) n (

q m

1

x n z ,...., x 1 z ); (b ...; );... (b

: ) a (

); ( β ; ... . ); ( β : ) (

n 1

n 1



     

 

     

 

  

   



 _{1 n}

) n ( n q 1

q 2

2 p

) n ( n m 1

m 2

2 2

2 1

6 ,...., z

); (b );...; (b : λ

2) -m

1 (

,...., λ

1 -m

1

: ) a (

); ( β ....; . ); ( β ; λ

2) -m

1 (

,..., λ

1) -m

1 (

: ) (

z 

 

   

  

Where U1;...U r

V

S [x₁....x_r] be the multivariable polynomial which was formed by generalized the

general class of polynomial as defined below

r 1...U

U

V

S [x1, ..., xr] =



 

 V k ....U k U

0 k ,... k

r r 1 1

r 1

) ....k k A(V, V)

( _{Uk ...U k 1 r}

1 1 1



 _

×

! k x ... ! k x

r k r

1 k 1

r 1

. (3.3)

Where U1 ... Ur are arbitrary positive integers and the coefficient A(V; k1, ... kr) (V, k1 >

0, i = 1, ...., r) are arbitrary constants, real or complex.

So by using (3.3) and the multivariable Hypergeometric function defined in (1.9) we occurred (3.1) and its pdf form (3.2).

Now assumed the following condition to be satisfied.

(xiii) Re (a, b, r) > 0, Re (α2, β2) > 0

(xiv) Min (, p, q) > 0 (j = 1, 2, ...,n) (not all zero simultaneously)

(xv) (a) 1 > 0, 2 > 0

(b) 1 + [V/U] > 0, 2 + [V/U] > 0

(c) 1 > 0, 2 + [V/U] > 0

where

1 = [ Re (a) + p + _

     





n

1 j

j

q ] and 2 = Re (a) + p + 



n

1 j

j

q .

Proofs of Theorem 2 and Theorem 3: The equations (2.1) of theorem 2 and (3.1) of theorem 3, can be derived by following the same lines as given in the proof of first integral and making appropriate modifications and changes.

References:

1. DHILLON, S.S.: A Study of Generalization of Special Function of Mathematical

Physics and Applications Ph.D. Thesis, Bundelkhand Univ. India (1989).

2. GOULD, L H.W. AND HOPPER, A.T.: Operational formulas connected with two

generalisation of Hermite polymials, Duke Math. J 29 (1962), pp 51-63.

3. KARLSONS, P.W.: Reduction of Certain generalised Kampe de feriet functions Math,

Scand. 32 (1973B), pp 265-268.

4. KRALL, H.L. AND FRINK, O.: A new class of orthogonal polynomials; the Bessel

Polynomials, Trans, Amer. Math. Soc. 65 (1949), pp 110-115.

5. SINGH. R.O. AND SHRTVASTAVA, K.N.: A note on generalistion of Lagnerre and

Humbert Polynomials. Recerca (Napoli) (2) 14, Settembre dicuembre, 11-21; Errata, Ibid, (2) (15), Maggio-agosto, 63 (1964).

6. SINGH, A.: A Study of Special Function Sof Mathematics Physics and Their application

7. SRIVASTAVA, H.M. AND DOAUST, M.C. Certain generalised Neumann expansion associated with Kampe de Feriet funcion. Nederl. akad. Wetencsch. Proc. Ser. A, 72-Indag. Nath, 31 (1969), pp 449-457.

8. SRIVASTVA, H.M. AND GARG, M.: Some integrals involving general class of