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

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

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

The first few Meixner–Pollaczek polynomials are

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

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

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

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

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

and the orthogonality relation is given by^[1]

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

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

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

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

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

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

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

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

• Sieved Pollaczek polynomials

• 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