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
DOI:
https://doi.org/10.1090/S0002-9947-1973-0352456-9
MathSciNet review:
0352456
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Abstract | References | Similar Articles | Additional Information
Abstract: Let have radius of convergence
and no singularities other than poles on the circle
. A complete solution is obtained for the infinite order differential equation
. It is shown that
possesses a solution if and only if the function g has a polynomial expansion in terms of the Appell polynomials generated by
. The solutions of
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
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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- [2] J. D. Buckholtz, Appell polynomial expansions and biorthogonal expansions in Banach spaces, Trans. Amer. Math. Soc. (to appear). MR 0333210 (48:11535)
- [3] -, Appell polynomials whose generating function is meromorphic on its circle of convergence, Bull. Amer. Math. Soc. 79 (1973), 469-472. MR 0315128 (47:3677)
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Additional Information
DOI:
https://doi.org/10.1090/S0002-9947-1973-0352456-9
Keywords:
Appell polynomials,
polynomial expansions,
infinite order differential equations
Article copyright:
© Copyright 1973
American Mathematical Society