CERTAIN INTEGRAL PROPERTIES OF GENERALIZED CLASS OF POLYNOMIALS AND GENERALIZED CONTOUR INTEGRAL ASSOCIATED WITH FEYNMAN INTEGRALS

(1)

International Journal of Physics and Mathematical Sciences ISSN: 2277-2111 (Online) An Online International Journal Available at http://www.cibtech.org/jpms.htm 2011 Vol. 1 (1) October-December, pp.1-8/Praveen Agarwal and Mehar Chand

Research Article

1

CERTAIN INTEGRAL PROPERTIES OF GENERALIZED CLASS OF POLYNOMIALS AND GENERALIZED CONTOUR INTEGRAL

ASSOCIATED WITH FEYNMAN INTEGRALS

*Praveen Agarwal¹ and Mehar Chand²

1Department of Mathematics, Anand Intenational College of Engineering, Jaipur-303012, India

2Department of Mathematics, Malwa College of IT and Management, Bathinda-151001, India

*Author for Correspondence ABSTRACT

The object of the present paper is to discuss certain integral properties of a general class of polynomials and I-function, proposed by Inayat-Hussain which contains a certain class of Feynman integrals. We establish certain new double integral relations pertaining to a product involving general class of polynomials and I-function. These double integral relations are unified in nature and act as key formulae from which we can obtain as their special cases, double integral relations concerning a large number of simpler special function and polynomials. For the sake of illustration, we record here some special cases of our main results which are also new and of interest by themselves. The results established here are basic in nature and are likely to find useful applications in several fields.

Subject Classification: (MSC 2010) 33C60, 33C45.

Key Words: Feynman Integrals, I-Function, General Class Of Polynomials

INTRODUCTION

The I-function will be defined and represented as follows [2]

   

 

, , ,

, 1, 1, 1

, ; , , , 2

1, 1,

I

 

   

  

 

  

 

 

  

 

aj j aji ji

m n z n n pi z d

p q ri i bj j m bji ji m qi iL

(1) where

 

   

1

1 1

1

1 1 1

   

 

   

    

 

 

  

      

  

 

      

m n

bj j aj j

j j

q p

r i i

bji ji aji ji

i j m j n

(2)

and

m n p q

, , _i, _i are integers satisfy ⁰^

n

^

p

i^,1^

m

^

q i

i



^^1,...,

r 

, r is finite,

  

_j, _j, _ji,



_ji are positive numbers and

a b a b

_j, _j, _ji, _ji are complex numbers. I-function which is a generalized form of the well known Fox's H-function [4, p.10, Eqn.(2.1.1)]. In the sequel the I-function will be studied under the following conditions of existence:

where

(I)

0, arg

2  

ⁱ



i

A z A

(3)

(II)

0, arg

2  

ⁱ



i

A z A

and

Re( B  1)  0

(4)

where

(2)

2

1 1 1 1

, (1, 2,..., )

   

     

         

i i

p q

n m

i j ji j ji

j j n j j m

A i r

⁽⁵⁾

and

   

1 1 1 1

1 , 1, 2,...

2

     

           

i i

q p

m n

j ji j ji i i

j j m j j n

B b b a a p q i r

(6)

The general class of polynomials

S

n ,...,n^{m ,...,m}₁¹ _r^r

  x

will be defined and represented as follows [3, p.185, Eqn.

(7)]:

 

1

1 1 r r i

,..., i i i

n ,..., ,

1 i

1 r

( n ) [n / m ] [n / m ] m

... x

0 0 ^ !



  

 



r

i i r

m m r n n l

i

l l

x A

l l l

S

⁽⁷⁾

where

n ,...,n

₁ _r 0 1 2

, , ,...;m ,...,m

₁ _rare arbitrary positive integers, the coefficients

A

n ,l_i _i

 n ,l

i i ^⁰



are arbitrary constants, real or complex.

S

n ,...,n^{m ,...,m}₁¹ _r^r

  x

yeilds a number of known polynomials as its special cases. These includes, among other, the Jacobi polynomials, the Bessel Polynomials, the Hermite Polynomials, the Lagurre Polynomials, the Brafman Polynomials and several others [5, p. 158-161].

Main Results

We shall establish the following results:

(A)

 

  

   

  ₁  ₁

1 1

0 0

1 1 1 1 1

1 1 1 1 1 1

 

 



 

            

 

               

       

 

^r^r ⁱ ⁱ ^jj j^j ^,n,m ^jiji ^jijiⁿm ^,pi,qi

a b

a , , a ,

m ,...,m m ,n

n ,...,n p ,q ;r b , , b ,

x y y xy x ty y t

S I dxdy

xy xy x y xy xy

 

 

   

^ ^

 

   

1

1 1

1 1 1

1 1 1 1

1 1

0 0

 

 



  

 

 

 

       



    

 



i i

1 1 r r r i

i i i

n ,l i

i i

1 r

b, , a ,j j , aji, ji

m,n z ,n n ,pi

pi ,qi ;r b ,j j ,m, b ,ji ji m ,qi, a b l ,i I

n l l

... t a l

l l l

n / m n / m

m A

! ⁽⁸⁾

The above result is valid under the conditions (3), (4); Re



^a^{ }^b ^b^j ^^j



^^{0 1}

^

^ ^j^^{m and}

^

^{a,b are}

positive. Also 0 x 1 and 0 y1.

Proof of the above result- In the left hand side of equation (8) put the value of i i

 

m ,n p ,q ;r

I x and

S

n ,...,n^{m ,...,m}₁¹ _r^r

  x

from (1) and (7) respectively, interchanging the order of integration and summation then making the use of known result [1, p.145], we get the result (8) after little simplification.

(B)

 

₁¹

 

_^ _{ }^{ }¹ _^ ¹

1 1

0 0

 

 



^



 

   

  

 

 

^r^r ⁱ ⁱ j^j ^jj ,m^,n ^jiji ^jiji ⁿm ^,pi,qi

a , , a ,

m ,...,m

b a m ,n

n ,...,n p ,q ;r b , , b ,

s t t s S s I t dsdt

 

 

 

   

^ ^

 

   

1

1 0

1 1

1 1 1

1 1 1 1

1 1

0 0

 

 





  



  

 

 

 

       



     

 



ⁱ ⁱ



ⁱ

1 1 r r r i

a b l

i i i

n ,l i

i i

1 r

b, , a ,j j , aji, ji

m,n z ,n n ,pi dz

pi ,qi ;r b ,j j ,m, b ,ji ji m ,qi, a b l ,i I

n l l

... t a l z z

l l l

n / m n / m m

! A

(9)



a^{ }b bj ^j



^^{0 1}

^

^ j^m and

^

a,b are positive. Also 0   s and 0   t .

(3)

3 Proof of the above result: In the left hand side of equation (9) put the value of i i

 

m ,n p ,q ;r

I x and

S

n ,...,n^{m ,...,m}₁¹ _r^r

  x

(C)

    

₁¹

   

_^ _{ }^{ }¹ _^ ¹

1 1

0 0

1 1 ^r 1 1 ^j ^j ^,n ^ji ^ji ⁿ ^,pi

r i i j j ,m ji ji m ,qi

a , , a ,

a b a m ,...,m m ,n

n ,...,n p ,q ;r b , , b ,

f st s t t S t s I t ^ ^ dsdt

 



   

       

 

 

 

 

  

 

   

^ ^

 

   

1

1 0

1 1

1 1 1

1 1 1 1 1

1 1

1

0 0

i

i i

1 1 r r r i i i i a b l

n ,l i

i i

1 r

b, , a ,j j , aji, ji

m,npi ,qi ;r z b ,j j ,m, b ,ji,nji m ,qin, ,pia b l ,i dz I

n l l

... t a l f z z

l l l

n / m n / m m

! A

 

 

  



  

 

 

  

       



      

 

 

(10)



a^{ }b bj ^j



^^{0 1}



^ j^m and



a,b are positive. Also 0  and 0s 1   . t 1

Proof of the above result: In the left hand side of equation (10) put the value of i i

 

m ,n p ,q ;r

I x and

S

n ,...,n^{m ,...,m}₁¹ _r^r

  x

(D)

 

   

  ₁  ₁

1 1

0 0

1 1 1 1 1

j j ji ji

,n n ,pi

r

r i i j j ji ji

,m m ,qi

a b

a , , a ,

m ,...,m m ,n

n ,...,n p ,q ;r b , , b ,

x y y x y y ty

S I dxdy

xy xy x xy xy



 

 



            

       

      

     

 

 

 

   

^ ^

 

   

1

1 1

1 1 1

1 1

1

0 0 ⁱ ⁱ

1 1 r r r i

i i n ,l

i i

1 r

a l , , a ,i j j , a ,ji ji

m,n t ,n n ,pi

pi ,qi ;r b ,j j ,m, b ,ji ji m ,qi, a b l ,i I

n l

... b

l l l

n / m n / m

m A

!

  

  



    

 

 

 

        



    

 



⁽¹¹⁾



a b  bj j



^{0 1}



 jm



and a,b, are positive. Also 0 x 1 and 0y1.

Proof of the above result: In the left hand side of equation (11) put the value of i i

 

m ,n p ,q ;r

I x and

S

n ,...,n^{m ,...,m}₁¹ _r^r

  x

Special Cases

(I) By applying the our results given in (A), (B), (C) and (D) to the case of Hermite polynomials [5] by

setting ²

 

² ¹

2

n/

n n

S x x H

x

 

  

 

in which

m ,...,m

₁ _r 2

; n ,...,n

₁ _r 

n;r

1;An ,l_i _i  

 

¹ ^l, we have the following interesting results.

(A1)

 

  

 

   

  ₁  ₁

1 1

1 1 2

0 0

1 1 1 1 1 1

1 1 1 1 1 1 1

2 1

j j ,n ji jin , pi

i i j j ,m ji jim ,qi

a b n /

a , , a ,

m ,n

n p ,q ;r b , , b ,

H I dxdy

xy xy x y xy x ty xy

xy

 

 



 

 

 

  

           

 

                 

        

  

 

 

(4)

4

 

   

 

   

^ ^

 

   

2 1 1

1 1 1

1 1 1 1

1 1

1 2

0

l l m,npi ,qi ;r t b ,jb, , a ,j ,mj, b ,jji,n, aji mji, ,qiji n, ,pia b l ,

n tl a l I

l l !

n / _ _

 

  

 

 

 

       

     



(12)

The conditions of convergence of the above result can be easily obtained from those of (8) (B1)

 

_^ _{ }^{ }¹ _^ ¹

1 1

1 2 1

0 0

1 2

j j ,n ji ji n ,pi

i i j j ,m ji ji m ,qi

a , , a ,

b a n / m ,n

n p ,q ;r b , , b ,

s t t s H I t dsdt

s

 

 



^



 

      

    

 

 

 

   

 

     

^ ^

 

   

2 1

0

1 1

1 1 1

1 1 1 1

1 1

1 2

0

l a b l

l m,n z b, , a ,j j ,n, aji, ji n ,pi dz

pi ,qi ;r b ,j j ,m, b ,ji ji m ,qi, a b l ,

n I

a l z z

l l !

n / _ _

 





  

  

 

 

 

       

     





⁽¹³⁾

The conditions of convergence of the above result can be easily obtained from those of (9) (C1)

    

   

_^ _{ }^{ }¹ _^ ¹

1 1

2 1 1 2

0 0

1 1 1 1

2 1

j j ,n ji jin ,pi

a , , a ,

a n / b a n / m,n

n p ,q ;r b , , b ,

f st s t t H I t dsdt

t s

 

 



       

 

    

    

 

 

 

      

^ ^

   

^ ^

 

   

1 1

2

0

1 1

1 1 1

1 1 1 1 1

1 1

2 0

l a b l

l m,npi ,qi ;r z b ,jb, , a ,j ,mj, b ,jji,n, aji mji, ,qiji n, , pia b l , dz

n a l f z z I

l l !

n /

_ _

 

  

 

  

 

  

       

 

  

     





(14) The conditions of convergence of the above result can be easily obtained from those of (10)

(D1)

 

   

 

   

  ₁  ₁

1 1

1 1 2

0 0

1 1 1 1 1

2 1

a n / b

a , , a ,

m ,n

n p ,q ;r b , , b ,

x y y y ty

H I dxdy

xy xy x x y xy

xy



 

 



 

 

 

          

            

     

  

 

 

 

   

 

   

^ ^

 

   

2 1 1

1 1 1

1 1

2 0

l l m ,n t a l , , a ,j j ,n, aji, ji n ,pi

pi ,qi ;r b ,j j ,m, b ,ji ji m ,qi, a b l ,

n I l ! b l

n / _ _ _

  

    

 

 

 

        

     



(15)

The conditions of convergence of the above result can be easily obtained from those of (11)

(II) By applying the our results given in (A), (B), (C) and (D) to the case of Laguerre polynomials [5] by setting

S

n²

  x



L

^{ }n^

  x

in which

m ,...,m

₁ _r 1

; n ,...,n

₁ _r 

n;r

1;

 

1 1

i i

n ,l

l

A n

n



  

  

  

, we have the following interesting results.

(A2)

 

  

 

   

  ₁  ₁

1 1

0 0

1 1 1 1 1

1 1 1 1 1 1

j j ,n ji jin ,pi

a b

a , , a ,

m ,n

n p ,q ;r b , , b ,

L I dxdy

xy xy x y xy xy

 



 



 

            

 

               

       

 

 

   

^ ^

 

   

0

1 1

1 1 1

1 1 1 1

1 1

n l l

l l

b, , a ,j j , aji, ji

m,n t ,n n ,pi

n n I

t a l

n l !

 

 





  

 

 

 

       

   

    

  



⁽¹⁶⁾

(5)

5

 

^{ }

 

_^ _{ }^{ }¹ _^ ¹

1 1

0 0

a , , a ,

b a m ,n

n p ,q ;r b , , b ,

s t t s L^ s I t ^ ^ dsdt

 



^



 

   

  

 

 

 

     

   

^ ^

 

   

1

0 0

1 1

1 1 1

1 1 1 1

1 1

n

a b l l

l l

b, , a ,j j , a ,ji ji

m ,n z ,n n ,pi dz

n n I

a l z z

n l !

 

 

 





  



  

 

 

 

       

   

    

  

 

⁽¹⁷⁾

    

^{ }

   

_^ _{ }^{ }¹ _^ ¹

1 1

0 0

1 1 1 1 ^j ^j ^,n ^ji ^ji ⁿ ^,pi

a , , a ,

a b a m ,n

n p ,q ;r b , , b ,

f st s t t L^ t s I t ^ ^ dsdt

 



   

       

 

 

 

      

1

0 0

1 1

1

n l a b l

l l

n n

a l f z z

n l !



  



   

     

  

 

 

   

^ ^

¹ ¹

   

1 1 1

1 1 1 1 1

1 1

b, , a ,j j , aji, ji

m,npi ,qi ;r z b ,j j ,m, b ,ji,nji m ,qin, ,pia b l , dz I

 

 

  

 

    

       

(18) The conditions of convergence of the above result can be easily obtained from those of (10)

(D2)

 

   

  ₁  ₁

1 1

0 0

1 1 1 1 1

j j ,n ji jin ,pi

a b

a , , a ,

m,n

n p ,q ;r b , , b ,

x y y x y y ty

L I dxdy

xy xy x xy xy



 



 



            

            

     

 

 

   

^ ^

 

   

0

1 1

1 1 1

1 1

1

n l

l l

a l , , a ,j j , aji, ji

m,n t ,n n ,pi

pi ,qi ;r b ,j j ,m, b ,ji ji m ,qi, a b l ,

n n I n b l !

  

  



 

    

 

 

 

        

   

    

  



⁽¹⁹⁾

The conditions of convergence of the above result can be easily obtained from those of (11)

(III) If we put r1,I-function reduces to the familiar Fox's H-function [4, p.10, Eqn. (2.1.1)], then the results (A), (B), (C) and (D) reduces to the following form:

(A3)

 

  

   

 

 ₁

1

1 1

0 0

1 1 1 1 1

1 1 1 1 1 1

j j ,p

r

r j j ,q

a b

m ,...,m m ,n a ,

n ,...,n p ,q b , ,

S H dxdy

xy xy x y xy xy





 

            

 

               

       

 

 

 

 

^ ^

 

1 1 1 1

1 1

0 0 ⁱ ⁱ 1

1 1 r r r i

i i i m ,n

n ,l i p ,q

i i

1 r

b, , a ,j j p z b ,j j ,q, a b l ,i

n l l

... t a l H

l l l

n / m n / m m

! A







 



  

 

  

 

 



    

 



⁽²⁰⁾

 

₁¹

 

_^ _^¹

1

1 1

0 0

j j ,p

r

r j j ,q

m ,...,m a ,

b a m ,n

n ,...,n p,q b , ,

s t t s S s H t ^ dsdt





 

   

  

 

 

 

 

   

^ ^

 

1 1

1 0

1 1

1

1 1

0 0 1

i i i

1 1 r r r i

a b l m ,n

i i

n ,l i p ,q

i i

1 r

b, , a ,j j ,p

z dz

b ,j j ,q, a b l ,i

n l

... a l z z H

l l l

n / m n / m m

! A



 



   

 



  

 

  

 

 



    

 

 

⁽²¹⁾