Products

# Double Product Formulae

(1)\begin{align} \prod_{i=q}^l \prod_{j=r}^m ij = \left(\prod_{i=q}^l i \right)^{(m-r+1)} \left(\prod_{j=r}^m j\right)^{(l-q+1)} \end{align}

(2)
\begin{align} \prod_{i=q}^l \prod_{j=r}^m i^j = \prod_{i=q}^l i ^{\sum_{j=r}^m j} = \prod_{i=q}^l i^{\frac{m(m+1) - r(r-1)}{2}} \end{align}

(3)
\begin{align} \prod_{i=q}^l \prod_{j=r}^m a^{ij} = \prod_{i=q}^l a^{(m-r+1)i \cdot \sum_{j=r}^m j} = \prod_{i=q}^l a^{\frac{(m-r+1)i \cdot [m(m+1) - r(r-1)]}{2}} \end{align}