Meixner–Pollaczek polynomials should not be confused with Meixner polynomials.
In mathematics, the Meixner–Pollaczek polynomials are a family of orthogonal polynomials P(x,φ) introduced by, which up to elementary changes of variables are the same as the Pollaczek polynomials P(x,a,b) rediscovered by in the case λ=1/2, and later generalized by him.
They are defined by
| (λ) | |
| P | |
| n |
(x;\phi)=
| (2λ)n | |
| n! |
ein\phi{}2F1\left(\begin{array}{c}-n,~λ+ix\ 2λ\end{array};1-e-2i\phi\right)
| λ | |
| P | |
| n |
(\cos\phi;a,b)=
| (2λ)n | |
| n! |
ein\phi{}2F1\left(\begin{array}{c}-n,~λ+i(a\cos\phi+b)/\sin\phi\ 2λ\end{array};1-e-2i\phi\right)
The first few Meixner–Pollaczek polynomials are
| (λ) | |
| P | |
| 0 |
(x;\phi)=1
| (λ) | |
| P | |
| 1 |
(x;\phi)=2(λ\cos\phi+x\sin\phi)
| (λ) | |
| P | |
| 2 |
(x;\phi)=x2+λ2+(λ2+λ-x2)\cos(2\phi)+(1+2λ)x\sin(2\phi).
The Meixner–Pollaczek polynomials Pm(λ)(x;φ) are orthogonal on the real line with respect to the weight function
w(x;λ,\phi)=|\Gamma(λ+ix)|2e(2\phi-\pi)x
| infty | |
| \int | |
| -infty |
| (λ) | |
| P | |
| n |
| (λ) | |
| (x;\phi)P | |
| m |
(x;\phi)w(x;λ,\phi)dx=
| 2\pi\Gamma(n+2λ) | |
| (2\sin\phi)2λn! |
\deltamn, λ>0, 0<\phi<\pi.
The sequence of Meixner–Pollaczek polynomials satisfies the recurrence relation[2]
| (λ) | |
| (n+1)P | |
| n+1 |
(x;\phi)=2l(x\sin\phi+
| (λ) | |
| (n+λ)\cos\phir)P | |
| n |
(x;\phi)-(n+2λ-1)Pn-1(x;\phi).
The Meixner–Pollaczek polynomials are given by the Rodrigues-like formula[3]
| (λ) | ||
| P | (x;\phi)= | |
| n |
| (-1)n | |
| n!w(x;λ,\phi) |
| dn | |
| dxn |
w\left(x;λ+\tfrac12n,\phi\right),
The Meixner–Pollaczek polynomials have the generating function[4]
| infty | |
| \sum | |
| n=0 |
tn
| (λ) | |
| P | |
| n |
(x;\phi)=(1-ei\phit)-λ+ix(1-e-i\phit)-λ-ix.