Meixner–Pollaczek polynomials

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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 Pλ
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}{}_2F_1(-n,\lambda+ix;2\lambda;1-e^{-2i\phi})
P_n^{\lambda}(\cos \phi;a,b) = \frac{(2\lambda)_n}{n!}e^{in\phi}{}_2F_1(-n,\lambda+i(a\cos \phi+b)/\sin \phi;2\lambda;1-e^{-2i\phi})

They 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 is given by

\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}

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