In mathematics, the Meixner–Pollaczek polynomials are a family of

orthogonal polynomials In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonal In mathematics, orthogonality (mathematics), orthogonality is the generalization of the geom ...

''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)^.

References

* * * * {{DEFAULTSORT:Meixner-Pollaczek polynomials Orthogonal polynomials