梅西納-珀拉澤克多項式

梅西納-珀拉澤克多項式是一個以超幾何函數定義的正交多項式。

${\displaystyle P_{n}^{(\lambda )}(x;\phi )={\frac {(2\lambda )_{n}}{n!}}e^{in\phi }{}_{2}F_{1}(-n,\lambda +ix;2\lambda ;1-e^{-2i\phi })}$

${\displaystyle P_{n}^{\lambda }(\cos \phi ;a,b)={\frac {(2\lambda )_{n}}{n!}}e^{in\phi }{}_{2}F_{1}(-n,\lambda +i(a\cos \phi +b)/\sin \phi ;2\lambda ;1-e^{-2i\phi })}$

Meixner-Pollaczek Polynomials animation

Meixner-Pollaczek Polynomials animation

正交性

${\displaystyle \int _{-\infty }^{\infty }P_{n}^{(\lambda )}(x;\phi )P_{m}^{(\lambda )}(x;\phi )w(x;\lambda ,\phi )dx={\frac {2\pi \Gamma (n+2\lambda )}{(2\sin \phi )^{2\lambda }n!}}\delta _{mn}}$

極限關係

連續雙哈恩多項式→梅西納-珀拉澤克多項式

${\displaystyle \lim _{t\to \infty }{\frac {S_{n}((x-t)^{2};\lambda +it,\lambda -it,tcos\phi }{t^{n}N!}}={\frac {P_{n}^{(\lambda )}(x;\phi }{(sin\phi )^{n}}}}$

連續哈恩多項式→梅西納-珀拉澤克多項式

${\displaystyle \lim _{t\to \infty }{\frac {S_{n}((x+t);\lambda +it,tan\phi ,\lambda -it,ttan\phi }{t^{n}N!}}={\frac {P_{n}^{(\lambda )}(x;\phi }{(cos\phi )^{n}}}}$

梅西納-珀拉澤克多項式→拉蓋爾多項式

梅西納-珀拉澤克多項式→埃爾米特多項式

參考文獻

• Koekoek, Roelof; Lesky, Peter A.; Swarttouw, René F., Hypergeometric orthogonal polynomials and their q-analogues, Springer Monographs in Mathematics, Berlin, New York: Springer-Verlag, 2010, ISBN 978-3-642-05013-8, MR 2656096, doi:10.1007/978-3-642-05014-5
• Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, René F., Pollaczek Polynomials, Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (編), NIST Handbook of Mathematical Functions, Cambridge University Press, 2010, ISBN 978-0521192255, MR2723248
• Meixner, J., Orthogonale Polynomsysteme Mit Einer Besonderen Gestalt Der Erzeugenden Funktion, J. London Math. Soc., 1934, s1–9: 6–13, doi:10.1112/jlms/s1-9.1.6
• Pollaczek, Félix, Sur une généralisation des polynomes de Legendre, Les Comptes rendus de l'Académie des sciences, 1949, 228: 1363–1365 [2015-01-27], MR 0030037, （原始內容存檔於2017-08-06）