阿斯基-威爾遜多項式

2nd order Askey-Wilson polynomials

2nd order Askey-Wilson polynomials

阿斯基-威爾遜多項式（Askey–Wilson polynomials）是一個以基本超幾何函數表示的正交多項式:

${\frac {a^{n}p_{n}(x;a,b,c,d,|q)}{(ab,ac,ad;q)_{n}}}=_{4}\phi _{3}(q^{-n},abcdq^{n-1},ae^{i\theta },ae^{-i\theta };ab,ac,ad;q,q)$

其中$x=cos(\theta )$ 阿斯基-威爾遜多項式是威爾遜多項式的q模擬.

極限關係

阿斯基-威爾遜多項式→連續雙q哈恩多項式

在阿斯基-威爾遜多項式中，令$d=0$即得連續雙哈恩多項式 $p_{n}(x;a,b,c,0|q)=p_{n}(x;a,b,c|q)$

阿斯基-威爾遜多項式→連續q哈恩多項式

在阿斯基-威爾遜多項式中作代換$\theta \to \theta +\phi$,$a\to ae^{i\theta }$,$b\to be^{i\theta }$,$c\to ce^{-i\theta }$,$d\to de^{-i\theta }$即得連續q哈恩多項式： $p_{n}(cos(\theta +\phi );ae^{i\theta },be^{i\theta },ce^{-i\theta },de^{-i\theta }|q)=p_{n}(cos(\theta +\phi ),a,b,c,d;q)$

阿斯基-威爾遜多項式→大q雅可比多項式

參考文獻

Roelof KoekoeK,Peter Lesky Rene Swarttouw,Hypergeometric Orthogonal Polynomials and Their q-Analogues, Springer 2010*Askey, Richard; Wilson, James, Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Memoirs of the American Mathematical Society, 1985, 54 (319): iv+55, ISBN 978-0-8218-2321-7, ISSN 0065-9266, MR 0783216, doi:10.1090/memo/0319 

• Gasper, George; Rahman, Mizan, Basic hypergeometric series, Encyclopedia of Mathematics and its Applications 96 2nd, Cambridge University Press, 2004, ISBN 978-0-521-83357-8, MR 2128719 
• Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, René F., Askey-Wilson class, 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 
• Koornwinder, Tom H., Askey-Wilson polynomial, Scholarpedia, 2012, 7 (7): 7761, doi:10.4249/scholarpedia.7761