Bateman polynomials
In mathematics, the Bateman polynomials are a family Fn of orthogonal polynomials introduced by Bateman (1933). The Bateman–Pasternack polynomials are a generalization introduced by Pasternack (1939).
Bateman polynomials are given by
where Pn is a Legendre polynomial.
Pasternack (1939) generalized the Bateman polynomials to polynomials Fm
n with
Carlitz (1957) showed that the polynomials Qn studied by Touchard (1956) , see Touchard polynomials, are the same as Bateman polynomials up to a change of variable: more precisely
Bateman and Pasternack's polynomials are special cases of the symmetric continuous Hahn polynomials.
Examples
The polynomials of small n read
- ;
- ;
- ;
- ;
- ;
- ;
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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