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International Journal of Mathematics and Statistics Invention (IJMSI)

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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 …

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.

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  • 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 www.ijmsi.org …(3) 21 | P a g e
  • 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 www.ijmsi.org 22 | P a g e
  • 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 www.ijmsi.org ji  B s) ji 23 | P a g e
  • 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   www.ijmsi.org 24 | P a g e
  • 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 www.ijmsi.org 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). www.ijmsi.org 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. REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] Fox C., The G and H-functions as symmetrical Fourier kernels, Trans. Amer. Math. Soc. 98 (1961), 395-429. Grosche, C. and Steiner, F., Hand book of Feynman path integrals, Springer tracts in modern physics, Vol.145, Springer-Verlag, 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). Südland,N., Bauman,B. and Nonnenmacher,T.F., Open problem; who know about the Aleph ()-functions? Fract. Calc. Appl. Anal. 1(4) (1998); 401-402. Südland,B., Baumann,B. and Nonnenmacher,T.F., Fractional driftless, Fokker-Planck equation with power law diffusion coefficients. In V.G. Gangha, E.W. Mayr, W.G. Varozhtsov, editors, Computer Algebra in Scientific Computing (CASC Konstanz 2001), Springer, Berlin (2001); 513-525. Szego, C., Orthogonal Polynomial, Amer. Math. Soc. Colloq. Publ. 23 fourth edition, Amer. Math. Soc. Providence, Rhode Island, 1975. www.ijmsi.org 27 | P a g e

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