International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.

1. International Journal of Mathematics and Statistics Invention (IJMSI)
E-ISSN: 2321 – 4767 || P-ISSN: 2321 - 4759
www.ijmsi.org || Volume 2 || Issue 1 || February - 2014 || PP-21-27
Some Integral Properties of Aleph Function, General Class of
Polynomials Associated With Feynman Integrals
1,
Yashmeen Khan, and 2, Pallavi Sharma
1,
2,
Department of Mathematics, Government College, Kota, Rajasthan
Department of Mathematics, Kautilya Institute of Technology and Engineering, Jaipur, Rajasthan
ABSTRACT: The aim of the present paper is to discuss certain integral properties of Aleph function and
general class of polynomials. The exact partition of a Gaussian model in statistical mechanics and several other
functions as its particular cases. During the course of finding, we establish certain new double integral
relations pertaining to a product involving a general class of polynomials and the Aleph function.
KEY WORDS AND PHRASES: Feynman integrals, Aleph function, General Class of Polynomials, Hermite
Polynomials, Laguerre Polynomials.
I.
INTRODUCTION
The conventional formulation may fail pertaining to the domain of quantum cosmology but Feynman
path integrals apply interesting by Feynman path integrals are reformulation of quantum mechanics on more
fundamental than the conventional formulation in term of operator. Feynman integral are useful in the study and
development of simple and multiple variable hypergeometric series which in turn are useful in statistics
mechanics.
The Aleph ()-functions, introduced by Südland et al.[9], however the notation and complete definition
is presented here in the following manner in terms of the Mellin-Barnes type integrals
  z]  

1
2 
m, n
y   r
i i

L
 z]  
m, n
x , y   r
i i i
  s) z
x  y   r
i i i

z


m, n
For all z  0 where  
(s)
(a  A 
   a  A 
j
j 1, n i
ji
ji n  1, x  r
i
(b  B 
   b  B 
j
j 1, m i
ji
ji m  1, y  r
i
…(1)
ds
 1 and
m
n


m, n
x  y   r
i i i
 s) 
b
j1
x
r






i 1

1  a
j1
i
a
j n  1
the integration path L  L
j
y

i
j
 B s)
i 
ji
R
 A
ji
 A s
j
…(2)
i

s
j
1  b
j m  1
ji
 B
ji
s
extends from   i  to   i   and is such that the poles, assumed
to be simple, of   1  a  A s), j = 1,…,n do not coincide with the pole of   b  B s), j= 1,…,m the
j
j
j
j
parameter xi, yi are non-negative integers satisfying:
0  n  x 1  m  y  
i
i
The parameters A  B  A
j
j
ji
B
ji
i
 0 for i  1,..., r
 0 and a  b  a
j
j
ji
b
ji
 C. The empty product in (2) is interpreted
as unity. The existence conditions for the defining integral (1) are given below
  0   arg (z) | 


2


  1 r
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…(3)
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2. Some Integral Properties Of Aleph Function, General Class Of…

  0   arg (z) | 

 and R {    1  0

2
…(4)

where
n





B   
j
 

m
A

j
j1

j1
x
y


A
j n  1
j



B
x
 y

  b  a     b   a

j
j

j
j
j1
j1
j n  1
 j m  1
For detailed account of the Aleph ()-function see (9) and (10).
The general class of polynomial introduced by Srivastava (9)
m
n
  n)
[n/m]
S
m

 p] 
n
mk
B
k!
k0
n, k
p
k
j
j m  1




…(5)
 1
   x  y    1 r


 2

…(6)
n  0,1,2,...
Main Results:
We shall establish the following results:
(A)
1
1
0 0
 n/m]


 1 p


q 


 1  pq

  n)
mk
B
k!
k0
. 

i 1
y
i 1
 
 m
1  pq
 
S

(1  p)(1  q)  n
 
  r
i
 1 p
 m, n
wq  

x  y   r
i i i
 1  pq

 1 q

w  dp dq

 1  pq

k
k    w
n, k
m, n  1
x

 1 q


 1  pq
  1   ;1), (a j  A j 1, n   i  a ji  A ji  n  1, x i  r

 b B 
   b  B 
  1  k      1
 j j 1, m i ji ji m  1, y i  r i


w 


…(2.1)
provided that
T
Re[     b     0   arg w | 
j
j
2
m is an arbitrary positive integer and the coefficient B
 n, k  0) are arbitrary constant, real or complex.
n, k
Proof. We have
m
S
n
 1 p

wq  

1  pq


m, n
x
i
 y  r
i i
m

2 
L
b
j1
x
r


i 1
i

(-n)
mk
B
k !
k0
n, k
 1 p


wq 


1  pq


k
n

1
[n/m]
 1 q

w  dp dq 

1  pq


j
 B s
j
y
i

j n  1
1  a
j1
a
ji
 A
ji
s
j
 A s
j
i

1  b
j m  1
ji
 B
ji
s
 1 q


w 


 1  pq

s
ds
…(2.2)
Multiplying both sides of (2.2) by


 1 p
  1 q  

1  pq
w 

 
 and integrating with respect to p and q between 0 and 1 for both
1  pq
1  pq   (1  p)(1  q) 

 
the variable and we get the result
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3. Some Integral Properties Of Aleph Function, General Class Of…

(B) 
0

 1
f(   w  
0
 n/m]


  n)

mk
B
k!
k0
. 
i 1
y
i 1
  
n
m, n
x  y  r
i i i
 w] d  dw
k   
  1   ;1), (a j  A j 1, n   i  a ji  A ji  n  1, x i  r

 b B 
   b  B 
   1  k   1 
 j j 1, m i ji ji m  1, y i  r i

m, n  1
x
S
    k 1
f(   
0
n, k
m
 1
w
  r
i

  d


…(2.3)
provided that
Re(     b     0  m is an arbitrary positive integer and coefficients B
j
i
n, k
 n, k  0) are arbitrary
constants, real or complex.
Proof. We have
m
S
n
 
  n)
[n/m]
m, n

 w) 
x  y  r
i i i
mk
B
k!
k0
m
n


1
2 
  b  B s)
j
j1
L
x
r

k

n, k

i 1

j
j1
y
a
j n  1
 A
ji
Multiplying both side (2.4) by f(   w  
 1
s
  1  a  A s  (w)
i

i
j
 1  b
j m  1
 1
w
…(2.4)
ds
i

s)
ji
j
 B s)
ji
ji
and integrating with respect to  and w between 0 and 
for both side the variable and get the result (2.3).
1
 n/m]


1
0 0
(C)
   w  1  w 
  n)
1   
 1
w
m

S
1
mk
B
k!
k0
 1
n, k
k    
f(   1   
n
m, n
 w(1    
k     1
x  y  r
i i i
1  w  d  dw
d
0
  1   ;1), (a j  A j 1, n   i  a ji  A ji  n  1, x i  r

…(2.5)

1  

x
y
  r  b  B 
   b  B 
   1  k      1 
i 1 i 1 i 
j
j 1, m
i
ji
ji m  1, y  r i

i


provided that Re() >0, Re() >0, m is an arbitrary positive integer and coefficient Bn,k (n,k ≥ 0) are arbitrary
constant, real or complex.
Proof. We have
m, n  1
m
S
n
 w  1    
 n/m]
m, n
x  y  r
i i i
1  w  

  n)
2 
L
  b  B s)
j
j1
x
r

i 1

i
n, k
k
w
1   
k
n


B
k!
k0
m
1
mk
j


j
q
i
j n  1
  1  a  A s)
j1
a
ji
 A
ji
s)
j
1  w 
s
ds
…(2.6)
i

 1  b
j m  1
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ji
 B s)
ji
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4. Some Integral Properties Of Aleph Function, General Class Of…
Multiplying both side of (2.6) by    w  1   
 1
1  w 
 1

w
and integrating with respect to  and w
between 0 and 1 for both the variable and get the result.
1
0 0
(D)
 n/m]


 q(1  p) 


 (1  pq) 
1
  n)
mk
. 
i 1
y

1
 q(1  p)  m, n
 wq(1  p) 

  x  y  r 
 dp dq
i i i 
(1  pq) 
1  pq


m
S
 1  p)
n
  k  z        1
n, k
  1  k     1 ), (a j  A j 1, n   i  a ji  A ji  n  1, x i  r

 b B 
   b  B 
   1  k       1 
 j j 1, m i ji ji m  1, y i  r i

m, n
x
 1 q 


 1  pq 
k  z       1
B
k!
k0

i 1
  r
i
T
provided that Re(       b     0   arg w | 
j
i

w 


…(2.7)
  m is an arbitrary integer and the coefficient B n,k
2
(n,k ≥ 0) are arbitrary constant, real or complex.
Proof. We have
m
S
n
 q(1  p)  m, n
 wq(1  p) 

  x  y  r 

i i i   1  pq)
1  pq 


 n/m]

2 
L
  b  B s)
j
j1
x
r


i 1
i
B
n, k
 q(1  p) 


 1  pq 
k
n


mk
k!
k0
m
1
  n)
j

j
y
i

  1  a  A s)
j1
a
j n  1
ji
 A
ji
i

s)
j
 1  b
j m  1
ji
 B s)
 wq(1  p) 


  1  pq) 
s
ds
ji
…(2.8)


  q(1  p) 

 1 q 
1
 and integrating with respect to p and q
Multiplying both side by  



  1  pq 
 1  pq   1  p) 


between 0 and 1for both the variable and get the result (2.7).
3. Particular Cases
By applying our result given in (2.1), (2.3), (2.5), (2.7) to the case of Hermite polynomial (8) and (11)
and by setting
 1

n
n
2 p

In which case m = 2, Bn,k = (1)k.
2
S  p]  p
1
(A)
1
0 0
. 
n/2
H

 1 p
  1 q

q 

 
 1  pq   1  pq
m, n
x  y  r
i i i









 1 p

qw 

(1  p)(1  q)  1  pq

 1 q

w  dp dq H

 1  pq

 1  pq)
n





2

n/2



1

 1 p 


 wq 


 1  pq 

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5. Some Integral Properties Of Aleph Function, General Class Of…
 n/2]
  n)


2k
  1
k
k
w
k   
k!
k0
m, n  1
x  1 y  1  r
i
i
i
 1   1   a  A 
   a A 

j
j 1, n i
ji
ji n  1, x  r
i
. w
(b  B 
   b  B 
   1  k     1 
j
j 1, m
i
ji
ji m  1, y  r i

i

valid under the same conditions as obtained from (A).

(B)
0
  n)

. 
  1
k
  1  n/2
i 1
y
i 1
f(   
0
m, n  1
x


2k
k!
k0
p 1
f(   w  w
0
 n/2]


  r
i




H
n




 1  m, n

  x  y   r  w  dw d 
i i i
2 n 
    k 1
d k   
 1   1   a  A 
    a  A 
j
j 1, n
i
ji
ji n  1, x  r
i
(b  B 
   b  B 
   1  k   1 
j
j 1, m
i
ji
ji m  1, y  r i
i

 


valid under the same conditions as required for (B).
1
(C)
1
   w  1   
0 0
. H
 n/2]
  n)


n
. 
 1
w
  n/2
1
2k
k
  1   k    
m, n  1
x
1  w 
1   
n/2

 m, n
1

  x , y   r 1  w  dw d 
i i i
 2 w 1    
k!
k0
 1
i 1
y
i 1
  r
i
    1   
k     1
0
  1   1   a j  A j 1, n    i  a ji  A ji  n  1, x i  r

 b B 
   b  B 
   1  k     1 
 j j 1, m i ji ji n  1, y i  r i


1  


valid under the same condition as required for (C).
(D)
1
1
0 0
 q(1  p) 


 1  pq 
H
 n/2]


  n)
2k
n





2


 1 q


 1  pq



1

q(1  p) 

1  pq 

k
  1     1 
k!
k0




1
1 n  2
 1  p)
m, n
x  y  r
i i i
q
x
i 1
y
 1  pq)
 wq(1  p)


 1  pq
m, n  1
i 1
  r
i
n/2
n/2

 dp dq


  1  k     1   a j  A j 1, n   i  a ji  A ji  n  1, x i  r
w

(b  B 
   b  B 
   1  k     1 
 j j 1, m i ji ji m  1, y i  r i





valid under the same conditions as obtained from (D).
For
the
B
n  r
1




 n   r  1)
n, k
Laguerre
polynomials
([8]and
[11])
setting
'
r
n
n
S  x)  L  x) in
which
case
the result (A), (B),(C) and D reduced to the following formulae
k
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25 | P a g e

6. Some Integral Properties Of Aleph Function, General Class Of…
(1)
1

 1 p
  1 q

q 

 
 1  pq   1  pq
1
0 0
 n/2]
  n)


n  r
1




 n   r  1)
k
k!
k0


 r  1 p
 m, n
 1 q

 1  pq)

L 

wq  
w  dp dq

 n


x  y  r
i i i  1  pq
 (1  p)(1  q) 
 1  pq


k
w
k  
k
 1   1   a  A 
    a  A 
j
j 1, n
i
ji
ji n  1, x  r
i


(b


m, n  1
. 




x  1 y  1   r
i
i
i
j
B 
   b  B 
   1  k     1 
j 1, m
i
ji
ji m  1, y  r i
i

w


valid under the same conditions as required for (A).

(2)

0
0
  n)
n


 1
f(   w  w
k
k!
k0
.

r
m, n
n
 1
x  y  r
i i i
L  
n  r
1




 n   r  1)

f(   
0
k
 w] dw d 
    k 1
k   
 1   1   a  A 
   a A 
j
j 1, n i
ji
ji n  1, x  r
i


(b


m, n  1
x  1 y  1   r
i
i
i
j
B 
   b  B 
   1  k   1 
j 1, m i
ji
ji m  1, y  r i
i




valid under the same condition as required for (B).
1
(3) 
1
  w    1   
0
0
  n)
n


. 
i 1
y
 1
w

r
L
n
 w  1    
1
k   
k'
0
f    1   
 1   1   a  A 
   a  A 
j
j 1, n i
ji
ji n  1, x  r
i
m, n  1
x
1  w 
n  r
1




 n   r  1)
k
k!
k0
 1
i 1
  r
i
m, n
x , y  r
i i i
1  w  dw d 
  k   1


 b B 
   b  B 
    1  k     1 
 j j 1, m i ji ji n  1, y i  r i





valid under the same condition as required for (C).
(4)
1

 q(1  p) 


 1  pq 
1
0 0
L
n


n
 q(1  p) 


 (1  pq 
  n)
k
k!
k0
.
r
 1 q


 1  pq
i 1
y

1
 1  p)
m, n
x  y  r
i i i
n  r
1




 n   r  1)
 wq(1  p)


 1  pq

 dp dq


  1
k'
 1  k     1   a  A 
   a  A 
j
j 1, n
i
ji
ji n  1, x  r
i
m, n  1
x




i 1
  r
i


(b


j
B 
   b  B 
   1  k     1 
j 1, m
i
ji
ji m  1, y  r i
i

w


valid under the same condition as obtained for (D).
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26 | P a g e

7. Some Integral Properties Of Aleph Function, General Class Of…
II.
CONCLUSION
The results obtained here are basic in nature and are likely to find useful applications in the study of
simple and multiple variable hypergeometric series which in turn are useful in statistical mechanics, electrical
networks and probability theory. These integrals reformulation of quantum mechanics are more fundamental
than the conventional formulation in terms of operators.
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[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
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Berlin Heidelberg, New York (1998).
Mathai.A.M. and Saxena,R.K., The H-function with applications in Statistics and other disciplines, Wiley Eastern, New Delhi,
1978.
Saxena, Ram K and Pogany, Tiber K., On fractional integration formula for Aleph functions, Vol. 218(3), 2011 Applied Math.
Comput. Ilsevier.
Saxena, R.K. and Kumar, R., A basic analogue of generalized H-function, Matematiche (Catania) 50 (1995), 263-271.
Saxena, R.K., Mathai, A.M. and Haubold, H.J., Unified fractional kinetic equation and a diffusion equation, Astrophys. Space Sci.
290 (2004), 299-310.
Srivastava, H.M., Indian J. Math. 14(1972), 1-6.
Srivastava, H.M. and N.P. Singh, The integration of certain products of the multivariable H-function with a general class of
polynomials. Rend. Circ. Mat. Palermo, 2(32), 157-187 (1983).
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1(4) (1998); 401-402.
Südland,B., Baumann,B. and Nonnenmacher,T.F., Fractional driftless, Fokker-Planck equation with power law diffusion
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