仿射q克拉夫楚克多项式

仿射q克拉夫楚克多项式是以基本超几何函数定义的正交多项式[1]

${\displaystyle K_{n}^{aff}(q^{-x};p;N;q)=\;_{2}\phi _{1}\left({\begin{matrix}q^{-n}&0&q^{-x}\\pq&q^{-N}\end{matrix}};q,q\right)}$

${\displaystyle n=0,1,2,\cdots N}$

极限关系

Q哈恩多项式→ 量子Q克拉夫楚克多项式：

${\displaystyle \lim _{a\to \infty }Q_{n}(q^{-}{x};a;p,N|q)=K_{n}^{qtm}(q^{-}{x};p,N;q)}$

仿射q克拉夫楚克多项式→ 小q拉盖尔多项式：

${\displaystyle \lim _{a\to 1}=K_{n}^{aff}(q^{x-N};p,N|q)=p_{n}(q^{x};p,q)}$

图集

AFFINE Q-KRAWTCHOUK POLYNOMIALS ABS COMPLEX 3D MAPLE PLOT

AFFINE Q-KRAWTCHOUK POLYNOMIALS IM COMPLEX 3D MAPLE PLOT

AFFINE Q-KRAWTCHOUK POLYNOMIALS RE COMPLEX 3D MAPLE PLOT

AFFINE Q-KRAWTCHOUK POLYNOMIALS ABS DENSITY MAPLE PLOT

AFFINE Q-KRAWTCHOUK POLYNOMIALS IM DENSITY MAPLE PLOT

AFFINE Q-KRAWTCHOUK POLYNOMIALS RE DENSITY MAPLE PLOT

参考文献

1. Roelof Koekoek, Hypergeometric Orthogonal Polynomials and its q-Analogues, p501,Springer,2010

• 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 |contribution-url=缺少标题 (帮助), 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 
• Stanton, Dennis, Three addition theorems for some q-Krawtchouk polynomials, Geometriae Dedicata, 1981, 10 (1): 403–425, ISSN 0046-5755, MR 0608153, doi:10.1007/BF01447435