Appell Polynomials and Differential Equations of Infinite Order

Abstract

Let $\Phi (z) = \Sigma _0^\infty {\beta _j}{z^j}$ have radius of convergence $r\;(0 < r < \infty )$ and no singularities other than poles on the circle $|z| = r$. A complete solution is obtained for the infinite order differential equation $( \ast )\;\Sigma _0^\infty {\beta _j}{u^{(j)}}(z) = g(z)$. It is shown that $(\ast )$ possesses a solution if and only if the function g has a polynomial expansion in terms of the Appell polynomials generated by $\Phi$. The solutions of $( \ast )$ are expressed in terms of the coefficients which appear in the Appell polynomial expansions of g. An alternate method of solution is obtained, in which the problem of solving $( \ast )$ is reduced to the problem of finding a solution, within a certain space of entire functions, of a finite order linear differential equation with constant coefficients. Additionally, differential operator techniques are used to study Appell polynomial expansions.