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 P^λ
_n(x,a,b) rediscovered by Pollaczek (1949) in the case λ=1/2, and later generalized by him.

They are defined by

P_{n}^{(\lambda )}(x;\phi )={\frac {(2\lambda )_{n}}{n!}}e^{in\phi }{}_{2}F_{1}\left({\begin{array}{c}-n,~\lambda +ix\\2\lambda \end{array}};1-e^{-2i\phi }\right)

P_{n}^{\lambda }(\cos \phi ;a,b)={\frac {(2\lambda )_{n}}{n!}}e^{in\phi }{}_{2}F_{1}\left({\begin{array}{c}-n,~\lambda +i(a\cos \phi +b)/\sin \phi \\2\lambda \end{array}};1-e^{-2i\phi }\right)

Examples

The first few Meixner–Pollaczek polynomials are

P_{0}^{(\lambda )}(x;\phi )=1

P_{1}^{(\lambda )}(x;\phi )=2(\lambda \cos \phi +x\sin \phi )

P_{2}^{(\lambda )}(x;\phi )=x^{2}+\lambda ^{2}+(\lambda ^{2}+\lambda -x^{2})\cos(2\phi )+(1+2\lambda )x\sin(2\phi ).

Properties

Orthogonality

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

w(x;\lambda ,\phi )=|\Gamma (\lambda +ix)|^{2}e^{(2\phi -\pi )x}

and the orthogonality relation is given by[1]

\int _{-\infty }^{\infty }P_{n}^{(\lambda )}(x;\phi )P_{m}^{(\lambda )}(x;\phi )w(x;\lambda ,\phi )dx={\frac {2\pi \Gamma (n+2\lambda )}{(2\sin \phi )^{2\lambda }n!}}\delta _{mn},\quad \lambda >0,\quad 0<\phi <\pi .

Recurrence relation

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

(n+1)P_{n+1}^{(\lambda )}(x;\phi )=2{\bigl (}x\sin \phi +(n+\lambda )\cos \phi {\bigr )}P_{n}^{(\lambda )}(x;\phi )-(n+2\lambda -1)P_{n-1}(x;\phi ).

Rodrigues formula

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

P_{n}^{(\lambda )}(x;\phi )={\frac {(-1)^{n}}{n!\,w(x;\lambda ,\phi )}}{\frac {d^{n}}{dx^{n}}}w\left(x;\lambda +{\tfrac {1}{2}}n,\phi \right),

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

Generating function

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

\sum _{n=0}^{\infty }t^{n}P_{n}^{(\lambda )}(x;\phi )=(1-e^{i\phi }t)^{-\lambda +ix}(1-e^{-i\phi }t)^{-\lambda -ix}.

References

Koekoek, Lesky, & Swarttouw (2010), p. 213.
Koekoek, Lesky, & Swarttouw (2010), p. 213.
Koekoek, Lesky, & Swarttouw (2010), p. 214.
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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[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.