Bateman polynomials

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

Bateman polynomials can be defined by the relation

${\displaystyle F_{n}\left({\frac {d}{dx}}\right)\operatorname {sech} (x)=\operatorname {sech} (x)P_{n}(\tanh(x)).}$

where P_n is a Legendre polynomial. In terms of generalized hypergeometric functions, they are given by

${\displaystyle F_{n}(x)={}_{3}F_{2}\left({\begin{array}{c}-n,~n+1,~{\tfrac {1}{2}}(x+1)\\1,~1\end{array}};1\right).}$

Pasternack (1939) generalized the Bateman polynomials to polynomials F^m
_n with

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

These generalized polynomials also have a representation in terms of generalized hypergeometric functions, namely

${\displaystyle F_{n}^{m}(x)={}_{3}F_{2}\left({\begin{array}{c}-n,~n+1,~{\tfrac {1}{2}}(x+m+1)\\1,~m+1\end{array}};1\right).}$

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

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

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

The polynomials of small n read

${\displaystyle F_{0}(x)=1}$;

${\displaystyle F_{1}(x)=-x}$;

${\displaystyle F_{2}(x)={\frac {1}{4}}+{\frac {3}{4}}x^{2}}$;

${\displaystyle F_{3}(x)=-{\frac {7}{12}}x-{\frac {5}{12}}x^{3}}$;

${\displaystyle F_{4}(x)={\frac {9}{64}}+{\frac {65}{96}}x^{2}+{\frac {35}{192}}x^{4}}$;

${\displaystyle F_{5}(x)=-{\frac {407}{960}}x-{\frac {49}{96}}x^{3}-{\frac {21}{320}}x^{5}}$;

The Bateman polynomials satisfy the orthogonality relation¹²

${\displaystyle \int _{-\infty }^{\infty }F_{m}(ix)F_{n}(ix)\operatorname {sech} ^{2}\left({\frac {\pi x}{2}}\right)\,dx={\frac {4(-1)^{n}}{\pi (2n+1)}}\delta _{mn}.}$

The factor ${\displaystyle (-1)^{n}}$ occurs on the right-hand side of this equation because the Bateman polynomials as defined here must be scaled by a factor ${\displaystyle i^{n}}$ to make them remain real-valued for imaginary argument. The orthogonality relation is simpler when expressed in terms of a modified set of polynomials defined by ${\displaystyle B_{n}(x)=i^{n}F_{n}(ix)}$, for which it becomes

${\displaystyle \int _{-\infty }^{\infty }B_{m}(x)B_{n}(x)\operatorname {sech} ^{2}\left({\frac {\pi x}{2}}\right)\,dx={\frac {4}{\pi (2n+1)}}\delta _{mn}.}$

The sequence of Bateman polynomials satisfies the recurrence relation³

${\displaystyle (n+1)^{2}F_{n+1}(z)=-(2n+1)zF_{n}(z)+n^{2}F_{n-1}(z).}$

The Bateman polynomials also have the generating function

${\displaystyle \sum _{n=0}^{\infty }t^{n}F_{n}(z)=(1-t)^{z}\,_{2}F_{1}\left({\frac {1+z}{2}},{\frac {1+z}{2}};1;t^{2}\right),}$

which is sometimes used to define them.⁴