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
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_{qto 1}C_{n}(q^{-n};a(1-q);q)=C_{n}(x;a)}
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}qright)left(1-{q}^{-n}{q}^{2}right)left(1-{q}^{-n}{q}^{3}right)left(1-{q}^{-x}right)left(1-{q}^{-x}qright)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-qright)^{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,4right){it {pochhammer}}left(-x,4right)}{{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 {nxleft(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
展开之：
