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<div>{{distinguish|Meixner polynomials}}<br />
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.<br />
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They are defined by <br />
:<math>P_n^{(\lambda)}(x;\phi) = \frac{(2\lambda)_n}{n!}e^{in\phi}{}_2F_1(-n,\lambda+ix;2\lambda;1-e^{-2i\phi})</math><br />
:<math>P_n^{\lambda}(\cos \phi;a,b) = \frac{(2\lambda)_n}{n!}e^{in\phi}{}_2F_1(-n,\lambda+i(a\cos \phi+b)/\sin \phi;2\lambda;1-e^{-2i\phi})</math><br />
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They are orthogonal on the real line with respect to the weight function<br />
:<math> w(x; \lambda, \phi)= |\Gamma(\lambda+ix)|e^{(2\phi-\pi)x}</math><br />
and the orthogonality is given by<br />
:<math>\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}</math><br />
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==See also==<br />
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*[[Sieved Pollaczek polynomials]]<br />
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==References==<br />
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*{{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}}<br />
*{{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}}<br />
*{{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}}<br />
*{{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}}<br />
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{{DEFAULTSORT:Meixner-Pollaczek polynomials}}<br />
[[Category:Orthogonal polynomials]]</div>64.121.114.179