Meixner–Pollaczek polynomials

Not to be confused with Meixner 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

$P_n^{(\lambda )}(x;\varphi )=\frac{(2\lambda {)}_n}{n!}{e}^{in\varphi }{}_2{F}_1(\begin{array}{c}-n,\lambda +ix\\ 2\lambda \end{array};1-e^{-2i\varphi })$

$P_n^{\lambda }(\cos \varphi ;a,b)=\frac{(2\lambda {)}_n}{n!}{e}^{in\varphi }{}_2{F}_1(\begin{array}{c}-n,\lambda +i(a\cos \varphi +b)/\sin \varphi \\ 2\lambda \end{array};1-e^{-2i\varphi })$

Examples

The first few Meixner–Pollaczek polynomials are

$P_0^{(\lambda )}(x;\varphi )=1$

$P_1^{(\lambda )}(x;\varphi )=2(\lambda \cos \varphi +x\sin \varphi )$

$P_2^{(\lambda )}(x;\varphi )=x^2+{\lambda }^2+({\lambda }^2+\lambda -x^2)\cos (2\varphi )+(1+2\lambda )x\sin (2\varphi ).$

Properties

Orthogonality

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

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

and the orthogonality relation is given by

$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{P}_n^{(\lambda )}(x;\varphi )P_m^{(\lambda )}(x;\varphi )w(x;\lambda ,\varphi )dx=\frac{2\pi \mathrm{\Gamma }(n+2\lambda )}{(2\sin \varphi {)}^{2\lambda }n!}{\delta }_{mn},\lambda >0,0<\varphi <\pi .$

Recurrence relation

The sequence of Meixner–Pollaczek polynomials satisfies the recurrence relation

$(n+1)P_{n+1}^{(\lambda )}(x;\varphi )=2(x\sin \varphi +(n+\lambda )\cos \varphi )P_n^{(\lambda )}(x;\varphi )-(n+2\lambda -1)P_{n-1}(x;\varphi ).$

Rodrigues formula

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

$P_n^{(\lambda )}(x;\varphi )=\frac{(-1{)}^n}{n!w(x;\lambda ,\varphi )}\frac{d^n}{d{x}^n}w(x;\lambda +\frac{1}{2}n,\varphi ),$

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

Generating function

The Meixner–Pollaczek polynomials have the generating function

$\sum _{n=0}^{\mathrm{\infty }}{t}^n{P}_n^{(\lambda )}(x;\varphi )=(1-e^{i\varphi }t{)}^{-\lambda +ix}(1-e^{-i\varphi }t{)}^{-\lambda -ix}.$

See also

• Sieved Pollaczek polynomials

References

• 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