ON SOME INTEGRAL PROPERTIES OF ALEPH FUNCTION, MULTIVARIABLE’S GENERAL CLASS OF POLYNOMIALS ASSOCIATED WITH FEYNMAN INTEGRALS

© Copyright 2014 | Centre for Info Bio Technology (CIBTech) 109

ON SOME INTEGRAL PROPERTIES OF ALEPH FUNCTION, MULTIVARIABLE’S GENERAL CLASS OF POLYNOMIALS

ASSOCIATED WITH FEYNMAN INTEGRALS

*Sanjay Bhattar¹ and Rakesh Kumar Bohra²

1Department of Mathematics, Malaviya National Institute of Technology, Jaipur, 302017 (India)

2Department of Mathematics, Global College of Technology, Jaipur, 302022 (India)

*Author for Correspondence

ABSTRACT

The idea of the present paper is to discuss certain integral properties of Aleph function and multivariable’s general class of polynomials. We establish certain new double integral relations pertaining to a product to a multivariable’s general class of polynomials and Aleph function.

Keywords: Feynman Integrals, Multivariable’s General Class of Polynomials, Aleph Function

INTRODUCTION

The Aleph (χ) - function, introduced by Sudland et al., (1998), however the notation and complete definition is presented here in the following manner in terms on the Mellin- Barnes type integrals

1, 1, ,

1, 1, ,

( , ) [ ( , )]

, , ( )

, , ; ( , ) [ ( , )] , , ;

[ ] | 1 ( )

2

j j n i ji ji n x ri

i i i j j m i ji ji m y ri i i i

a A a A

m n m n s

x y r b B b B L x y r

z _ z ^_{ } s z ds

 





  

   



  ⁽¹⁾

For all z0where   ( 1)and

1 1

, , , ;

1 1 1

( ) (1 )

( )

( ) (1 )

i i i i

m n

j j j j

j j

m n

x y r r x m

i ji ji ji ji

i j n j

b B s a A s

s

a A s b B s

 

 

   

    

 

 

    

  

(2)

The integration path L^Li,^R extends from  i to i , and is such that the poles, assumed to be simple of  (1 a_jA s_j ),j=1,...,n do not coincide with the pole of (b_j B s_j ),j=1,...,m the parameter

,

x yi iare non-negative integers satisfying:0 n xi,0 m yi i, ˃0 for i= 1,...,r.

The Aj ,Bj ,Aji ,Bji ˃0 and ^aj ,bj ,^aji ,bjiC. The empty product in (2) is interpreted as unity. The existence conditions for the defining integral (1) are giving below

0,arg( )z 2 ,l 1, ...,r

l l

     (3)

0, 0

arg( )z 2 and R l{ }

l l

    

(4) Where

1 1 1 1

x y

n m l l

Aj Bj Ajl Bjl

l l

j j j n j m

      ^      ^

(5)

1( ) , 1, ...,

1 1 1 1 2

x y

n m l l

bj aj bjl ajl x y l r

l l l l

j j j n j m

 

 

 

             

(6) For detailed account of Aleph (χ)-function see Sudland et al., (2001 and 1998). Feynman integrals (Edwards, 1922; Grosche and Steiner, 1998).

The general Class of polynomials ₁¹,...,^{,..., r}



1, ...,



r

m m

r

n n

S x x of r variables defined and represented as follows (Srivastava, 1985):

© Copyright 2014 | Centre for Info Bio Technology (CIBTech) 110

 

¹

1 1

1

1,..., ,

1

..., 1, ..., ( ) ,

...

r

r

i i i

i i r

r

r

n n

m m r

m k k

n k

m m

i

k k i i

r

n n

n A x

S x x k



  

 

 

 





 

 ⁽⁷⁾

Where n m_i, _i 1,...;m_i  0 , i1, 2,..., ;r the coefficients A k k( ,_{1 2},...,k_r) , (k_i0)are arbitrary constant, real or complex. The general class of polynomials (Srivastava and Garg, 1987) is capable of reducing to a number of familiar multivariable polynomials by suitable specializing the arbitrary coefficients

1 2

( , ,..., _r) , ( _i 0)

A k k k k 

The general class of multivariable polynomials is defined by Srivastava and Garg (1987)

 

^{1 1 1}

1

1 1 1

,...., .... 1

1 .... 1

,...., 0

1

, ...., ( ) ( , , ...., ) ...

r r r r

r r r

k k

h k h k L

h h r

L r h k h k r

k k

r

X X

S X X L A L k k

k k

  

  

   (8)

Where h h₁, ₂,...,h_rare arbitrary positive integers andA L k k( ; ₁, ₂,...,k_r) , ( ;L h_i N i; 1, 2,..., )r Coefficients are arbitrary constant, real or complex.

Evidently the case r =1 of the polynomials (8) would correspond to the polynomials given by Srivastava (1972)

 

^, ,

0

( )

{ (0,1, 2, ...)}

L

h L k

L k

h L k

k

S X L A X L N

k



     (9)

MAIN RESULTS: 1

We shall establish the following results (A)

1 1 1 1 1 1,..., 1 , 1

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

0 0

m m

p q q pq S r p wq m n q w dpdq

L x y r

pq pq p q pq i i i pq

 

 

              

 

                   

=

,...,

1 11,..., 0 ( ) 1 1,..., ( ; 1, ..., ) ( 1 ( )) ( 1 ) k

m k m kr r L r w i r

Lm k m kr r A L k kr k ki

k kr i i i

 

  

  

  

     

 

1 ;1 , , 1, , 1, ;

, 1

, ;

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

aj Aj n i aji Aji n x ri

m n r w

xi yi i r bj Bj m i bji Bji m y ri i i ki

 

 

   

    

    

 

     

             

(10)

The above result is valid under the conditions (3), (4); Re 0 , arg , (1 )

[ b / ] 2

j

T

j j m

          and a,b are positive. Also 0˂p˂1 and 0˂q˂1.

Proof:-We have

,..., (1 ) , (1 )

1

1 , , ; 1

m mr p m n q

SL pq wq x yi i i r pqw

 

     

     

    =

1 1

1 1 1

....

.... 1

,...., 0 1

1

( ) ( , , ...., ) 1

1

i r r

r r

r i

h k h k L r k

h k h k r

k k i

L A L k k p wq

k pq

  

   

    

 

       

1 1

1 1 1

( ) (1 )

1 1

( )

2 1

( ) (1 )

i

m n

j j j j

j j s

x m

L r

i ji ji ji ji

i j n j

b B s a A s

q w ds

a A s b B s pq

 

  

   

    



     

 

   

⁽¹¹⁾

© Copyright 2014 | Centre for Info Bio Technology (CIBTech) 111 Multiplying both sides of (11) by

1 1 1

1 1 (1 )(1 )

p q pq

pqq pq p q

 

        

     

         

 

and integrating with respect to p and q between 0 and 1 for both the variable and get the result.

(B)

,...,

 

, ,

1 1 1

( ) [ ]

, , ; 0 0

m mr m n

f w w S w d dw

L x yi i i r

 

     

     

 

¹

,...,

1 1 ( ) 1 1,..., ( ; 1, ..., ) ( ) ( ) 1

,..., 0 1 1 0

1

r k m kk m kkrr r L L m k m kr rA L k kr ir wkkii ir ki f i i

 

  

  

 

        

  

     

 

1 ;1 , , 1, , 1, ;

, 1

, ;

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

aj Aj n i aji Aji n x ri

m n d

xi yi i r bj Bj m i bji Bji m y ri i ir ki

 

   

   

    

    

 

     

             

(12)

The above result is valid under the conditions (3), (4) ; Re 0 , arg , (1 )

[ b / ] 2

j

T

j j m

          and a,b are positive. Also 0˂˂ and 0˂w˂.

Proof:-We have

,..., ,

 

1 [ ]

, , ;

m mr m n

SL x y r w

i i i

   = ^{1 1} _{1 1}

 

1

....

.... 1

,...., 0 1

( ) ( , , ...., ) 1

r r i

r r

r i

h k h k L r k

h k h k r

k k i

L A L k k

k 

  

   

 

 

    

 

 

1 1

1 1 1

( ) (1 )

1 ( )

2 ( ) (1 )

i

m n

j j j j

j j s

x m

L r

i ji ji ji ji

i j n j

b B s a A s

w ds

a A s b B s

 

  

   

    

    

 

   

⁽¹³⁾

Multiplying both sides of (13) by

1 1

( ) ( ) ( )

f  w   w 

  

 

 and integrating with respect to and w between 0 and for both the variable and get the result.

(C)

 

1 1 ( ) (1 ) 1(1 ) 1 1,..., [ (1 )] , (1 )

, , ; 0 0

m mr m n

w w w S w w d dw

L x yi i i r

  

           

 

 

¹

,..., 1

1 1 ( ) 1 1,..., ( ; 1, ..., ) ( ) ( ) 1

,..., 0 1 1 0

1

r k k

m kk m kkrr r L L m k m kr rA L k kr ir wkii ir ki f i i

 

  

  

 

       

  

(1 ,1), ( , ) ,[ ( , )]

1, 1, ;

, 1 (1 )

, , ;

1 1

( , ) ,[ ( , )] , [1 ;1]

1, 1, ; 1

aj Aj n i aji Aji n x r

m n i

xi yi i r r

bj j m i bji ji m yi r i i ki

 

  

     

 

   

 

   

        

(14)

© Copyright 2014 | Centre for Info Bio Technology (CIBTech) 112 The above result is valid under the conditions (3), (4); Re 0 , arg , (1 )

[ b / ] 2

j

T

j j m

          and a,b are positive. Also 0˂˂1and 0˂w˂1.

Proof:-We have

   

,..., ,

1 (1 ) 1

, , ;

m mr m n

SL w x y r w

i i i

  

  = ^{1 1} _{1 1}

 

1

....

.... 1

,...., 0 1

( ) ( , , ...., ) 1 1

r r i

r r

r i

h k h k L r k

h k h k r

k k i

L A L k k w

k 

  

   

 

   

 

     

 

 

1 1

1 1 1

( ) (1 )

1 (1 )

2 ( ) (1 )

i

m n

j j j j

j j s

x m

L r

i ji ji ji ji

i j n j

b B s a A s

w ds

a A s b B s

 

  

   

    



    

 

   

(15)

Multiplying both sides of (15) by

1 1

( w) (1 ) (1 w) w

    

   

 

 and integrating with respect to and w between 0 and 1for both the variable and get the result.

(D)

1 1 (1 ) 1 1 1,..., (1 ) , (1 )

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

0 0

a m m

q p q r q p m n wq p

SL x y r dpdq

pq pq p pq i i i pq

 

 

              

 

                 

,...,

1 1 1

( ) 1 1,..., ( ; 1, ..., ) ( 1)

,..., 0 1

m k m kr r L

Lm k m kr rA L k kr k

k kr i

 

    



   

 

1 1 ,1 , , 1, , 1, ;

, 1

, ;

1, 1

, , ( , ) , ,1

, 1, 1, ; 1

r

a i ki aj Aj n i aji Aji n x ri

m n w

xi yi i r r

bj Bj m i bji Bji m y ri i a i ki

 

 

   

     

          

    

  

  

 

             

(16)

The above result is valid under the conditions (3), (4); Re 0 , arg , (1 )

[ b / ] 2

j

T

j j m

          and a,b are positive. Also 0˂p˂1and 0˂q˂1.

Proof:-We have

,..., (1 ) , (1 )

1

1 , , ; 1

m mr q p m n wq q

SL pq xi yi i r pq

 

     

     

   = ^{1 1}

1 1 1

....

.... 1

,...., 0 1

(1 )

( ) ( , , ...., ) 1[ ]

1

r r i

r r

r i

h k h k L r

k

h k h k r

k k i

q p

L A L k k

k pq

  

   

  

 

     

1 1

1 1 1

( ) (1 )

1 (1 )

[ ]

2 1

( ) (1 )

i

m n

j j j j

j j s

x m

L r

i ji ji ji ji

i j n j

b B s a A s

wq q pq ds

a A s b B s

 

  

   

    



     

 

   

⁽¹⁷⁾

Multiplying both sides of (17) by

(1 ) 1 1

1 1 (1 )

q p a q

pq q pq p

 

       

     

        

 

and integrating with respect to p and q between 0 and 1 for both the variable and get the result.

© Copyright 2014 | Centre for Info Bio Technology (CIBTech) 113 MAIN RESULTS: 2

If we use the multivariable’s general Class of polynomials ₁¹,...,^{,..., r}



1, ...,



r

m m

r

n n

S x x defined and represented as follows (Sudland et al., 2001) then we also get new results.

(A1)

1 1 (1 ) 1 (1 ) 1,..., (1 ) , (1 )

,..., ;

1 1 (1 )(1 ) 1 1 , , 1

0 0

m m

p q pq r p m n q

q S wq w dpdq

n n x y r

pq pq p q r pq i i i pq

 

 

              

 

                   

= ^{1/ 1 /}

 

... , ( )

0 0 1 1

1

n m n m

r r

r r nikim ki iAn ki i wki ki

k kr i i



   

 

            

 

     

     

 

1 ;1 , , 1, , 1, ;

, 1

, ;

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

aj Aj n i aji Aji n x ri

m n w

xi yi i r bj Bj m i bji Bji m y ri i ir ki

 

 

   

    

    

 

     

             

(18)

The above result is valid under the conditions (3), (4); Re 0 , arg , (1 )

[ b / ] 2

j

T

j j m

          and a,b are positive. Also 0˂p˂1 and 0˂q˂1.

Proof:-We have

,..., (1 ) , (1 )

1

,..., 1 , , ; 1

1

m mr p m n q

S wq w

n nr pq xi yi i r pq

 

     

     

    =

1 1

1

0 0 1 ,

( ) 1

... ( )

1

r

r i i i

i i r

n n

m m r m k k

k k i i n k

n p

A wq

k pq

  

   

 

     

1 1

1 1 1

( ) (1 )

1 1

( )

2 1

( ) (1 )

i

m n

j j j j

j j s

x m

L r

i ji ji ji ji

i j n j

b B s a A s

q w ds

a A s b B s pq

 

  

   

    

 

     

 

   

⁽¹⁹⁾

Multiplying both sides of (19) by

1 1 1

1 1 (1 )(1 )

p q pq

pqq pq p q

 

        

     

         

 

and integrating with respect to p and q between 0 and 1 for both the variable and get the result.

(B1)

,...,

 

, ,

1 1 1

( ) [ ]

,..., ;

, , ,

0 0 1

m mr m n

f w w S w d dw

n nr x yi i i r

 

     

     

   

/ / 1

1 1

... , ( ) 1

0 0 1 1 0

1

nk m nkr mr ir nikim ki i An ki iwki ir ki f ir ki

r

 

  

         

       

        

 

     

     

 

1 ;1 , , 1, , 1, ;

, 1

, ;

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

aj Aj n i aji Aji n x ri

m n r d

xi yi i r bj Bj m i bji Bji m y ri i i ki

 

   

   

    

    

 

     

             

(20)

© Copyright 2014 | Centre for Info Bio Technology (CIBTech) 114 The above result is valid under the conditions (3), (4) ; Re 0 , arg , (1 )

[ b / ] 2

j

T

j j m

          and a,b are positive. Also 0˂˂ and 0˂w˂.

Proof:-We have

,..., ,

 

1 [ ]

,..., , , ;

1

m mr m n

S w

n nr xi yi i r

   =

1 1

1

0 0 1 ,

( )

... ( )

r

r i i i

i i r

n n

m m r m k k

k k i i n k

n A

k 

  

  

 

    

 

 

1 1

1 1 1

( ) (1 )

1 ( )

2 ( ) (1 )

i

m n

j j j j

j j s

x m

L r

i ji ji ji ji

i j n j

b B s a A s

w ds

a A s b B s

 

  

   

    

    

 

   

⁽²¹⁾

Multiplying both sides of (21) by

1 1

( ) ( ) ( )

f  w   w 

  

 

 and integrating with respect to and w between 0 and for both the variable and get the result.

(C1)

 

1 1 ( ) (1 ) 1(1 ) 1 1,..., [ (1 )] , (1 )

,..., , , ;

0 0 1

m mr m n

w w w S w w d dw

n nr x yi i i r

  

           

 

   

/ / 1 1

1 1

... , ( ) 1

0 0 1 1 0

1

nk m nkr mr ir nikim ki i An ki i wki ir ki f ir ki d

r

 

   

        

             

 

     

(1 ,1), ( , ) ,[ ( , )]

1, 1, ;

, 1 (1 )

, , ;

1 1

( , ) ,[ ( , )] , [1 ;1]

1, 1, ; 1

aj Aj n i aji Aji n x r

m n i

xi yi i r r

bj j m i bji ji m yi r i i ki

 

  

     

 

   

 

   

        

(22)

The above result is valid under the conditions (3), (4) ; Re 0 , arg , (1 )

[ b / ] 2

j

T

j j m

          and a,b are positive. Also 0˂˂1and 0˂w˂1.

Proof:-We have

   

,..., ,

1 (1 ) 1

,..., , , ;

1

m mr m n

S w w

n nr xi yi i r

  

  =

1 1

1

0 0 1 ,

( )

... [ (1 )]

r

r i i i

i i r

n n

m m r m k k

k k i i n k

n A w

k 

  

  

  

    

 

 

1 1

1 1 1

( ) (1 )

1 (1 )

2 ( ) (1 )

i

m n

j j j j

j j s

x m

L r

i ji ji ji ji

i j n j

b B s a A s

w ds

a A s b B s

 

  

   

    



    

 

   

(23)

Multiplying both sides of (23) by

1 1

( w) (1 ) (1 w) w

    

   

 

 and integrating with respect to and w between 0 and 1for both the variable and get the result.

(D1)

© Copyright 2014 | Centre for Info Bio Technology (CIBTech) 115

1 1 (1 ) 1 1 1,..., (1 ) , (1 )

,..., ;

(1 ) 1 (1 ) 1 (1 ) , , (1 )

0 0

a m m

q p q S r q p m n wq p dpdq

n n x y r

pq pq p r pq i i i pq

 

 



             

 

                 

= ^{1/ 1 /}

 

... , ( 1)

0 0 1

1

n m n m

r

r r ni m ki i An ki i

k k i ki

r



   

 

           

 

    

   

 

1 1 ,1 , , 1, , 1, ;

, 1

, ;

1, 1

, , ( , ) , ,1

, 1, 1, ; 1

r

a i ki aj Aj n i aji Aji n x ri

m n w

xi yi i r r

bj Bj m i bji Bji m y ri i a i ki

 

 

   

     

          

    

  

  

 

             

(24)

The above result is valid under the conditions (3), (4) ; Re 0 , arg , (1 )

[ b / ] 2

j

T

j j m

          and a,b are positive. Also 0˂p˂1and 0˂q˂1.

Proof:-We have

,..., (1 ) , (1 )

1

,..., 1 , , ; 1

1

m mr q p m n wq q

Sn nr pq x yi i i r pq

 

     

     

   =

1 1

1

0 0 1 ,

( ) (1 )

... [ ]

1

r

r i i i

i i r

n n

m m r m k k

k k i i n k

n q p

k A pq

  

   

 

    

  

 

1 1

1 1 1

( ) (1 )

1 (1 )

[ ]

2 1

( ) (1 )

i

m n

j j j j

j j s

x m

L r

i ji ji ji ji

i j n j

b B s a A s

wq q pq ds

a A s b B s

 

  

   

    



     

 

   

⁽²⁵⁾

Multiplying both sides of (25) by

(1 ) 1 1

1 1 (1 )

q p a q

pq q pq p

 

       

     

        

 

and integrating with respect to p and q between 0 and 1 for both the variable and get the result.

SPECIAL CASES 1 If we choosing

1 2 ... r

    =1 in equation (18), (20), (22) and (24) then Aleph –function reduce to I-function (Agrawal, 2011)

(A2)

1 1 1 1 1 1,..., 1 , 1

,..., ;

1 1 (1 )(1 ) 1 1 , 1

0 0

m m

p q q pq S r p wq Im n q w dpdq

n n x y r

pq pq p q r pq i i pq

 

              

 

                   

= ^{1/ 1 /}

 

... , ( )

0 0 1 1

1

nk m nkr mr ir nikim ki iAn ki iwki ir ki

r



   

 

           

 

     

     

 

1 ;1 , , 1, , 1, ;

, 1

, ;

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

aj Aj n aji Aji n x ri

m nx y r r w

i i bj Bj m bji Bji m y ri i ki



 

    

    

 

             

(26)

© Copyright 2014 | Centre for Info Bio Technology (CIBTech) 116 The above result is valid under the conditions (3), (4); Re 0 , arg , (1 )

[ b / ] 2

j

T

j j m

          and a,b are positive. Also 0˂p˂1 and 0˂q˂1

(B2)

,...,

 

, ,

1 1 1

( ) [ ]

,..., ;

, ,

0 0 1

m mr m n

f w w S I w d dw

n nr x y ri i

 

   

     

   

/ / 1

1 1

... , ( ) 1

0 0 1 1 0

1

nk m nkr mr ir nikim ki i An ki i wki ir ki f ir ki

r

 

  

         

       

        

 

     

     

 

1 ;1 , , 1, , 1, ;

, 1

, ;

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

aj Aj n aji Aji n x ri

m nx y r r d

i i bj Bj m bji Bji m y ri i ki



 

 

    

    

 

             

(27)

The above result is valid under the conditions (3), (4) ; Re 0 , arg , (1 )

[ b / ] 2

j

T

j j m

          and a,b are positive. Also 0˂˂ and 0˂w˂.

(C2)

 

1 1 ( ) (1 ) 1(1 ) 1 1,..., [ (1 )] , (1 )

,..., , ;

0 0 1

m mr m n

w w w S w I w d dw

n nr x y ri i

  

         

 

   

/ / 1 1

1 1

... , ( ) 1

0 0 1 1 0

1

nk m nkr mr ir nikim ki i An ki iwki ir ki f ir ki d

r

 

   

        

             

 

     

(1 ,1), ( , ) ,[( , )]

1, 1, ;

, 1

(1 )

, ;

1 1

( , ) ,[( , )] ,[1 ;1]

1, 1, ; 1

a A a A

j j n ji ji n x r

m n i

xi yi r r

b b k

j j m ji ji m yi r i i





   

 

   

 

    

        

(28)

The above result is valid under the conditions (3), (4) ; Re 0 , arg , (1 )

[ b / ] 2

j

T

j j m

          and a,b are positive. Also 0˂˂1and 0˂w˂1.

(D2)

1 1 (1 ) 1 1 1,..., (1 ) , (1 )

,..., ;

(1 ) 1 (1 ) 1 (1 ) , (1 )

0 0

a m m

q p q r q p m n wq p

S I dpdq

n n x y r

pq pq p r pq i i pq

 

              

 

                 

= ^{1/ 1 /}

 

... , ( 1)

0 0 1

1

n m n m

r

r r ni m ki i An ki i

k k i ki

r



   

 

           

 

    