罗杰斯-斯泽格多项式维基百科,自由的 encyclopedia 罗杰斯-斯泽格多项式(英语:Rogers–Szegő polynomials)是1926年匈牙利数学家斯泽格首先研究的在单位圆上的正交多项式,以Q阶乘幂定义如下; h n ( x ; q ) = ∑ k = 0 n ( q ; q ) n ( q ; q ) k ( q ; q ) n − k x k {\displaystyle h_{n}(x;q)=\sum _{k=0}^{n}{\frac {(q;q)_{n}}{(q;q)_{k}(q;q)_{n-k}}}x^{k}} Rogers-Szego Polynomials Rogers-Szego Polynomials 前面几个罗杰斯-斯泽格多项式为: h 1 ( x ; q ) = 1 + x {\displaystyle h_{1}(x;q)=1+x} h 2 ( x ; q ) = 1 + ( 1 − q 2 ) ∗ x ( 1 − q ) + x 2 {\displaystyle h_{2}(x;q)=1+{\frac {(1-q^{2})*x}{(1-q)}}+x^{2}} h 3 ( x ; q ) = 1 + ( 1 − q 3 ) ∗ x ( 1 − q ) + ( 1 − q 3 ) ∗ x 2 ( 1 − q ) + x 3 {\displaystyle h_{3}(x;q)=1+{\frac {(1-q^{3})*x}{(1-q)}}+{\frac {(1-q^{3})*x^{2}}{(1-q)}}+x^{3}} h 4 ( x ; q ) = 1 + ( 1 − q 4 ) ∗ x ( 1 − q ) + ( 1 − q 3 ) ∗ ( 1 − q 4 ) ∗ x 2 ( ( 1 − q ) ∗ ( 1 − q 2 ) ) + ( 1 − q 4 ) ∗ x 3 ( 1 − q ) + x 4 {\displaystyle h_{4}(x;q)=1+{\frac {(1-q^{4})*x}{(1-q)}}+{\frac {(1-q^{3})*(1-q^{4})*x^{2}}{((1-q)*(1-q^{2}))}}+{\frac {(1-q^{4})*x^{3}}{(1-q)}}+x^{4}} h 5 ( x ; q ) = 1 + ( 1 − q 5 ) ∗ x ( 1 − q ) + ( 1 − q 4 ) ∗ ( 1 − q 5 ) ∗ x 2 ( ( 1 − q ) ∗ ( 1 − q 2 ) ) + ( 1 − q 4 ) ∗ ( 1 − q 5 ) ∗ x 3 ( ( 1 − q ) ∗ ( 1 − q 2 ) ) + ( 1 − q 5 ) ∗ x 4 ( 1 − q ) + x 5 {\displaystyle h_{5}(x;q)=1+{\frac {(1-q^{5})*x}{(1-q)}}+{\frac {(1-q^{4})*(1-q^{5})*x^{2}}{((1-q)*(1-q^{2}))}}+{\frac {(1-q^{4})*(1-q^{5})*x^{3}}{((1-q)*(1-q^{2}))}}+{\frac {(1-q^{5})*x^{4}}{(1-q)}}+x^{5}}
罗杰斯-斯泽格多项式(英语:Rogers–Szegő polynomials)是1926年匈牙利数学家斯泽格首先研究的在单位圆上的正交多项式,以Q阶乘幂定义如下; h n ( x ; q ) = ∑ k = 0 n ( q ; q ) n ( q ; q ) k ( q ; q ) n − k x k {\displaystyle h_{n}(x;q)=\sum _{k=0}^{n}{\frac {(q;q)_{n}}{(q;q)_{k}(q;q)_{n-k}}}x^{k}} Rogers-Szego Polynomials Rogers-Szego Polynomials 前面几个罗杰斯-斯泽格多项式为: h 1 ( x ; q ) = 1 + x {\displaystyle h_{1}(x;q)=1+x} h 2 ( x ; q ) = 1 + ( 1 − q 2 ) ∗ x ( 1 − q ) + x 2 {\displaystyle h_{2}(x;q)=1+{\frac {(1-q^{2})*x}{(1-q)}}+x^{2}} h 3 ( x ; q ) = 1 + ( 1 − q 3 ) ∗ x ( 1 − q ) + ( 1 − q 3 ) ∗ x 2 ( 1 − q ) + x 3 {\displaystyle h_{3}(x;q)=1+{\frac {(1-q^{3})*x}{(1-q)}}+{\frac {(1-q^{3})*x^{2}}{(1-q)}}+x^{3}} h 4 ( x ; q ) = 1 + ( 1 − q 4 ) ∗ x ( 1 − q ) + ( 1 − q 3 ) ∗ ( 1 − q 4 ) ∗ x 2 ( ( 1 − q ) ∗ ( 1 − q 2 ) ) + ( 1 − q 4 ) ∗ x 3 ( 1 − q ) + x 4 {\displaystyle h_{4}(x;q)=1+{\frac {(1-q^{4})*x}{(1-q)}}+{\frac {(1-q^{3})*(1-q^{4})*x^{2}}{((1-q)*(1-q^{2}))}}+{\frac {(1-q^{4})*x^{3}}{(1-q)}}+x^{4}} h 5 ( x ; q ) = 1 + ( 1 − q 5 ) ∗ x ( 1 − q ) + ( 1 − q 4 ) ∗ ( 1 − q 5 ) ∗ x 2 ( ( 1 − q ) ∗ ( 1 − q 2 ) ) + ( 1 − q 4 ) ∗ ( 1 − q 5 ) ∗ x 3 ( ( 1 − q ) ∗ ( 1 − q 2 ) ) + ( 1 − q 5 ) ∗ x 4 ( 1 − q ) + x 5 {\displaystyle h_{5}(x;q)=1+{\frac {(1-q^{5})*x}{(1-q)}}+{\frac {(1-q^{4})*(1-q^{5})*x^{2}}{((1-q)*(1-q^{2}))}}+{\frac {(1-q^{4})*(1-q^{5})*x^{3}}{((1-q)*(1-q^{2}))}}+{\frac {(1-q^{5})*x^{4}}{(1-q)}}+x^{5}}