Polynomial Expansions for Solutions of $D_x^r u(x,t) = D_t u(x,t),r = 2,3,4, \cdots $

Abstract

This paper is an extension of results by P. C. Rosenbloom and D. V. Widder [Trans. Amer. Math. Soc., 92 (1959), pp. 220–266] concerning the expansion of a solution $u(x,t)$ of the heat equation, $D_x^2 u(x,t) = D_t u(x,t)$, in a series of polynomial solutions. It is found that a polynomial expansion \[ \sum_{n = 0}^\infty {a_n v_{r,n} (x,t)} \] converges in an infinite strip $| t | < \sigma $, where the polynomials \[ v_{r,n} (x,t) = n!\sum\limits_{k + rl = n} {\frac{{x^k }}{{k!}}\frac{{t^l }}{{l!}}} \] satisfy the partial differential equation $D_x^r u(x,t) = D_t u(x,t)$. Furthermore it is found that there exists a solution of $D_x^r u(x,t) = D_t u(x,t)$ which has a Maclaurin expansion in a strip $| t | < \sigma $ and which reduces to $f(x)$ for $t = 0$ if and only if $f(x)$ is an entire function of special growth. All the proofs need only elementary calculus and make no use of fundamental solutions.