New Finite Integrals of Generalized Meliin-Barnes Type of Contour Integrals

© 2012. Praveen Agarwal & Mehar Chand.This is a research/review paper, distributed under the terms of the Creative Commons Attribution-Noncommercial 3.0 Unported License http://creativecommons.org/licenses/by-nc/3.0/), permitting all non commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Global Journal of Science Frontier Research Mathematics & Decision Sciences Volume 12 Issue 3 Version 1.0 March 2012 Type : Double Blind Peer Reviewed International Research Journal Publisher: Global Journals Inc. (USA) Online ISSN: 2249-4626 & Print ISSN: 0975-5896

New Finite Integrals of Generalized Meliin-Barnes Type of Contour Integrals

By Praveen Agarwal & Mehar Chand

Anand International College of Engineering

Abstract - In the present paper, we obtain three new finite integral formulas. These formulas involve the product of a general class of polynomials and the generalized Meliin - Barnes type of contour integrals. Mainly we are using series representation of the H -function given by Agarwal [14], Agarwal and Jain [13]. These integral formulas are unified in nature and act as the key formulas from which we can obtain as their special cases. By giving suitable values to the parameters, our main integral formulas are reduces to the Fox H-function, the G-function and generalized wright hypergeometric function.

Keywords : H -function, general class of polynomial, generalized wright hypergeometric function.

GJSFR-F Classication : (MSC 2000) 33C45, 33C60

New Finite Integrals of Generalized Meliin-Barnes Type of Contour Integrals Strictly as per the compliance and regulations of :


Praveen Agarwal & Mehar Chand

where

j

j

m nA

j j j jj 1 j 1q p

Bj j j j

j m 1 j n 1

(b ) { (1 a )}

{ (1 b )} (a ) (1.2)

It may be noted that the contains fractional powers of some of the gamma

function and , , ,m n p q are integers such that 1 ,1m q n p1, 1,

,j jp q are positive real

numbers and 1, 1,

,j jn m qA B may take non-integer values, which we assume to be positive

for standardization purpose. 1,j p

and 1,j q

are complex numbers.

The nature of contour L , sufficient conditions of convergence of defining integral

(1.1) and other details about the H -function can be seen in the papers [9, 10]. The

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© 2012 Global Journals Inc. (US)

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E-mail : [email protected] Author : Department of mathematics, Malwa College of IT and Management, Bathinda-151001, India.

behavior of the H -function for small values of |z| follows easily from a result given by

Rathie [3]:

1.

A.A

. In

ayat-

Huss

ain

, N

ew p

roper

ties

of hyper

geo

met

ric

seri

es d

eriv

able

fro

m F

eynm

an

inte

gra

ls:

I. T

ransf

orm

ati

on a

nd r

eeduca

tion f

orm

ula

e, J

.Phys.

A:M

ath

.Gen

.20 (

1987), 4

109-4

117.

2.

A.A

. In

ayat-

Huss

ain

, N

ew p

roper

ties

of hyper

geo

met

ric

seri

es d

eriv

able

fro

m F

eynm

an

inte

gra

ls:

II.A

gen

eraliza

tion of

the

H-funct

ion,

J.P

hys.

A.M

ath

.Gen

.20 (1

987), 4119-4128.

Keywords : H -function, general class of polynomial, generalized wright hypergeometric function.

I. INTRODUCTION

H -function is defined and represented in the following manner [10].

j j j j jm,n m,n 1,n n 1,pp,q p,q

Lj j j j j1,m m 1,q

a , ;A , a , 1H z H z z d

2 ib , ;B , b , z 0 (1.1)

Abstract - In the present paper, we obtain three new finite integral formulas. These formulas involve the product of a general class of polynomials and the generalized Meliin- Barnes type of contour integrals. Mainly we are using series representation of the H - function given by Agarwal [14 ], Agarwal and Jain [13 ]. These integral formulas are unified in nature and act as the key formulas from which we can obtain as their special cases. By giving suitable values to the parameters, our main integral formulas are reduces to the Fox H-function, the G-function and generalized wright hypergeometric function.

In 1987, Inayat-Hussain [1,2] was introduced generalization form of Fox's H-function, which is popularly known as H -function. Now H -function stands on fairly firm footing through the research contributions of various authors [1-3, 9- 10, 13-15].

Author : Department of mathematics, Anand Internation College of Engineering, Jaipur-302012, India.

Ref.

m,np,qH z o |z| ;

Where

1min Re ,| | 0j

j mj

bz

(1.3)

1

1 1 1 1| | | | | | | | 0

q qm n

j j j j j jj j m j j n

B b B a A A , 0 | |z (1.4)

The following function which follows as special cases of the H -function will be

required in the sequel [10]

1,1, 1,, 1

1, 1,

, ; 1 , ,;

, ; 0,1 , 1 , ,

j j j j j jpp pp qp q

j j j j j jq q

a A a Az H z

b B b B (1.5)

The general class of polynomials 1

1

,...,,..., [ ]r

r

m mn nS x will be defined and represented as

follows [6, p.185, eqn. (7)]:

1 11

11

[ / ] [ / ],...,

,..., ,0 0 1

( )[ ] ...

!

r ri i ir

r i i

r

n m n m ri m l lm m

n n n ll l i i

nS x A x

l (1.6)

where 1 1,..., 0,1, 2,...; ,...r rn n m m are arbitrary positive integers, the coefficients

, ( , 0)i in l i iA n l are arbitrary constants, real or complex. 1

1

,...,,..., [ ]r

r

m mn nS x yields a number of known

polynomials as its special cases. These includes, among other, the Jacobi polynomials, the

Bessel Polynomials, the Lagurre Polynomials, the Brafman Polynomials and several

others [8, p. 158-161].

The following formulas [11, p.77, Eqs. (3.1), (3.2) & (3.3)] will be required in our

investigation.

12

1/20

( 1/ 2)( 1)2 (4 )

p

p

b pax c dx

x pa ab c,

0; 0; 4 0;Re( ) 1/ 2 0a b c ab p (1.7)

12

2 1/20

1 ( 1/ 2)( 1)2 (4 )

p

p

b pax c dx

x px b ab c,

0; 0; 4 0;Re( ) 1/ 2 0a b c ab p (1.8)

12

2 1/20

( 1/ 2)( 1)(4 )

p

p

b b pa ax c dx

x px ab c,

0; 0; 4 0;Re( ) 1/ 2 0a b c a b p (1.9)

II. MAIN INTEGRALS

First Integral


)

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1

1

1 1

1

12 2 2,,...,,,...,

10

[ / ] [ / ]

,1/20 0 1

...!2 (4 )

i

r

r

r ri i

i i

r

u v wr m nm m

p qn n ii

n m n m ri m l i

n lul l i i

b b bax c S y ax c H z ax c dx

x x x

n yA

la ab c (4 )

i

i i

l

v lab c

6.

H.M

.Sriv

as ta

va,

A

multilin

ear

gen

eratin

g

functio

n

for

the

Konhauser

sets of

bio

rthogonal

poly

nom

ials su

ggested

by th

e Laguerre p

oly

nom

ials, P

aci_c J

.Math

.117, (1

985), 1

83-1

91.

10.K

.C.

Gupta

, R

. Jain

and R

. A

garw

al,

On ex

istence

conditio

ns

for

a gen

eralized

Mellin

-Barn

es type in

tegra

l Natl A

cad S

ci Lett. 3

0(5

-6) (2

007), 1

69-1

72.

Ref.

1,1, 1 1,1, 1

1, 11,

1 / 2 , ;1 , , , , ( , )

(4 ) , , ( , , ) , , ;1

r

i i j j j j j n pim n np q w r

j j j j j m q i iim

u v l w a A azH

ab c b b B u v l w(2.1)

The above result will be converge under the following conditions

i. 0; 0; 4 0a b c ab and ,iv w are positive integers.

ii.1

1Re min 02

j

j mj

bu w

iii. 11| arg |2

z , where 1 is given by equation (1.4)

Second Integral

1

1

1 1

1

12 2 2,,...,,,...,2

10

[ / ] [ / ]

,1/20 0 1

1

...!2 (4 )

i

r

r

r ri i

i i

r

u v wr m nm m

p qn n ii

n m n m ri m l

n lul l i i


x x xx

nA

lb ab c

1,1, 1 1,1, 1

1, 11,

(4 )

1/ 2 , ;1 , , , , ( , )

(4 ) , , ( , , ) , , ;1

i

i i

li

v l

r


j j j j j m q i iim

y

ab c

u v l w a A azH

ab c b b B u v l w (2.2)


i. 0; 0; 4 0a b c ab and ,iv w are positive integers.

ii.1

1Re min 02

j

j mj

bu w

iii. 11| arg |2


Third Integral

1

1

1 1

1

12 2 2,,...,,,...,2

10

[ / ] [

,1/20 0 1

...!(4 )

i

r

r

i i

i i

r

u v wr m nm m

p qn n ii

n m ri m l

n lul l i i

b b b ba ax c S y ax c H z ax c dx

x x xx

nA

lab c

/ ]

1,1, 1 1,1, 1

1, 11,

(4 )

1/ 2 , ;1 , , , , ( , )

(4 ) , , ( , , ) , , ;1

ir r

i i

ln mi

v l

r


j j j j j m q i iim

y

ab c

u v l w a A azH

ab c b b B u v l w(2.3)

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i. 0; 0; 4 0a b c a band ,iv w are positive integers.

ii.1

1Re min 02

j

j mj

bu w

Notes

iii. 11| arg |2


Proof : To prove the first integral, we express H - function occurring on the L.H.S. of

equation (2.1) in terms of Mellin-Barnes type of contour integral given by equation (1.1)

and the general class of polynomials 1

1

,...,,..., [ ]r

r

m mn nS x in series form with the help of equation

(1.6) and then interchanging the order of integration and summation, we get:

11 1

1

12[ / ] [ / ]

, 20 0 1 0

1... ( )! 2

ri iir r

i i i

i i

r

u v l wn m n m r

i m l ln l i

l l i i L

n bA y ax c dx z d

l i x (2.4)

Further using the result (1.7) the above integral becomes

1 1

1

1

[ / ] [ / ]1

,1/2 10 0 1

1

1/ 2( ) 1...! 22 (4 ) (4 ) 41

ir ri i

i i i ir

rln m n m r i iii m l i

n lu v l wrl l i i L

i ii

u v l wn y zA d

l ia ab c ab c ab cu v l w

(2.5)

(3.1) If we put 1,j jA B

H - function reduces to Fox’s H-function [7, p. 10, Eqn.

(2.1.1)], then the equation (2.1), (2.2) and (2.3) takes the following form.

1

1

1 1

1

12 2 2,..., ,

,..., ,10

[ / ] [ / ]

,1/20 0 1

...!2 (4 )

i

r

r

r ri i

i i

r

u v wr

m m m nn n i p q

i

n m n m ri m l i

n lul l i i


x x x

n yA

la ab c

1 1,, 11, 1

11,

1 / 2 , ;1 , ,

(4 )(4 ) , , , ;1

i

i i

rl

i i j ji pm np qv l w r

j j i iiq

u v l w azH

ab cab c b u v l w

(3.1.1)

1

1

1 1

1

12 2 2,..., ,

,..., ,210

[ / ] [ / ]

,1/20 0 1

1

...!2 (4 )

i

r

r

r ri i

i i

r

u v wr

m m m nn n i p q

i

n m n m ri m l

n lul l i i


x x xx

nA

lb ab c

1 1,, 11, 1

11,

1 / 2 , ;1 , ,

(4 )(4 ) , , , ;1

i

i i

rl

i i j ji pi m np qv l w r

j j i iiq

u v l w ay zH

ab cab c b u v l w

(3.1.2)


)

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1

1

1 1

1

12 2 2,..., ,

,..., ,210

[ / ] [

,1/20 0 1

...!(4 )

i

r

r

i i

i i

r

u v wr

m m m nn n i p q

i

n m ri m l

n lul l i i

b b b ba ax c S y ax c H z ax c dx

x x xx

nA

lab c

/ ] 1 1,, 11, 1

11,

1 / 2 , ;1 , ,

(4 )(4 ) , , , ;1

ir r

i i

rln m i i j ji pi m n

p qv l w r

j j i iiq

u v l w ay zH

ab cab c b u v l w

(3.1.3)

7.

H.M

. Sriv

asta

va,

K.C

. G

upta

and

S.P

. G

oyal,

The

H-fu

nctio

n

of

one

and

two

varia

bles w

ith a

pplica

tions, S

outh

Asia

n P

ublish

ers, New

Deh

li, Madra

s (1982).

Then interpreting with the help of (1.1) and (2.5) provides first integral.

The proof of second and third integral can be developing on the lines similar to those given with first integral with the help of the result (1.8) and (1.9) respectively.

Ref.

The Conditions of validity of (3.1.1), (3.1.2) and (3.1.3) easily follow from those

given in (2.1), (2.2) and (2.3).

(3.2) By applying the results given in (2.1), (2.2) and (2.3) to the case of Hermite

polynomials [4, 5] by setting 2 /2 1( )2

nn nS x x H

x in which

1 1 ,,..., 2; ,..., ; 1; , , ( 1)i i

lr r i i n lm m n n n r v v y y A , we have the following interesting results.

/ 212 2 2 2

,,

0

[ /2]2

1/20

1 12

( 1)!2 (4 ) (4 )

nu v v wm np qn

lnll

u vll

b b b bax c y ax c H ax c H z ax c dx

x x y x x

n yH

la ab c ab c

, 1 1, 1,1, 1

1, 1,

1 / 2 , ;1 , , , , ,

(4 ) , , , , , , ;1

j j j j jm n n n pp q w

j j j j jm m q

u vl w a A azab c b b B u vl w

(3.2.1) / 212 2 2 2

,,2

0

[ /2]2

1/20

1 1 12

( 1)!2 (4 ) (4 )

nu v v wm np qn

lnll

ul

b b b bax c y ax c H ax c H z ax c dx

x x y x xx

n y

lb ab c ab c

, 1 1, 1,1, 1

1, 1,

1 / 2 , ;1 , , , , ,

(4 ) , , , , , , ;1

j j j j jm n n n pp qvl w

j j j j jm m q

u vl w a A azH

ab c b b B u vl w

(3.2.2) / 212 2 2 2

,,2

0

[ /2]2

1/20

1 12

( 1)!(4 ) (

nu v v wm np qn

lnll

ul

b b b b ba ax c y ax c H ax c H z ax c dx

x x y x xx

n y

lab c

, 1 1, 1,1, 1

1, 1,

1 / 2 , ;1 , , , , ,

4 ) (4 ) , , , , , , ;1

j j j j jm n n n pp qvl w

j j j j jm m q

u vl w a A azH

ab c ab c b b B u vl w

(3.2.3)


given in (2.1), (2.2) and (2.3).

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((3.3) By applying the our results given in (2.1), (2.2) and (2.3) to the case of

Lagurre polynomials [4, 5] by setting 2 ( ) [ ]n nS x L x in which

1 1 ,1,..., 1; ,..., ; 1; , ,

1i ir r i i n l

l

nm m n n n r v v y y A

n, we have the following interesting

results.

12 2 2,,

0

[ /2]2

1/20

1! 12 (4 ) (4 )

u v wm np qn

lnl

u vll l

b b bax c L y ax c H z ax c dx

x x x

n ynnla ab c ab c

, 11, 1

1 / 2 ,

(4 )m np q w

u vl wzH

ab c1, 1,

1, 1,

;1 , , , , ,

, , , , , , ;1

j j j j jn n p

j j j j jm m q

a A a

b b B u vl w(3.3.1)

Notes

12 2 2,,2

0

[ /2]2

1/20

, 11, 1

1

1! 12 (4 ) (4 )

1/ 2

(4 )

u v wm np qn

lnl

u vll l

m np q w

b b bax c L y ax c H z ax c dx

x x xx

n yn

nlb ab c ab c

u vzH

ab c1, 1,

1, 1,

, ;1 , , , , ,

, , , , , , ;1

j j j j jn n p

j j j j jm m q

l w a A a

b b B u vl w (3.3.2)

12 2 2,,2

0

[ /2]2

1/20

, 11, 1

1! 1(4 ) (4 )

1

(4 )

u v wm np qn

lnl

u vll l

m np q w

b b b ba ax c L y ax c H z ax c dx

x x xx

n ynnlab c ab c

zH

ab c1, 1,

1, 1,

/ 2 , ;1 , , , , ,

, , , , , , ;1

j j j j jn n p

j j j j jm m q

u vl w a A a

b b B u vl w (3.3.3)


given in (2.1), (2.2) and (2.3).

(3.4) If we put 1; 1j j j jA B , in (1.1) then the H -function reduces to the

general type of G-function [12] i.e., 1, 1,,

1, 1,

,1,1 , ,1

,1,1 , ,1

j jm n n n pp q

j jm m q

a aH z

b b

1,

1,

,1

,1

j p

j q

aG z

b, So using same

assumptions in the equations (2.1), (2.2) and (2.3) then they takes the following form.


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1

1

1 1

1

12 2 2,..., ,

,..., ,10

[ / ] [ / ]

,1/20 0 1

...!2 (4 )

i

r

r

r ri i

i i

r

u v wr

m m m nn n i p q

i

n m n m ri m l i

n lul l i i

b b bax c S y ax c G z ax c dx

x x x

n yA

la ab c

1 1,, 11, 1

11,

1 / 2 , ;1 , ,1

(4 )(4 ) ,1 , , ;1

i

i i

rl

i i ji pm np qv l w r

j i iiq

u v l w azG

ab cab c b u v l w

(3.4.1)

1

1

1 1

1

12 2 2,..., ,

,..., ,210

[ / ] [ / ]

,1/20 0 1

1

...!2 (4 )

i

r

r

r ri i

i i

r

u v wr

m m m nn n i p q

i

n m n m ri m l

n lul l i i

b b bax c S y ax c G z ax c dx

x x xx

nA

lb ab c

1 1,, 11, 1

11,

1 / 2 , ;1 , ,1

(4 )(4 ) ,1 , , ;1

i

i i

rl

i i ji pi m np qv l w r

j i iiq

u v l w ay zG

ab cab c b u v l w

(3.4.2)

1

1

12 2 2,..., ,

,..., ,210

i

r

r

u v wr

m m m nn n i p q

i

b b b ba ax c S y ax c G z ax c dx

x x xx

12.M

eijer, C.S

., On th

e G-fu

nctio

n, P

roc. N

at. A

cad. W

etensch

, 49, p

. 227 (1

946).

Ref.

1 1

1

[ / ] [

,1/20 0 1

...!(4 )

i i

i i

r

n m ri m l

n lul l i i

nA

lab c

/ ] 1 1,, 11, 1

11,

1 / 2 , ;1 , ,1

(4 )(4 ) ,1 , , ;1

ir r

i i

rln m i i ji pi m n

p qv l w r

j i iiq

u v l w ay zG

ab cab c b u v l w

(3.4.3)


given in (2.1), (2.2) and (2.3).

(3.5) If we put 1 1, 1, 1, 0, 1, 1 , 1j j j jn p m q q b a a b b , in (1.1) then the H -

function reduces to generalized wright hypergeometric function [16] i.e.

1, 1, 1,, 1

1, 1,

1 , ; , ;;

0,1 , 1 , ; , ;

j j j j j jp p pp q p q

j j j j j jq q

a A a AH z z

b B b B, using same assumptions in the equations

(2.1), (2.2) and (2.3) then they takes the following form.

1

1

1 1

1

12 2 21,,...,

,...,10

1,

[ / ]

1/20 1

, ;;

, ;

...!2 (4 )

i

r

r

i i

u v wr j j j pm m

p qn n ii j j j q

n m ri m l

ul i i

a Ab b bax c S y ax c z ax c dx

x x xb B

n

la ab c

[ / ] 1 1,1 1,

011,

1 / 2 , ;1 , , ;;

(4 )(4 ) , ; , , ;1

ir r

i i i ir

rln m i i j j ji pi

p qn l v l wrl

j j j i iiq

u v l w a Ay zA

ab cab c b B u v l w

(3.5.1)

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1

1

1 1

1

12 2 21,,...,

,...,210

1,

[ / ]

1/20

, ;1 ;, ;

...!2 (4 )

i

r

r

i i

u v wr j j j pm m

p qn n ii j j j q

n mi m l

ul i i

a Ab b bax c S y ax c z ax c dx

x x xx b B

n

lb ab c

[ / ] 1 1,1 1,

0 111,

1 / 2 , ;1 , , ;;

(4 )(4 ) , ; , , ;1

ir r

i i i ir

rln m r i i j j ji pi

p qn l v l wrl

j j j i iiq

u v l w a Ay zA

ab cab c b B u v l w

(3.5.2)

1

1

1 1

1

12 2 21,,...,

,...,210

1,

[ / ]

1/20

, ;;

, ;

...(4 )

i

r

r

i

u v wr j j j pm m

p qn n ii j j j q

n mi m l

ul

a Ab b b ba ax c S y ax c z ax c dx

x x xx b B

n

ab c

[ / ] 1 1,1 1,

0 111,

1 / 2 , ;1 , , ;;

! (4 )(4 ) , ; , , ;1

ir ri

i i i ir

rln m r i i j j ji pi

p qn l v l wrl i i

j j j i iiq

u v l w a Ay zA

l ab cab c b B u v l w

(3.5.3)


given in (2.1), (2.2) and (2.3).

iv. ACKNOWLEDGMENTS

The authors are thankful to the Professor H.M. Srivastava (University of Victoria,

Canada) for his kind help and suggestion in the preparation of this paper. 16.W

right, E

.M., (

1935a), T

he

asy

mpto

tic

expansi

on o

f th

e gen

eralize

d h

yper

geo

met

ric

funct

ion. J. London M

ath

. Soc.

10. 286-2

93

Ref.

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Scienc

eFr

ontie

rResea

rch

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olum

eXII

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e e

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20

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arch


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REFERENCES RÉFÉRENCES REFERENCIAS

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Notes

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