Q拉蓋爾多項式

q拉蓋爾多項式是一個以基本超幾何函數和Q階乘冪定義的正交多項式

q-Laguerre Polynomials

$\displaystyle L_{n}^{(\alpha )}(x;q)={\frac {(q^{\alpha +1};q)_{n}}{(q;q)_{n}}}{}_{1}\phi _{1}(q^{-n};q^{\alpha +1};q,-q^{n+\alpha +1}x)$

正交性

Q-拉蓋爾多項式滿足下列正交關係

$\int _{a}^{b}\!L_{n}^{(\alpha )}(x;q)*L_{m}^{(\alpha )}(x;q)*(x^{\alpha })/(-x;q)_{\infty }\,dx={\frac {(q^{\alpha }+1;q)_{n}}{(q;q)_{n}*q^{n}}}$

極限關係

小q雅可比多項式→Q拉蓋爾多項式.

在校q雅可比多項式的定義中，令$a=q^{\alpha }$以及$x\to -b^{-1}q^{-1}$,並令$b\to \infty$,即得q拉蓋爾多項式。

Q梅西納多項式→Q拉蓋爾多項式;

令Q梅西納多項式中$b=q^{\alpha }$,以及$q^{-x}=cq^{\alpha }x$,然後取$c\to \infty$即得Q拉蓋爾多項式。

$\lim _{c\to \infty }M_{n}(cq^{\alpha }x;q^{\alpha },c;q)={\frac {(q;q)_{n}}{(q^{\alpha +1};q)_{n}}}$

圖集

下列 :$\displaystyle L_{5}^{(3.7)}(x+iy;q)$圖，以q 為可變參數。

Q-LAGUERRE POLYNOMIALS ABS COMPLEX 3D MAPLE PLOT

Q-LAGUERRE POLYNOMIALS IM COMPLEX 3D MAPLE PLOT

Q-LAGUERRE POLYNOMIALS RE COMPLEX 3D MAPLE PLOT

Q-LAGUERRE POLYNOMIALS ABS DENSITY MAPLE PLOT

Q-LAGUERRE POLYNOMIALS RE DENSITY MAPLE PLOT

Q-LAGUERRE POLYNOMIALS IM 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 
• Moak, Daniel S., The q-analogue of the Laguerre polynomials, J. Math. Anal. Appl., 1981, 81 (1): 20–47, MR 0618759, doi:10.1016/0022-247X(81)90048-2