International Journal of Mathematics and Mathematical Sciences
Volume 2003 (2003), Issue 33, Pages 2119-2137
doi:10.1155/S0161171203207080
On polynomials of Sheffer type arising from a Cauchy problem
Department of Mathematics, King's College at the University of Western
Ontario, 266 Epworth Avenue, London N6A 2M3, Ontario, Canada
Received 7 July 2002
Copyright © 2003 D. G. Meredith. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
A new sequence of eigenfunctions is developed and studied in
depth. These theta polynomials are derived from a recent analytic
solution of the canonical Cauchy problem for parabolic equations,
namely, the inverse heat conduction problem. By appealing to the
methods of the operator calculus, it is possible to categorize the
new functions as polynomials of binomial and Sheffer types. The
connection of the new set with the classical polynomials of
Laguerre is carefully examined. Some integral relations involving
the Laguerre polynomials and the theta polynomials are presented
along with a number of binomial identities. The inverse
heat conduction problem is revisited and an analytic solution
depending on the generalized theta polynomials is presented.