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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Appell polynomials and differential equations of infinite order

Author: J. D. Buckholtz
Journal: Trans. Amer. Math. Soc. 185 (1973), 463-476
MSC: Primary 30A62; Secondary 34A35
MathSciNet review: 0352456
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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 $ \vert z\vert = 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.

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Keywords: Appell polynomials, polynomial expansions, infinite order differential equations
Article copyright: © Copyright 1973 American Mathematical Society