Q梅西纳多项式

Q梅西纳多项式以基本超几何函数定义如下：

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

极限关系

Q梅西纳多项式→Q拉盖尔多项式;

令Q梅西纳多项式中$b=q^{\alpha }$,以及$q^{-x}=cq^{\alpha }x$,然后取$c\to \infty$即得Q拉盖尔多项式。

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

图集

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

参考文献

• 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. (编), NIST Handbook of Mathematical Functions, Cambridge University Press, 2010, ISBN 978-0521192255, MR2723248