Q梅西納多項式

Q梅西納多項式以基本超幾何函數定義如下：

$M_{n}(q^{-x};b,c;q)=_{2}\Phi _{1}(q^{-n},q^{-x};bq;q;-q^{n+1}/c)$

極限關係

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

QMEIXNER POLYNOMIALS ABS COMPLEX 3D MAPLE PLOT

QMEIXNER POLYNOMIALS IM COMPLEX 3D MAPLE PLOT

QMEIXNER POLYNOMIALS RE COMPLEX 3D MAPLE PLOT

QMEIXNER POLYNOMIALS ABS DENSITY MAPLE PLOT

QMEIXNER POLYNOMIALS IM DENSITY MAPLE PLOT

QMEIXNER 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