International Journal of Mathematics and Mathematical Sciences
Volume 2003 (2003), Issue 33, Pages 2119-2137

On polynomials of Sheffer type arising from a Cauchy problem

D. G. Meredith

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.


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.