Meixner–Pollaczek 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
:$P_n^(x;\phi) = \frace^_2F_1\left(\begin -n,~\lambda+ix\\ 2\lambda \end; 1-e^\right)$
:$P_n^(\cos \phi;a,b) = \frace^_2F_1\left(\begin-n,~\lambda+i(a\cos \phi+b)/\sin \phi\\ 2\lambda \end;1-e^\right)$

Examples

The first few Meixner–Pollaczek polynomials are
:$P_0^(x;\phi)=1$
:$P_1^(x;\phi)=2(\lambda\cos\phi + x\sin\phi)$
:$P_2^(x;\phi)=x^2+\lambda^2+(\lambda^2+\lambda-x^2)\cos(2\phi)+(1+2\lambda)x\sin(2\phi).$

Properties

Orthogonality

The Meixner–Pollaczek polynomials ''P''_m^(λ)(''x'';φ) are orthogonal on the real line with respect to the weight function
:$w(x; \lambda, \phi)= , \Gamma(\lambda+ix), ^2 e^$
and the orthogonality relation is given by
:$\int_^P_n^(x;\phi)P_m^(x;\phi)w(x; \lambda, \phi)dx=\frac\delta_,\quad \lambda>0,\quad 0<\phi<\pi.$

Recurrence relation

The sequence of Meixner–Pollaczek polynomials satisfies the recurrence relation
:$(n+1)P_^(x;\phi)=2\bigl(x\sin\phi + (n+\lambda)\cos\phi\bigr)P_n^(x;\phi)-(n+2\lambda-1)P_(x;\phi).$

Rodrigues formula

The Meixner–Pollaczek polynomials are given by the Rodrigues-like formula
:$P_n^(x;\phi)=\frac\fracw\left(x;\lambda+\tfrac12n,\phi\right),$
where ''w''(''x'';λ,φ) is the weight function given above.

Generating function

The Meixner–Pollaczek polynomials have the generating functionKoekoek, Lesky, & Swarttouw (2010), p. 215.
:$\sum_^t^n P_n^(x;\phi) = (1-e^t)^(1-e^t)^.$

See also

* Sieved Pollaczek polynomials

References

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{{DEFAULTSORT:Meixner-Pollaczek polynomials
Orthogonal polynomials