Bateman polynomials

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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

$F_{n}\left({\frac {d}{dx}}\right)\cosh ^{{-1}}(x)=\cosh ^{{-1}}(x)P_{n}(\tanh(x))={}_{3}F_{2}(-n,n+1,(x+1)/2;1,1;1)$

where P_n is a Legendre polynomial.

Pasternack (1939) generalized the Bateman polynomials to polynomials 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 Q_n studied by Touchard (1956) , see Touchard_polynomials, are the same as Bateman polynomials up to a change of variable: more precisely

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

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

Examples

The polynomials of small n read

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

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

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

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

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

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

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 F_n(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, ISSN 0008-414X, MR 0079021

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