# Q-series 의 공식 모음

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## 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

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

## 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