ON GENERALIZED HEAT POLYNOMIALS

We consider the generalized heat equation of n th order + r @r -Ur @__u If the initial temperature is an even power function then the heat transform @t with the source solution as the kernel gives the heat polynomial. We discuss various properties of the heat polynomial and its Appell transform. Also, we give series representation of the heat transform when the initial temperature is a power function.

In this paper we shall establish various properties of the polynomial solutions th and its Appell transforms of the generalized heat equation of the n order, 2 2u n-1 @u a @u --r + --u temperature in terms of Laguerre polynomials and confluent hypergeometric Most of the results derived here are similar to the ones found in [4 & 5], which are for the less general equation @2u x @x which in turn is a generalization of the ordinary heat equation, [7]   @2u @u a-x --These known results can be considered as special cases of our more general results, when a 0 and n I. 2.
PRELIHINARY RESULTS.(2.2). (2.3) 2 + a and I (z), the usual modified Bessel function of the first We shall call the functioon U to be the source solution of the heat equation If U is considered as the kernel, then for a suitable f, its heat transform F rkF(r't) fO U(s'r:t)skf(s)ds' 1 where k +u and F(r,O) f(r), the initial temperature.Numerous properties of the heat transform have been given in [6].We note.thatits inversion is given by rkf(r) U(s,ir:t)(s/i)kF(is,t)ds.
derive a generating function for P2n,p(r,t) we l-4yt Let t > O. Using (2.5) and (2.Now we give a generating function for W2n,/a(r,t), the Appell transform of for the case t > O.
from its representation given in (2.6).The lemma is then proved on the same lines as 2n-k P2n,(r,-t) i P2n,g(ir,t) utely when [z[ t.Using (2.8), we have abso z 1 n=0 W2n'(r't) H(0,r't)t-k n=0 (z/t) P2n,(r'-t) 2 r k l H,(0,r:t+4z), due to Lena 2 d ming e of the definition of H given by (2.9).giving integral representation for W2n,(r,t).Also we give other generating fctions for the fction P2n,(r,t) d its Appell trsfo W2n,y(r,t).We shall simply write d the results, which c be proved folling a similar alysis as ed for the L 2 d 3 ove.N we shall prove iortt property of the sets of fctions P2n,(r,t) d W2n,(r,t) d sh tt they fo a biorthogonal syst.
The integral on the right handside of (3.6) with a change of variable can be written as, [3,p.I88].

4z2t
PROOF.Note that the series converges for Iz2t < s, using the asymptotic estimates of the functions P2n,/a and W2n,/a therefore, A similar result can also be proved involving W2n,.

SERIES REPHESENTATION
In this section we shall establish a series representation of the heat transform F(r,t) in terms of Laguerre polynomials and confluent hypergeometric functions.
ACKNOWLEDGEMENT.This research is partially supported by a grant from Natural Sciences and Engineering Research Council of Canada. I.
LEMA 3.For t_> O, Iz[ < t, and k tt-u + , If we expd the right hd side of (3.4) by Taylor series in pers of zco=paring this series with the series on the left hand side of (3.4),

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on finite sums involving the functions P2n,/a and W2n,/a.LE 7.For t > O, /a > O, and a complex z, n Z (-l)m fn+/a] (2.5), z(-I fn+.]    2n, m=O m tn-mj z r, t n-O n n-k Z a , P (ir,-t) Z a P (r,t), = O, throughout, most of the results derived