查看“︁梅西纳-珀拉泽克多项式”︁的源代码

[[File:Meixner-Pollaczek 2.gif|thumb|Meixner-Pollaczek  Polynomials  animation]]
[[File:Meixner-Pollaczek 3.gif|thumb|Meixner-Pollaczek  Polynomials  animation]]
'''梅西纳-珀拉泽克多项式'''是一个以[[超几何函数]]定义的正交多项式。

:<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>
:<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>

==正交性==
:<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>

==极限关系==
;[[连续双哈恩多项式]]→梅西纳-珀拉泽克多项式

<math>\lim_{t \to \infty}\frac{ S_{n}((x-t)^2;\lambda+it,\lambda-it,tcos\phi         }{ t^{n}N! }=\frac{ P_{n}^{(\lambda)}(x;\phi  }{(sin\phi)^n    }                            </math>

;[[连续哈恩多项式]]→梅西纳-珀拉泽克多项式

<math>\lim_{t \to \infty}\frac{ S_{n}((x+t);\lambda+it,tan\phi,\lambda-it,ttan\phi         }{ t^{n}N! }=\frac{ P_{n}^{(\lambda)}(x;\phi  }{(cos\phi)^n    }                            </math>

;梅西纳-珀拉泽克多项式→[[拉盖尔多项式]]

;梅西纳-珀拉泽克多项式→[[埃尔米特多项式]]

==参考文献==
*{{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 | accessdate=2015-01-27 | archive-date=2017-08-06 | archive-url=https://web.archive.org/web/20170806003945/http://gallica.bnf.fr/ark:/12148/bpt6k31801/f1363 | dead-url=no }}

[[Category:正交多项式]]