Bateman polynomials

In mathematics, the Bateman polynomials are a family F_n of orthogonal polynomials introduced by Bateman (1933). The Bateman–Pasternack polynomials are a generalization introduced by Pasternack (1939).

Bateman polynomials are given by

where P_n is a Legendre polynomial.

Pasternack (1939) generalized the Bateman polynomials to polynomials Fm
n with

Carlitz (1957) showed that the polynomials Q_n studied by Touchard (1956) , see Touchard polynomials, are the same as Bateman polynomials up to a change of variable: more precisely

Bateman and Pasternack's polynomials are special cases of the symmetric continuous Hahn polynomials.

The polynomials of small n read

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