Integrals Involving the Product of Two Different Wright’s Generalized Hypergeometric Functions

Integrals Involving the Product of Two Different Wright’s Generalized Hypergeometric Functions

G. P. Gautam, Yasmeen* and Sulakshana Bajaj*

*Department of Mathematics, Govt. College, Kota (Raj.) India.

Abstract:- The present paper contains four interesting results with proper validity conditions, for the product of two different Wrights Generalized Hypergeometric functions. Some special cases of the general integrals lead to interesting results for Generalized Hypergeometric Functions pFq .

Keywords :- Wrights Generalized Hypergeometric Function, Generalized Gaussian Hypergeometric Function of one variable, Legendre function, pochhammer symbol.

A.M.S. classification (2010): 33C45, 33C20, 33C60, 42C10.

The series in equation (1.2) is convergent for all values of z and it defines an integral function of z if

q

p

q

p

1 j j is positive. (1.4)

j 1 j 1

If =0 then series in (1.2) is convergent for |z|<-1 and it defines an analytic function of z for |z|< -1. Here

1. INTRODUCTION

   The Wrights Generalized Hypergeometric function pq (z), in contour integral form, is defined as

   p

   j

   j

   j

   j 1

   q

   j 1

   j

   j

   (1.5)

   a ,…., a ;

   p a

   1 j 1

   j j s sz s

   The generalized hypergeometric function pFq with p numerator

   parameters and q denominator parameters in contour integral form [2; p.43 (6)] is defined by

   1, 1

   p, p

   z

   ds …..(1.1)

   p q b ,…., b

   ;

   2 i D

   q b

   • s p p s

   1, 1

   q, q

   j j

   aj

   aj ssz

   j 1

   j 1

   a1,…., ap ;

   F

   1

   z

   j 1 ds

   …..(1.6)

   q p q b ,…., b ;

   2 i q

   Where D is a contour in the complex s plane which runs from

   bj j 1

   1

   q

   L bj s

   j 1

   s i to s i ( is an arbitrary real number) so that the points s = 0, 1, 2,.. resp. lie to the right of D.

   Wrights generalized hypergeometric function in series form [2;

   p.50 (21)], denoted by pq (z) is defined as:

   L is a contour runs from i to +i, no aj (j=1,.,p) is zero or a negative integer; arg(z) .

   n

   n

   Generalized Gaussian Hypergeometric function p Fq [1; p.73 (2)] , in

   p

   p

   A

   ,…., A ;

   Aj jn z

   series form, is defined by :

   '

   1, 1

   p, p

   z j 1

   …(1.2)

   1, 1 ,…., Bq,q ;

   n0

   q P

   B n

   n

   n

   j jnn!

   A ,…., A ;

   Aj z

   j 1

   F ' 1

   P Q

   P Q

   P z j 1

   Q

   Q

   Here p & q are non-negative integers ; 1, .., p ; 1,.., q are positive constants ; A1,..,Ap ; B1,.,Bq are complex constants such that

   B1 ,…., BQ ;

   (1.7)

   n0

   Bj n!

   n

   n

   j 1

   Aj jn 0, 1, 2,… ( j 1,…., p;

   n 0,1, 2…) (1.3)

   and the pochhammer symbol (a)n [2;p.21 (14)] is defined by

   (a n)

   ( u 1)

   u (1+r+)(1+ ) u,r

   (a)

   (a)

   ; a 0, 1, 2, 3………..

   (1.8)

   a(b 1) 1 u!( u 1) r 0

   (2+r+ )

   I ,

   (2.4)

   n a(a 1)(a 2)…….(a n 1) ; n 1, 2, 3……

   1 ; n 0

   Where

2. SOME AUXILIARY RESULTS

   We require four integrals [3; (1), (2), (20) and (3); p. 284 288] in the following forms:

   u,r

   I

   I

   ,

   Theorem (3.1)

   (u)r ( u r 1)

   r !( r 1)

3. MAIN INTEGRALS

   (1) If a > 0, b -1, Re() > -1, Re ()> -1, then

   a t (a t) a bt 2t

   (i) If a>o, b -1, Re ( )>-1, d >o, n=0,1,2,. and Re ( ) + n Re (c) > -1

   (ii) Each of '& – functions occurring in (3.1) satisfy

   P

   P

   (a bt)2

   ( , )

   u

   a bt

   dt

   convergence conditions (1.4), (1.5) and (1.1) then

   0

   a t (a t)

   ( , ) a bt 2t

   o (a bt)2 Pu a bt .

   ( u 1) (1+ )(1- + ) (2.1)

   a(b 1) 1 u! (2+u+ + )(1-u- + )

   ' z(a t)c x(a t)d

   . P Q (a bt)c p q (a bt)d dt

   (2) If a > 0, b 1, Re() > 1, Re () > 1, then

   a t (a t)

   P( , ) a bt 2t dt

   P

   zn

   ( A

   j j

   n) ( u 1)

   (a bt)2 u

   a bt

   = j 1 .

   0

   (1)u ( u 1)

   (1+)(1- +)

   Q

   j j

   j j

   n 0 n ! (B n) u! a(b 1)

   j 1

   1

   a(b 1)

   1 u!

   (1-u- +)(2+u+ +)

   (2.2)

   p 2

   ((ap ,p )), (1+ + cn,d ), (1 + + cn,d) ;

   q 2 ((b , )), (1-u – + cn,d), (2+ u + + cn,d); x

   a t (a t)

   a bt 2t a bt 2t

   q q

   …(3.1)

   3

   P( , ) P( , ) dt

   Theorem (3.2)

   (a bt)2 u a bt a bt

   0

   ( u 1)

   (1+r+)(1+r- +) J ,r

   (2.3)

   If the conditions of theorem (3.1) with and replaced by

   & resp. hold good then

   a(b 1) 1 u!

   u

   u

   r 0

   (1+r-u- +)(2+r+u+ +)

   ,

   a

   a

   t (a t)

   2

   P ( , ) a bt 2t .

   Where

   o (a bt)

   a bt

   ,r

   (1)u ( 1)( ) ( 1)

   ' z(b 1)c tc x(b 1)d td

   J , r r

   . dt

   !r !( r 1)

   P Q

   (a bt)c

   p q

   (a bt)d

   (4) If a > 0, b 1, Re () > 1, Re() > 1, then

   P

   a t (a t) a bt 2t

   P( , ) dt

   P( , ) dt

   ( u 1)(1)u

   =

   ( A

   1

   j j

   n) zn

   .

   2 u

   a(b 1)1 u! Q

   0 (a bt)

   a bt

   n 0

   (B

   1

   j j

   n) n!

   ((ap ,p )), (1+ + cn,d ), (1- + + cn,d) ; ((ap ,p )), (1+r+ + cn,k ), (1+ +dn, l );

   . p 2q2 ((b , )), (1-u – + cn,d), (2+ u + + cn,d); x

   p 2q1 ((b ,

   )), (2 + r + + +cn + dn, k l) ; x

   q q

   ….3.2

   q q

   (3.4)

   Theorem (3.3)

   If the conditions of theorem (3.1) with and replaced by

   & resp. hold good then

   Where I u,r is defined in (2.4).

   ,

   ,

   Proof of (3.1) to (3.4) :

   Expressing the – function in the L.H.S. of (3.1) in series form by (1.2) and the -function in contour integral form by (1.1), we get

   a t (a t)

   ( , ) a bt 2t

   ( , ) a bt 2t

   a t (a t)

   a bt 2t

   o 2 Pu

   P

   . P ( , ) .

   (a bt)

   a bt

   a bt

   o (a bt)2 u

   a bt

   ' z(b 1)c tc x(b 1)d td P

   .P Q c

   p q d dt

   ( A

   n) zn (a t)nc

   (a bt)

   (a bt)

   .

   n 0

   j 1 Q

   (B

   j j

   .

   n) n!(a bt)nc

   ( u 1)

   . .

   P

   j j

   j j

   ( A n) zn

   1

   J .

   J .

   ,r

   ,

   j 1 j j

   p

   a(b 1)1 u!

   n0

   Q

   1

   1

   (Bj jn) n!

   r 0

   . 2

   (aj j s) (s)(x) (a t)

   j 1

   1

   1

   q

   sd

   .ds.dt

   ((a , )), (1+r+ + cn,d ), (1+ r- ++cn,d) ;

   D (bj j s) (a bt)

   . p p

   q2

   q2

   j 1

   x

   q

   q

   p 2

   ((bq , )), (1+r-u- + cn,d), (2+ r + u + + cn,d);

   .(3.3)

   (3.5)

   ,

   ,

   Where J ,r is defined in (2.3).

   Theorem (3.4)

   If

   (i) a>o, b -1, k > 0, l > 0,

   1. Re ()+ n Re (c) > -1 and Re (

      ) + n Re (d) > -1,

      Changing the order of t-integral and summation and again of the t-integral & contour integral and then evaluating the inner t- integral by (2.1), then on interpreting the contour integral into – function by (1.1), the RHS of (3.1) follows immediately.

      This completes the proof of (3.1).

      The remaining theorems (3.2), (3.3) and (3.4) can also be proved in a similar way, as above, by using (2.2), (2.3) and (2.4) respectively instead of (2.1).

   2. Each of ' & – functions occurring in (3.4) satisfy (1.4)

   & (1.5) then

   Our results (3.1), (3.2), (3.3) and (3.4) seem to be new.

4. SPECIAL CASES

a t (a t)

z(b 1)c tc (a t)d

If we replace

p q

(z) by

p Fq

(z) ;

' (z) by

F ' (z)

'

'

o (a bt)2 P Q

(a bt)cd .

and take d=1 in (3.1), (3.2),(3.3), we get the cases (4.1),

P Q

P Q

P Q

P Q

(4.2),(4.3) respectively. If we take k=1, l=0 in (3.4), We get (4.4)

x(b 1)k tk (a t)l

( , ) a bt 2t

& taking k=0, l=1 in (3.4), We get the case (4.5).

.p q

(a bt)k l

.Pu

a bt

dt

P

n

Cor. (4.1)

a

t (a t)

( , ) a bt 2t

( Aj jn) z

j 1

( u 1) u

.

Iu,r .

o (a bt)

2 Pu

a bt .

Q

j j

j j

n 0 (B

j 1

n) n!a(b 1)1 u!( u 1) r 0

,

z(a t)c x(a t)

. F '

F dt

P Q (a bt)c p q

(a bt)

a1,…,ap , 1+ r + + cn,1 + r – + + cn ;

P

( A )

zn ( u 1) (1 nc) (1 nc)

. p2 Fq2

b ,….,b ,1+ r + + cn u , 2 + r +u+ + +cn; x

j n 1 q

j 1

Q

n 0 (B )

n! a(b 1)1

u ! (2 u nc) (1 u nc)

.(4.3)

j 1

j n

a ,a ,….,a , 1+ +cn ,1- + +cn ;

,r

J

J

, is defined in (2.3).

Cor. (4.4)

F 1 2 p x

q2

q2

p2

b1,b 2 ,….,bq , 2+u+ + +cn, 1-u- + +cn;

.. (4.1)

a t (a t)

( , ) a bt 2t

' z(b 1)c t c (a t)d

o (a bt)2 Pu

a bt

P FQ

(a bt)cd .

Cor. (4.2)

. F x(b 1)t dt

a t (a t)

( , ) a bt 2t

p q (a bt)

o (a bt)2 Pu a bt .

z(b 1)c tc x(b 1)t

P

( A )

zn ( u 1) (1 dn)

'

j n

. p FQ

(a bt)c

. p Fq

(a bt) dt

j 1 .

Q

Q

n 0

Q

Q

(B )

n! a(b 1)1 u !

j 1 j n

r 0

r 0

P n u u (u)r ( u 1)r (1 r cn)

Q

( Aj )n z ( u 1)(1)

j 1

(1 nc) (1 cn)

.

. r ! ( r 1) (2 r cn dn).

n 0

(B ) n! a(b 1)1 u! (2 cn u ) (1 u nc)

j 1 j n

a1,…, ap ,1 r cn ;

. p1Fq1

b ,…,b , 2 r cn dn; x

a1,…..,ap , 1+ +cn,1- + +cn ;

F x

1 q

(4.4)

p2

q2 b ,……,b ,1-u- + +cn, 2+u+ + +cn;

1 q

..(4.2)

Cor. (4.5)

Cor.(4.3)

a t (a t)

( , ) a bt 2t ' z(b 1)c t c (a t )d

o 2 Pu

P FQ

cd .

a t (a t)

( , ) a bt 2t

( , ) a bt 2t

(a bt)

a bt

(a bt)

o (a bt)2 Pu

a bt

P

a bt .

x(a t)

. F '

z(b 1)c tc

F x(b 1)t dt

. p Fq (a bt) dt

P Q

(a bt)c

p q

(a bt)

P

( A )

zn ( u 1) (1 dn)

P

( A )

zn ( u 1) (1 r cn)

n 0

j 1 j n

Q

.

1

Q

j 1 j n

. j 1

(Bj )n

n! a(b 1) u!

n 0

(B )

n! a(b 1)1 u! (1 r cn u )

u (u) ( u 1) (1 r cn)

j 1 j n

. r r .

(1 r cn)

,r

r 0

r ! ( r 1) (2 r cn dn)

r 0

r 0

. (2 r u cn) J , .

a ,…, a ,1 dn ;

. F 1 p x

q1

q1

b ,…,b , 2 r cn dn;

p1

1

q

1

q

1

q

1

q

(4.5)

REFERENCES:-

1. Rainville, E.D. (1960). Special Functions. The Macmillan Co. Inc., New York. Reprinted by Chelsea Pub. Co. Bronx, New York, 1971.

2. Srivastava, H.M. and Manocha, H.L. (1984). A Treatise on Generating Functions. Halsted Press, Ellis Horwood Ltd., Chichester, U.K.John Wiley and Sons, New York, Chichester, Brisbane and Toronto.

3. Erdelyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F.G. (1954). Tables of Integral Transforms. Vol. II. Mc Graw-Hill, New York.