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 can be defined by the relation

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

Pasternack (1939) generalized the Bateman polynomials to polynomials Fm
n
with

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

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

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The polynomials of small n read

 ;
 ;
 ;
 ;
 ;
 ;

Properties

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Orthogonality

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The Bateman polynomials satisfy the orthogonality relation[1][2]

 

The factor   occurs on the right-hand side of this equation because the Bateman polynomials as defined here must be scaled by a factor   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  , for which it becomes

 

Recurrence relation

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The sequence of Bateman polynomials satisfies the recurrence relation[3]

 

Generating function

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The Bateman polynomials also have the generating function

 

which is sometimes used to define them.[4]

References

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  1. ^ Koelink (1996)
  2. ^ Bateman, H. (1934), "The polynomial  ", Ann. Math. 35 (4): 767-775.
  3. ^ Bateman (1933), p. 28.
  4. ^ Bateman (1933), p. 23.
  • Al-Salam, Nadhla A. (1967). "A class of hypergeometric polynomials". Ann. Mat. Pura Appl. 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, arXiv:math/9409230, 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, and Dublin Philosophical Magazine and Journal of Science, 28 (187): 209–226, doi:10.1080/14786443908521175, 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