Krattenthaler's Identity

Theorem

 * $\begin{vmatrix}

\left({x + q_2}\right) \left({x + q_3}\right) & \left({x + p_1}\right) \left({x + q_3}\right) & \left({x + p_1}\right) \left({x + p_2}\right) \\ \left({y + q_2}\right) \left({y + q_3}\right) & \left({y + p_1}\right) \left({y + q_3}\right) & \left({y + p_1}\right) \left({y + p_2}\right) \\ \left({z + q_2}\right) \left({z + q_3}\right) & \left({z + p_1}\right) \left({z + q_3}\right) & \left({z + p_1}\right) \left({z + p_2}\right) \end{vmatrix} = \left({x - y}\right) \left({x - z}\right) \left({y - z}\right) \left({p_1 - q_2}\right) \left({p_1 - q_3}\right) \left({p_2 - q_3}\right)$

where $\left\vert{\, \cdot \,}\right\vert$ denotes determinant.