Bateman polynomials

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

Bateman polynomials are given by

F_n\left(\frac{d}{dx}\right)\cosh^{-1}(x) = \cosh^{-1}(x)P_n(\tanh(x))

where Pn is a Legendre polynomial.

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

F_n^m\left(\frac{d}{dx}\right)\cosh^{-1-m}(x) = \cosh^{-1-m}(x)P_n(\tanh(x))

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

Q_n(x)=(-1)^n2^nn!\binom{2n}{n}^{-1}F_n(2x%2B1)

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

References