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: 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.

**[1]**R. P. Boas and R. C. Buck,*Polynomial expansions of analytic functions*, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete, N.F., Band 19, Academic Press, New York; Springer-Verlag, Berlin, 1964. MR**29**#218. MR**0037346 (12:248f)****[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)****[4]**W. T. Martin,*On expansions in terms of a certain general class of functions*, Amer. J. Math.**58**(1936), 407-420. MR**1507163****[5]**J. M. Whittaker,*Interpolator function theory*, Cambridge Univ. Press, Cambridge, 1935.

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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