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In mathematics, the '''Bateman polynomials''' are a family ''F''<sub>''n''</sub> of [[orthogonal polynomials]] introduced by {{harvs|txt|last=Bateman|authorlink=Harry Bateman|year=1933}}. The '''Bateman–Pasternack polynomials''' are a generalization introduced by {{harvtxt|Pasternack|1939}}.

Bateman polynomials can be defined by the relation
:<math>F_n\left(\frac{d}{dx}\right)\operatorname{sech}(x) = \operatorname{sech}(x)P_n(\tanh(x)).</math>
where ''P''<sub>''n''</sub> is a [[Legendre polynomial]]. In terms of [[generalized hypergeometric function]]s, they are given by 
:<math>F_n(x)={}_3F_2\left(\begin{array}{c}-n,~n+1,~\tfrac12(x+1)\\ 1,~1 \end{array}; 1\right).</math>

{{harvtxt|Pasternack|1939}} generalized the Bateman polynomials to polynomials ''F''{{su|b=''n''|p=''m''}} with

:<math>F_n^m\left(\frac{d}{dx}\right)\operatorname{sech}^{m+1}(x) = \operatorname{sech}^{m+1}(x)P_n(\tanh(x))</math>

These generalized polynomials also have a representation in terms of generalized hypergeometric functions, namely
:<math>F_n^m(x)={}_3F_2\left(\begin{array}{c}-n,~n+1,~\tfrac12(x+m+1)\\ 1,~m+1 \end{array}; 1\right).</math>

{{harvtxt|Carlitz|1957}} showed that the polynomials ''Q''<sub>''n''</sub> studied by {{harvtxt|Touchard|1956}} , see [[Touchard polynomials]], are the same as Bateman polynomials up to a change of variable: more precisely
:<math> Q_n(x)=(-1)^n2^nn!\binom{2n}{n}^{-1}F_n(2x+1)</math>

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

==Examples==
The polynomials of small ''n'' read
:<math>F_0(x)=1</math>;
:<math>F_1(x)=-x</math>;
:<math>F_2(x)=\frac{1}{4}+\frac{3}{4}x^2</math>;
:<math>F_3(x)=-\frac{7}{12}x-\frac{5}{12}x^3</math>;
:<math>F_4(x)=\frac{9}{64}+\frac{65}{96}x^2+\frac{35}{192}x^4</math>;
:<math>F_5(x)=-\frac{407}{960}x-\frac{49}{96}x^3-\frac{21}{320}x^5</math>;

==Properties==

===Orthogonality===

The Bateman polynomials satisfy the orthogonality relation<ref>Koelink (1996)</ref><ref>Bateman, H. (1934), [https://www.jstor.org/stable/1968493 "The polynomial <math>F_n(x)</math>"], ''Ann. Math.'' '''35''' (4): 767-775.</ref>
:<math>\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}.</math>
The factor <math>(-1)^n</math> occurs on the right-hand side of this equation because the Bateman polynomials as defined here must be scaled by a factor <math>i^n</math> 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 <math>B_n(x)=i^nF_n(ix)</math>, for which it becomes
:<math>\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}.</math>

===Recurrence relation===

The sequence of Bateman polynomials satisfies the recurrence relation<ref>Bateman (1933), p. 28.</ref>
:<math>(n+1)^2F_{n+1}(z)=-(2n+1)zF_n(z) + n^2F_{n-1}(z).</math>

===Generating function===

The Bateman polynomials also have the generating function
:<math>\sum_{n=0}^{\infty}t^nF_n(z)=(1-t)^z\,_2F_1\left(\frac{1+z}{2},\frac{1+z}{2};1;t^2\right),</math>
which is sometimes used to define them.<ref>Bateman (1933), p. 23.</ref>

==References==
{{Reflist}}

*{{cite journal|first1= Nadhla A. |last1=Al-Salam
|title=A class of hypergeometric polynomials
|journal=Ann. Mat. Pura Appl.
|year=1967 | volume=75|number=1 | pages=95–120 | doi=10.1007/BF02416800
|doi-access=free}}
*{{Citation | last1=Bateman | first1=H. | title=Some properties of a certain set of polynomials. | url=https://www.jstage.jst.go.jp/article/tmj1911/37/0/37_0_23/_article/-char/ja/ | jfm=59.0364.02 | year=1933 | journal=Tôhoku Mathematical Journal | volume=37 | pages=23–38 }}
*{{Citation | last1=Carlitz | first1=Leonard | title=Some polynomials of Touchard connected with the Bernoulli numbers | doi=10.4153/CJM-1957-021-9 | mr=0085361 | year=1957 | journal=[[Canadian Journal of Mathematics]] | issn=0008-414X | volume=9 | pages=188–190 | doi-access=free }}
*{{Citation | last1=Koelink | first1=H. T. | title=On Jacobi and continuous Hahn polynomials | doi=10.1090/S0002-9939-96-03190-5 | mr=1307541 | year=1996 | journal=[[Proceedings of the American Mathematical Society]] | issn=0002-9939 | volume=124 | issue=3 | pages=887–898| doi-access=free | arxiv=math/9409230 }}
*{{Citation | last1=Pasternack | first1=Simon | title=A generalization of the polynomial F<sub>n</sub>(x) | mr=0000698 | year=1939 | journal=London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science | volume=28 | issue=187 | pages=209–226| doi=10.1080/14786443908521175 }}
*{{Citation | last1=Touchard | first1=Jacques | title=Nombres exponentiels et nombres de Bernoulli | mr=0079021 | year=1956 | journal=[[Canadian Journal of Mathematics]] | issn=0008-414X | volume=8 | pages=305–320 | doi=10.4153/cjm-1956-034-1 | doi-access=free }}

[[Category:Orthogonal polynomials]]