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)) ={}_3F_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

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

Q_n(x)=(-1)^n2^nn!\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

F_0(x)=1;
F_1(x)=-x;
F_2(x)=\frac{1}{4}+\frac{3}{4}x^2;
F_3(x)=-\frac{7}{12}x-\frac{5}{12}x^3;
F_4(x)=\frac{9}{64}+\frac{65}{96}x^2+\frac{35}{192}x^4;
F_5(x)=\frac{407}{960}x-\frac{49}{96}x^3-\frac{21}{320}x^5;

References