Meixner–Pollaczek polynomials

In mathematics, the Meixner–Pollaczek polynomials are a family of orthogonal polynomials P(λ)
n
(x,φ) introduced by Meixner (1934), which up to elementary changes of variables are the same as the Pollaczek polynomials Pλ
n
(x,a,b) rediscovered by Pollaczek (1949) in the case λ=1/2, and later generalized by him.

They are defined by

Examples

The first few Meixner–Pollaczek polynomials are

Properties

Orthogonality

The Meixner–Pollaczek polynomials Pm(λ)(x;φ) are orthogonal on the real line with respect to the weight function

and the orthogonality relation is given by[1]

Recurrence relation

The sequence of Meixner–Pollaczek polynomials satisfies the recurrence relation[2]

Rodrigues formula

The Meixner–Pollaczek polynomials are given by the Rodrigues-like formula[3]

where w(x;λ,φ) is the weight function given above.

Generating function

The Meixner–Pollaczek polynomials have the generating function[4]

See also

References

  1. Koekoek, Lesky, & Swarttouw (2010), p. 213.
  2. Koekoek, Lesky, & Swarttouw (2010), p. 213.
  3. Koekoek, Lesky, & Swarttouw (2010), p. 214.
  4. Koekoek, Lesky, & Swarttouw (2010), p. 215.
  • Koekoek, Roelof; Lesky, Peter A.; Swarttouw, René F. (2010), Hypergeometric orthogonal polynomials and their q-analogues, Springer Monographs in Mathematics, Berlin, New York: Springer-Verlag, doi:10.1007/978-3-642-05014-5, ISBN 978-3-642-05013-8, MR 2656096
  • Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, René F. (2010), "Pollaczek Polynomials", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248
  • Meixner, J. (1934), "Orthogonale Polynomsysteme Mit Einer Besonderen Gestalt Der Erzeugenden Funktion", J. London Math. Soc., s1–9: 6–13, doi:10.1112/jlms/s1-9.1.6
  • Pollaczek, Félix (1949), "Sur une généralisation des polynomes de Legendre", Les Comptes rendus de l'Académie des sciences, 228: 1363–1365, MR 0030037
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