Meixner–Pollaczek polynomials

In mathematics, the Meixner–Pollaczek polynomials are a family of orthogonal polynomials P{{su|b=n|p=(λ)}}(x,φ) introduced by {{harvs|txt|authorlink=Josef Meixner|last=Meixner|year=1934}}, which up to elementary changes of variables are the same as the Pollaczek polynomials P{{su|b=n|p=λ}}(x,a,b) rediscovered by {{harvs|txt|authorlink=Felix Pollaczek|last=Pollaczek|year=1949}} 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 P_m^(λ)(x;φ) are orthogonal on the real line with respect to the weight function

: $w(x; \lambda, \phi)= |\Gamma(\lambda+ix)|^2 e^{(2\phi-\pi)x}$

and the orthogonality relation is given byKoekoek, Lesky, & Swarttouw (2010), p. 213.

: $\int_{-\infty}^{\infty}P_n^{(\lambda)}(x;\phi)P_m^{(\lambda)}(x;\phi)w(x; \lambda, \phi)dx=\frac{2\pi\Gamma(n+2\lambda)}{(2\sin\phi)^{2\lambda}n!}\delta_{mn},\quad \lambda>0,\quad 0<\phi<\pi.$

=Recurrence relation=

The sequence of Meixner–Pollaczek polynomials satisfies the recurrence relationKoekoek, Lesky, & Swarttouw (2010), p. 213.

: $(n+1)P_{n+1}^{(\lambda)}(x;\phi)=2\bigl(x\sin\phi + (n+\lambda)\cos\phi\bigr)P_n^{(\lambda)}(x;\phi)-(n+2\lambda-1)P_{n-1}(x;\phi).$

=Rodrigues formula=

The Meixner–Pollaczek polynomials are given by the Rodrigues-like formulaKoekoek, Lesky, & Swarttouw (2010), p. 214.

: $P_n^{(\lambda)}(x;\phi)=\frac{(-1)^n}{n!\,w(x;\lambda,\phi)}\frac{d^n}{dx^n}w\left(x;\lambda+\tfrac12n,\phi\right),$

where w(x;λ,φ) is the weight function given above.

=Generating function=

The Meixner–Pollaczek polynomials have the generating functionKoekoek, Lesky, & Swarttouw (2010), p. 215.

: $\sum_{n=0}^{\infty}t^n P_n^{(\lambda)}(x;\phi) = (1-e^{i\phi}t)^{-\lambda+ix}(1-e^{-i\phi}t)^{-\lambda-ix}.$

References

{{Citation | last1=Koekoek | first1=Roelof | last2=Lesky | first2=Peter A. | last3=Swarttouw | first3=René F. | title=Hypergeometric orthogonal polynomials and their q-analogues | publisher=Springer-Verlag | location=Berlin, New York | series=Springer Monographs in Mathematics | isbn=978-3-642-05013-8 | doi=10.1007/978-3-642-05014-5 | mr=2656096 | year=2010}}
{{dlmf|id=18.35|title=Pollaczek Polynomials|first=Tom H. |last=Koornwinder|first2=Roderick S. C.|last2= Wong|first3=Roelof |last3=Koekoek||first4=René F. |last4=Swarttouw}}
{{Citation | last1=Meixner | first1=J. | title=Orthogonale Polynomsysteme Mit Einer Besonderen Gestalt Der Erzeugenden Funktion | doi=10.1112/jlms/s1-9.1.6 | year=1934 | journal=J. London Math. Soc. | volume=s1-9 | pages=6–13}}
{{Citation | last1=Pollaczek | first1=Félix | title=Sur une généralisation des polynomes de Legendre | url=http://gallica.bnf.fr/ark:/12148/bpt6k31801/f1363 | mr=0030037 | year=1949 | journal=Les Comptes rendus de l'Académie des sciences | volume=228 | pages=1363–1365}}

Category:Orthogonal polynomials