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Meixner–Pollaczek polynomials

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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 ( λ ) ( x ; ϕ ) = ( 2 λ ) n n ! e i n ϕ 2 F 1 ( − n , λ + i x ; 2 λ ; 1 − e − 2 i ϕ ) P n λ ( cos ⁡ ϕ ; a , b ) = ( 2 λ ) n n ! e i n ϕ 2 F 1 ( − n , λ + i ( a cos ⁡ ϕ + b ) / sin ⁡ ϕ ; 2 λ ; 1 − e − 2 i ϕ )

They are orthogonal on the real line with respect to the weight function

w ( x ; λ , ϕ ) = | Γ ( λ + i x ) | 2 e ( 2 ϕ − π ) x

and the orthogonality is given by

∫ − ∞ ∞ P n ( λ ) ( x ; ϕ ) P m ( λ ) ( x ; ϕ ) w ( x ; λ , ϕ ) d x = 2 π Γ ( n + 2 λ ) ( 2 sin ⁡ ϕ ) 2 λ n ! δ m n

References

Meixner–Pollaczek polynomials Wikipedia
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