大q雅可比多項式

大q-雅可比多項式（英語：Big q-Jacobi polynomials）是一個以基本超幾何函數定義的正交多項式：

BIG Q JACOBI 2D Maple PLOT

$\displaystyle P_{n}(x;a,b,c;q)={}_{3}\phi _{2}(q^{-n},abq^{n+1},x;aq,cq;q,q)$

正交性

大q-雅可比多項式滿足下列正交關係

$\sum _{x}(p_{n}(x)*p_{m}(x)|x|v_{x}=h_{n}*\delta _{m}n$

極限關係

大q雅可比多項式→大q拉蓋爾多項式

令大q雅可比多項式中的$b=0$,即得大q拉蓋爾多項式

$P_{n}(x;a,0,c;q)=P_{n}(x;a,c;q)$

圖集

BIG Q JACOBI ABS COMPLEX3D Maple PLOT2

BIG Q JACOBI IM COMPLEX3D Maple PLOT

BIG Q JACOBI RE COMPLEX3D Maple PLOT

BIG Q JACOBI ABS COMPLEX DENSITY Maple PLOT

BIG Q JACOBI IM COMPLEX DENSITY Maple PLOT

BIG Q JACOBI RE COMPLEX DENSITY Maple PLOT

參考文獻

• Andrews, George E.; Askey, Richard, Classical orthogonal polynomials, Brezinski, C.; Draux, A.; Magnus, Alphonse P.; Maroni, Pascal; Ronveaux, A. (編), Polynômes orthogonaux et applications. Proceedings of the Laguerre symposium held at Bar-le-Duc, October 15–18, 1984., Lecture Notes in Math. 1171, Berlin, New York: Springer-Verlag: 36–62, 1985, ISBN 978-3-540-16059-5, MR 0838970, doi:10.1007/BFb0076530 
• 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., http://dlmf.nist.gov/18, 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