Polynomial Expansions for Solutions of the System $D_{X_1}^k U(X_1 , \cdots ,X_r ) = D_{X_k} U(X_1 , \cdots ,X_r )^ * $

Abstract

This paper is an extension of results by P. C. Rosenbloom and D. V. Widder [Traps. Amer. Math. Soc., 92(1959), pp. 220–266], [Duke Math. J., 29(1962), pp. 497–503] 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 \[\sum_{n = 0}^\infty {a_n P_{r,n} (X_1 , \cdots ,X_r )} \]converges in an infinite strip $|X_r | < \sigma $, where the polynomials \[P_{r,n} (X_1 , \cdots ,X_r ) = n!\sum_{k_1 + 2k_2 + \cdots + rk_r = n} {\frac{{X_1^{k_1 } }}{{k_1 !}} \cdots \frac{{X_r^{k_r } }}{{k_r !}}} \] satisfy the system $D_{X1}^k U(X_1 , \cdots ,X_r ) = D_{X_k} U(X_1 , \cdots ,X_r ) $. This paper includes several applications of classical initial-value problems of parabolic differential equations. For example, it is found that there exists a solution of $(A_r D_x^r + \cdots + A_2 D_x^2 )u(x,t) = D_t u(x,t)$ 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.