q查理耶多項式是一個以基本超幾何函數定義的正交多項式
令Q查理耶多項式 a→a*(1-q),並令q→1，即得查理耶多項式
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)}
Q查理耶多項式之第4項(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)}}} 展開之： 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 查理耶多項式的k=4項為
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}}}}
展開之
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}}}}
二者顯然相等 QED
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展開之：
