Bateman polynomials

In mathematics, the Bateman polynomials are a family F_n of orthogonal polynomials introduced by Template:Harvs. The Bateman–Pasternack polynomials are a generalization introduced by Template:Harvtxt.

Bateman polynomials can be defined by the relation

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

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

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

Template:Harvtxt generalized the Bateman polynomials to polynomials FTemplate:Su with

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

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

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

Template:Harvtxt showed that the polynomials Q_n studied by Template:Harvtxt , see Touchard polynomials, are the same as Bateman polynomials up to a change of variable: more precisely

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

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

Examples

The polynomials of small n read

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

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

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

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

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

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

Properties

Orthogonality

The Bateman polynomials satisfy the orthogonality relation

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

The factor $(-1)^{n}$ occurs on the right-hand side of this equation because the Bateman polynomials as defined here must be scaled by a factor $i^{n}$ 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 $B_{n}(x)=i^{n}F_{n}(ix)$, for which it becomes

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

Recurrence relation

The sequence of Bateman polynomials satisfies the recurrence relation

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

Generating function

The Bateman polynomials also have the generating function

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

which is sometimes used to define them.

References

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