連續q哈恩多項式

連續q-哈恩多項式以基本超幾何函數定義： $p_{n}(x;a,b,c|q)=a^{-n}e^{-inu}(abe^{2iu},ac,ad;q)_{n}*_{4}\Phi _{3}(q^{-n},abcdq^{n-1},ae^{i{t+2u}},ae^{-it};abe^{2iu},ac,ad;q;q)$

並且

$x=cos(t+u)$

極限關係

連續q哈恩多項式→Q梅西納-帕拉澤克多項式

${\frac {p_{n}(cos(\theta +\phi );a,0,0,a;q)}{(q;q)_{n}}}=P_{n}(cos(\theta +\phi );a|q)$

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： $p_{n}(cos(\theta +\phi );ae^{i\theta },be^{i\theta },ce^{-i\theta },de^{-i\theta }|q)=p_{n}(cos(\theta +\phi ),a,b,c,d;q)$

圖集

CONTINUOUS q hahn ABS COMPLEX3D Maple PLOT

CONTINUOUS q hahn IIM COMPLEX3D Maple PLOT

CONTINUOUS q hahn RE COMPLEX3D Maple PLOT

CONTINUOUS q hahn ABS densitu Maple PLOT

CONTINUOUS q hahn im densitu Maple PLOT

CONTINUOUS q hahn RE densitu Maple PLOT

參考文獻

• 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, doi:10.2277/0521833574 
• 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., Chapter 18: Orthogonal 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