大q拉蓋爾多項式

大q-拉蓋爾多項式是一個以基本超幾何函數定義的正交多項式:

$P_{n}(x;a,b;q)={\frac {1}{(b^{-1}*q^{-n};q,n)}}*_{2}\Phi _{1}(q^{-n},aqx^{-1};aq|q;{\frac {x}{b}})$

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

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

大q拉蓋爾多項式→小q拉蓋爾多項式

在大q拉蓋爾多項式中，令$x\to bqx$,並令$b\to \infty$即得小q拉蓋爾多項式 $\lim _{b\to \infty }P_{n}(bqx;a,b;q)=p_{n}(x;a|q)$

大→q拉蓋爾多項式→阿爾-薩拉姆-卡里茲多項式 I

圖集

BIG Q-LAGUERRE POLYNOMIALS ABS COMPLEX 3D MAPLE PLOT

BIG Q-LAGUERRE POLYNOMIALS IM COMPLEX 3D MAPLE PLOT

BIG Q-LAGUERRE POLYNOMIALS RE COMPLEX 3D MAPLE PLOT

BIG Q-LAGUERRE POLYNOMIALS ABS DENSITY MAPLE PLOT

BIG Q-LAGUERRE POLYNOMIALS IM DENSITY MAPLE PLOT

BIG Q-LAGUERRE POLYNOMIALS RE DENSITY 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., 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