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
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{\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
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{\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):
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{\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:
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{\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
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{\displaystyle {\frac {1}{24}}\,{\frac {{\it {pochhammer}}\left(-n,4\right){\it {pochhammer}}\left(-x,4\right)}{{a}^{4}}}}
展开之
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{\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 , 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 , MR 2723248