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

The first few Meixner–Pollaczek polynomials are

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

and the orthogonality relation is given by

The sequence of Meixner–Pollaczek polynomials satisfies the recurrence relation

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

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

The Meixner–Pollaczek polynomials have the generating function

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