Q查理耶多项式q查理耶多项式是一个以基本超几何函数定义的正交多项式 c n ( x ; a ; q ) = 2 ϕ 1 ( q − n , q − x ; 0 ; q , − q n + 1 / a ) {\displaystyle \displaystyle c_{n}(x;a;q)={}_{2}\phi _{1}(q^{-n},q^{-x};0;q,-q^{n+1}/a)} 极限关系 令Q查理耶多项式 a→a*(1-q),并令q→1,即得查理耶多项式 l i m q → 1 C n ( q − n ; a ( 1 − q ) ; q ) = C n ( x ; a ) {\displaystyle lim_{q\to 1}C_{n}(q^{-n};a(1-q);q)=C_{n}(x;a)} 验证Q查理耶多项式→查理耶多项式Q查理耶多项式之第4项(k=4): ( 1 − q − n ) ( 1 − q − n q ) ( 1 − q − n q 2 ) ( 1 − q − n q 3 ) ( 1 − q − x ) ( 1 − q − x q ) ( 1 − q − x q 2 ) ( 1 − q − x q 3 ) ( q n ) 4 q 4 a 4 ( 1 − q ) 5 ( 1 − q 2 ) ( 1 − q 3 ) ( 1 − q 4 ) {\displaystyle {\frac {\left(1-{q}^{-n}\right)\left(1-{q}^{-n}q\right)\left(1-{q}^{-n}{q}^{2}\right)\left(1-{q}^{-n}{q}^{3}\right)\left(1-{q}^{-x}\right)\left(1-{q}^{-x}q\right)\left(1-{q}^{-x}{q}^{2}\right)\left(1-{q}^{-x}{q}^{3}\right)\left({q}^{n}\right)^{4}{q}^{4}}{{a}^{4}\left(1-q\right)^{5}\left(1-{q}^{2}\right)\left(1-{q}^{3}\right)\left(1-{q}^{4}\right)}}} 展开之: 1 24 36 n x − 66 n x 2 + 36 n x 3 − 6 n x 4 − 66 n 2 x + 121 n 2 x 2 − 66 n 2 x 3 + 11 n 2 x 4 + 36 n 3 x − 66 n 3 x 2 + 36 n 3 x 3 − 6 n 3 x 4 − 6 n 4 x + 11 n 4 x 2 − 6 n 4 x 3 + n 4 x 4 a 4 {\displaystyle {\frac {1}{24}}\,{\frac {36\,nx-66\,n{x}^{2}+36\,n{x}^{3}-6\,n{x}^{4}-66\,{n}^{2}x+121\,{n}^{2}{x}^{2}-66\,{n}^{2}{x}^{3}+11\,{n}^{2}{x}^{4}+36\,{n}^{3}x-66\,{n}^{3}{x}^{2}+36\,{n}^{3}{x}^{3}-6\,{n}^{3}{x}^{4}-6\,{n}^{4}x+11\,{n}^{4}{x}^{2}-6\,{n}^{4}{x}^{3}+{n}^{4}{x}^{4}}{{a}^{4}}}} 另一方面 查理耶多项式的k=4项为 1 24 p o c h h a m m e r ( − n , 4 ) p o c h h a m m e r ( − x , 4 ) a 4 {\displaystyle {\frac {1}{24}}\,{\frac {{\it {pochhammer}}\left(-n,4\right){\it {pochhammer}}\left(-x,4\right)}{{a}^{4}}}} 展开之 1 24 n x ( 36 − 66 x + 36 x 2 − 6 x 3 − 66 n + 121 n x − 66 n x 2 + 11 n x 3 + 36 n 2 − 66 n 2 x + 36 n 2 x 2 − 6 n 2 x 3 − 6 n 3 + 11 n 3 x − 6 n 3 x 2 + n 3 x 3 ) a 4 {\displaystyle {\frac {1}{24}}\,{\frac {nx\left(36-66\,x+36\,{x}^{2}-6\,{x}^{3}-66\,n+121\,nx-66\,n{x}^{2}+11\,n{x}^{3}+36\,{n}^{2}-66\,{n}^{2}x+36\,{n}^{2}{x}^{2}-6\,{n}^{2}{x}^{3}-6\,{n}^{3}+11\,{n}^{3}x-6\,{n}^{3}{x}^{2}+{n}^{3}{x}^{3}\right)}{{a}^{4}}}} 二者显然相等 QED 图集 Q-CHARLIER ABS COMPLEX 3D MAPLE PLOT Q-CHARLIER IM COMPLEX 3D MAPLE PLOT Q-CHARLIER RE COMPLEX 3D MAPLE PLOT Q-CHARLIER ABS DENSITY MAPLE PLOT Q-CHARLIER IM DENSITY MAPLE PLOT Q-CHARLIER 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 |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