Q-series 의 공식 모음

개요

• 합공식의 q-analogue

여러가지 공식

\[\lim_{z\to\infty}\frac{(z)_{n}}{z^{n}}=(-1)^{n}q^{\frac{n(n-1)}{2}}\]

\[(q^{l+1};q)_{n}=\frac{(q;q)_{n+l}}{(q;q)_{l}}\] or \[(q^{l};q)_{n}=\frac{(q;q)_{n+l-1}}{(q;q)_{l-1}}\]

\[l\geq n,\quad (q^{-l};q)_{n}=(-1)^nq^{n(n-1)/2-nl}(q^{l-n+1};q)_n=(-1)^nq^{n(n-1)/2-nl}\frac{(q;q)_{l}}{(q;q)_{l-n}}\]

\[(-q)_n=(-q;q)_{n}=\frac{(q^2;q^2)_{n}}{(q;q)_{n}}\]

\[(-q;q)_{2n+1}=(-q)_{2n}(1+q^{2n+1})=\frac{(q^2;q^4)_{n}(q^4;q^4)_{n}}{(q;q^2)_{n}(q^2;q^2)_{n}}(1+q^{2n+1})\]

\[(q)_{2n}=(q;q^2)_{n}(q^2;q^2)_{n}\]

\[(-q)_{2n}=\frac{(q^2;q^2)_{2n}}{(q;q)_{2n}}=\frac{(q^2;q^4)_{n}(q^4;q^4)_{n}}{(q;q^2)_{n}(q^2;q^2)_{n}}\]

\[\frac{(-q)_{n}}{(q)_{2n}}=\frac{1}{(q;q^2)_{n}(q;q)_{n}}\]

\[(a)_{n+r}=(a)_{n}(aq^{n})_{r}\]

\[(-q;q^{2})_{n}=\frac{(-q;q)_{2n}}{(-q^{2};q^{2})_{n}}=\frac{(q^{2};q^{2})_{2n}(q^{2};q^{2})_{n}}{(q^{4};q^{4})_{n}(q;q)_{2n}}=\frac{(q^{2};q^{4})_{n}(q^{2};q^{2})_{n}}{(q;q)_{2n}}\]

\[(-q^2;q^{2})_{n}=\frac{(q^4;q^4)_{n}}{(q^2;q^2)_{n}}=\frac{1}{(q^2;q^4)_{n}}\]

\[W(q)=(-q)_{\infty}=\sum_{n\geq 0}\frac{q^{n(n+1)/2}}{(q)_{n}}=\frac{(q^{2};q^{2})_{\infty}}{(q;q)_{\infty}}\]

q-이항정리

• 가우스 공식

\[\prod_{r=0}^{n-1}(1+zq^r)=(1+z)(1+zq)\cdots(1+zq^{n-1})= \sum_{r=0}^{n} \begin{bmatrix} n\\ r\end{bmatrix}_{q}q^{r(r-1)/2}z^r\]

• 하이네 공식

\[\prod_{r=0}^{n-1}\frac{1}{1-zq^r}=\sum_{r=0}^{\infty} \begin{bmatrix} n+r-1\\ r\end{bmatrix}_{q} z^r\]

• q-이항정리 항목 참조

무한곱 공식

• 자코비 삼중곱(Jacobi triple product)\[\sum_{n=-\infty}^\infty z^{n}q^{n^2}= \prod_{m=1}^\infty \left( 1 - q^{2m}\right) \left( 1 + zq^{2m-1}\right) \left( 1 + z^{-1}q^{2m-1}\right)\]
• quintuple product identity

메모

• Math Overflow http://mathoverflow.net/search?q=

매스매티카 파일 및 계산 리소스

• https://docs.google.com/file/d/0B8XXo8Tve1cxNllHMDQyQzdlN1k/edit