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 Pλn(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 ) {\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)}
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 ) {\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)}
Examples The first few Meixner–Pollaczek polynomials are
P 0 ( λ ) ( x ; ϕ ) = 1 {\displaystyle P_{0}^{(\lambda )}(x;\phi )=1}
P 1 ( λ ) ( x ; ϕ ) = 2 ( λ cos ϕ + x sin ϕ ) {\displaystyle P_{1}^{(\lambda )}(x;\phi )=2(\lambda \cos \phi +x\sin \phi )}
P 2 ( λ ) ( x ; ϕ ) = x 2 + λ 2 + ( λ 2 + λ − x 2 ) cos ( 2 ϕ ) + ( 1 + 2 λ ) x sin ( 2 ϕ ) . {\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 ).}
Properties
Orthogonality The Meixner–Pollaczek polynomials Pm(λ)(x;φ) 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 {\displaystyle w(x;\lambda ,\phi )=|\Gamma (\lambda +ix)|^{2}e^{(2\phi -\pi )x}}
and the orthogonality relation is given by
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