In the context of (numerically) calculating reduced density matrices in the Lipkin-Meshkov-Glick model (a model introduced to describe atomic nuclei, which has however found many other applications as well), I need to carry out a sum over Clebsch-Gordan coefficients.

If it is possible to find a simplification that replaces the sum below by a (simple) explicit expression, the numerical effort would be greatly reduced (the angular momenta in question are large, therefore the sum over $m_1$ has many terms). The sum in question is:

$\sum_{m_1} \langle J M|j_1 m_1 j_2 m_2\rangle \langle j_1 m_1 j_2 m_2'|J M'\rangle$

I have found sum formulas for Clebsch-Gordan coefficients, but none applicable to this one. Any insight would therefore be greatly appreciated!