International Journal of Technical Research and Applications e-ISSN: 2320-8163, www.ijtra.com Volume 2, Issue 4 (july-aug 2014), PP. 122-126 122 | Page H - FUNCTION AND GENERAL CLASS OF POLYNOMIAL AND HEAT CONDUCTION IN A ROD. Dr. Rachna Bhargava Department of Mathematics Global College of Technology (GCT), Jaipur-302022, India Abstract - In this paper, first we evaluate a finite integral involving general class of polynomials and the product of two H -functions and then we make its application to solve boundary value problem on heat conduction in a rod under the certain conditions and further we establish an expansion formula involving about product of H -function. In view of generality of the polynomials and products of H -function occurring here in, on specializing the coefficients of polynomials and parameters of the H -function, our results would readily reduce to a large number of results involving known class of polynomials and simpler functions. Keywords: General Class of Polynomials, H Function, Jacobi polynomial and Leguerre polynomials. Mathematics Subject Classification : 33C60, 34B05 I. INTRODUCTION The general class of polynomials introduced by Shrivastava [7] and defined by [8] and [10] as follows: 0,1,2,... n x ! k A n) x] S k k n, k m, [n/m] 0 k m n         ….. (1) where m is an arbitrary positive integer the coefficient A n,k (n,k ≥ 0) are arbitrary constants, real or complex. H -function will be defined and represented as follows [2] and [4]:                              P 1, N j j a N 1, j A j j (a Q 1, M j B j j b M 1, j j b N M, Q P, N M, Q P, z H z] H             d z i 2 1 i i …(2) where  ≠ 0 and                                          j j P 1 N j j B j j Q 1 M j j A j j N 1 j j j M 1 j a b 1 a 1 b …(3) and also the H -function occurring in the paper was introduced by Inayat-Hussain [4] and studied by Bushman and Shrivastava [2]. The following series representation for the H -function was obtained by Rathie [5].                              P 1, N j j c N 1, j C j j (c Q 1, M j D j j d M 1, j j d N M, Q P, N M, Q P, z H z] H r h, h r r h, j j P 1 j j D r h, j j Q 1 M j j C r h, j j N 1 j r h, j j M h j 1 j M 1 h 0 r z ! r 1 c d 1 c 1 d                                                  r r r h, h d     …(4) The nature of contour L and series of various conditions on its parameters can be seen in the paper by Bushman and Shrivastava [2]. We shall also make use of the following behaviour of the z] H N M, Q P,  function for small value of f(z) as recorded by Saxena [6, p.112, eq.(2.3) and (2.4)]     | z | ( 0 z} H N M, Q P, for small z where (2) for ) / (d Re min j j M j 1      and (4) for ) / (b Re min j j M j 1      The following more general conditions given by 0 T T 2 1 (z arg 1 1 1        and 0 T T 2 1 (z arg 2 2 2        . where 0 B A T j 1 P 1 1 N j j j 1 Q 1 1 M j j j 1 N 1 j j 1 M 1 j 1                        and 0 D C T j 2 P 1 2 N j j j 2 Q 1 2 M j j j 2 N 1 j j 2 M 1 j 2                        .