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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^{(\lambda)}(x;\phi) = \frac{(2\lambda)_n}{n!}e^{in\phi}{}_2F_1\left(\begin{array}{c} -n,\lambda+ix\ 2\lambda \end{array}; 1-e^{-2i\phi}\right) :P_n^{\lambda}(\cos \phi;a,b) = \frac{(2\lambda)_n}{n!}e^{in\phi}{}_2F_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^{(\lambda)}(x;\phi)=1 :P_1^{(\lambda)}(x;\phi)=2(\lambda\cos\phi + x\sin\phi) :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; \lambda, \phi)= |\Gamma(\lambda+ix)|^2 e^{(2\phi-\pi)x} and the orthogonality relation is given by :\int_{-\infty}^{\infty}P_n^{(\lambda)}(x;\phi)P_m^{(\lambda)}(x;\phi)w(x; \lambda, \phi)dx=\frac{2\pi\Gamma(n+2\lambda)}{(2\sin\phi)^{2\lambda}n!}\delta_{mn},\quad \lambda0,\quad 0

Recurrence relation

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

Rodrigues formula

The Meixner–Pollaczek polynomials are given by the Rodrigues-like formula :P_n^{(\lambda)}(x;\phi)=\frac{(-1)^n}{n!,w(x;\lambda,\phi)}\frac{d^n}{dx^n}w\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 function :\sum_{n=0}^{\infty}t^n P_n^{(\lambda)}(x;\phi) = (1-e^{i\phi}t)^{-\lambda+ix}(1-e^{-i\phi}t)^{-\lambda-ix}.

References

References

  1. Koekoek, Lesky, & Swarttouw (2010), p. 213.
  2. Koekoek, Lesky, & Swarttouw (2010), p. 213.
  3. Koekoek, Lesky, & Swarttouw (2010), p. 214.
  4. Koekoek, Lesky, & Swarttouw (2010), p. 215.
Wikipedia Source

This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page.

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