Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 302 / 2 / b / Solution

With normalized states and $J_{-}^{†}=J_{+}$,Put $n=I-m$. Iterating from the highest-weight state givesHencewhere the factorial arguments are integers because $I\pm m\in {\mathbb{Z}}_{\ge 0}$.