# 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

${\displaystyle F_{n}\left({\frac {d}{dx}}\right)\cosh ^{-1}(x)=\cosh ^{-1}(x)P_{n}(\tanh(x))={}_{3}F_{2}(-n,n+1,(x+1)/2;1,1;1)}$

where Pn is a Legendre polynomial.

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

${\displaystyle F_{n}^{m}\left({\frac {d}{dx}}\right)\cosh ^{-1-m}(x)=\cosh ^{-1-m}(x)P_{n}(\tanh(x))}$

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

${\displaystyle Q_{n}(x)=(-1)^{n}2^{n}n!{\binom {2n}{n}}^{-1}F_{n}(2x+1)}$

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

## Examples

The polynomials of small n read

${\displaystyle F_{0}(x)=1}$;
${\displaystyle F_{1}(x)=-x}$;
${\displaystyle F_{2}(x)={\frac {1}{4}}+{\frac {3}{4}}x^{2}}$;
${\displaystyle F_{3}(x)=-{\frac {7}{12}}x-{\frac {5}{12}}x^{3}}$;
${\displaystyle F_{4}(x)={\frac {9}{64}}+{\frac {65}{96}}x^{2}+{\frac {35}{192}}x^{4}}$;
${\displaystyle F_{5}(x)={\frac {407}{960}}x-{\frac {49}{96}}x^{3}-{\frac {21}{320}}x^{5}}$;