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In mathematics, the Bateman polynomials are a family \( F_n \) 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 \( P_n \) is a Legendre polynomial.

Pasternack (1939) generalized the Bateman polynomials to polynomials \( F_m \)
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 \( Q_n \) 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

Al-Salam, Nadhla A. (1967). "A class of hypergeometric polynomials". Ann. Matem. Pura Applic. 75 (1): 95–120. doi:10.1007/BF02416800.
Bateman, H. (1933), "Some properties of a certain set of polynomials.", Tôhoku Mathematical Journal 37: 23–38, JFM 59.0364.02
Carlitz, Leonard (1957), "Some polynomials of Touchard connected with the Bernoulli numbers", Canadian Journal of Mathematics 9: 188–190, doi:10.4153/CJM-1957-021-9, ISSN 0008-414X, MR 0085361
Koelink, H. T. (1996), "On Jacobi and continuous Hahn polynomials", Proceedings of the American Mathematical Society 124 (3): 887–898, doi:10.1090/S0002-9939-96-03190-5, ISSN 0002-9939, MR 1307541
Pasternack, Simon (1939), "A generalization of the polynomial Fn(x)", London, Edinburgh, Dublin Philosophical Magazine and Journal of Science 28: 209–226, MR 0000698
Touchard, Jacques (1956), "Nombres exponentiels et nombres de Bernoulli", Canadian Journal of Mathematics 8: 305–320, doi:10.4153/cjm-1956-034-1, ISSN 0008-414X, MR 0079021

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