Q拉卡多項式

Q拉卡多項式是以基本超幾何函數定義的多項式：

$p_{n}(q^{-x}+q^{x+1}cd;a,b,c,d;q)={}_{4}\phi _{3}\left[{\begin{matrix}q^{-n}&abq^{n+1}&q^{-x}&q^{x+1}cd\\aq&bdq&cq\\\end{matrix}};q;q\right]$

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：

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

令Q拉卡多項式中$a=b$,$cq=q^{-N}$ 即得雙Q克拉夫楚克多項式

圖集

Q-Racah function abs complex 3D PLOT

Q-Racah function Im complex 3D PLOT

Q-Racah function re density PLOT

Q-Racah function abs density PLOT

Q-Racah function Im density PLOT

Q-Racah function re density PLOT

參考文獻

• Askey, Richard; Wilson, James, A set of orthogonal polynomials that generalize the Racah coefficients or 6-j symbols, SIAM Journal on Mathematical Analysis, 1979, 10 (5): 1008–1016, ISSN 0036-1410, MR 0541097, doi:10.1137/0510092 
• 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