Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2016 / Paper 1 / 18 / b / ii / Solution

Apply the recurrence repeatedly down to $m=2$:Including the base factor $1/(n-1)$ and rewriting the products as factorials givesThe formula also includes $m=2$ because $0!=1$. It is symmetric in $m,n$, consistent with the integration by substitution $x=1/t$. Equivalently it is the Euler beta function $B(m-1,n-1)$, although that identification is not needed for the recurrence derivation.