雙q哈恩多項式

雙q哈恩多項式是一個以基本超幾何函數定義的正交多項式

$R_{n}(\mu (x);\delta ,\gamma ,N|q)=_{3}\Phi _{2}(q^{-n},q^{-x},\gamma \delta {q}^{x+1};\gamma {q},q_{N};q,q)$

$n=1,2,3\cdots N$

極限關係

Q拉卡多項式→雙q哈恩多項式；

在Q拉卡多項式中令$\beta =0$以及$\alpha {q}=q^{-N}$,即得雙q哈恩多項式：

$R_{n}(\mu (x);q^{-N-1},0,\gamma ,\delta |q)=R_{n}(\mu (x);\gamma ,\delta ,N|q)$

雙q哈恩多項式→仿射克拉夫楚克多項式

在雙q哈恩多項式中令$\gamma =p$以及$\delta =0$即得仿克拉夫楚克多項式。

圖集

DUAL QHAHN ABS COMPLEX3D MAPLE PLOT

DUAL QHAHN IM COMPLEX3D MAPLE PLOT

DUAL QHAHN RE COMPLEX3D MAPLE PLOT

DUAL QHAHN ABS DENSITY MAPLE PLOT

DUAL QHAHN IM DENSITY MAPLE PLOT

DUAL QHAHN 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