羅傑斯-斯澤格多項式

Rogers-Szego Polynomials

Rogers-Szego Polynomials

羅傑斯-斯澤格多項式（英語：Rogers–Szegő polynomials）是1926年匈牙利數學家斯澤格首先研究的在單位圓上的正交多項式，以Q階乘冪定義如下；

$h_{n}(x;q)=\sum _{k=0}^{n}{\frac {(q;q)_{n}}{(q;q)_{k}(q;q)_{n-k}}}x^{k}$

前面幾個羅傑斯-斯澤格多項式為：

$h_{1}(x;q)=1+x$

$h_{2}(x;q)=1+{\frac {(1-q^{2})*x}{(1-q)}}+x^{2}$

$h_{3}(x;q)=1+{\frac {(1-q^{3})*x}{(1-q)}}+{\frac {(1-q^{3})*x^{2}}{(1-q)}}+x^{3}$

$h_{4}(x;q)=1+{\frac {(1-q^{4})*x}{(1-q)}}+{\frac {(1-q^{3})*(1-q^{4})*x^{2}}{((1-q)*(1-q^{2}))}}+{\frac {(1-q^{4})*x^{3}}{(1-q)}}+x^{4}$

$h_{5}(x;q)=1+{\frac {(1-q^{5})*x}{(1-q)}}+{\frac {(1-q^{4})*(1-q^{5})*x^{2}}{((1-q)*(1-q^{2}))}}+{\frac {(1-q^{4})*(1-q^{5})*x^{3}}{((1-q)*(1-q^{2}))}}+{\frac {(1-q^{5})*x^{4}}{(1-q)}}+x^{5}$

參考文獻

• 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 
• Szegő, Gábor, Beitrag zur theorie der thetafunktionen, Sitz Preuss. Akad. Wiss. Phys. Math. Ki., 1926, XIX: 242–252, Reprinted in his collected papers