大q雅可比多项式

大q-雅可比多项式（英语：Big q-Jacobi polynomials）是一个以基本超几何函数定义的正交多项式^[1]：

BIG Q JACOBI 2D Maple PLOT

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

正交性

大q-雅可比多项式满足下列正交关系

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

极限关系

大q雅可比多项式→大q拉盖尔多项式

令大q雅可比多项式中的${\displaystyle b=0}$,即得大q拉盖尔多项式

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

图集

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

参考文献

• Andrews, George E.; Askey, Richard, Classical orthogonal polynomials, Brezinski, C.; Draux, A.; Magnus, Alphonse P.; Maroni, Pascal; Ronveaux, A. (编), 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 |contribution-url=缺少标题 (帮助), Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (编), NIST Handbook of Mathematical Functions, Cambridge University Press, 2010, ISBN 978-0521192255, MR2723248

1. Roelof p438