Appell polynomials and differential equations of infinite order
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- by J. D. Buckholtz PDF
- Trans. Amer. Math. Soc. 185 (1973), 463-476 Request permission
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.References
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Additional Information
- © Copyright 1973 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 185 (1973), 463-476
- MSC: Primary 30A62; Secondary 34A35
- DOI: https://doi.org/10.1090/S0002-9947-1973-0352456-9
- MathSciNet review: 0352456