Let $C^{j\_3 m\_3}\_{j\_1 m\_1 j\_2 m\_2}$ be the standard Clebsch–Gordan coefficients of $\operatorname{SU}(2)$. They obey the orthogonality relation
$$ \sum\_{j\_3} \sum\_{m\_3} \left(C^{j\_3 m\_3}\_{j\_1 m\_1 j\_2 (m\_3 - m\_1)} \right)^2 = 1.$$
My question is about what can be said if I remove the sum over $j\_3$. Does there exist a bound for
$$ \max\_{j\_3} \sum\_{m\_3} \left(C^{j\_3 m\_3}\_{j\_1 m\_1 j\_2 (m\_3 - m\_1)} \right)^2 $$
in terms of $m\_1$, $j\_1$ and $j\_2$? Of course it is $\leq 1$, but I am interested in whether this expression decays in $j\_1$ and $j\_2$, or some combination thereof. Positive or negative statements, or a reference, would be very useful.



Single sum of squares of Clebsch–Gordan coefficients
Consider the sum of [Clebsch–Gordan coefficients](https://en.wikipedia.org/wiki/Clebsch%E2%80%93Gordan_coefficients),$^\ast$
$$J= \sum\_{m\_3=-j\_3}^{j\_3} \left(C^{j\_3,m\_3}\_{j\_1, m\_1; j\_2, (m\_3 - m\_1)} \right)^2$$
with $2j\_1,2j\_2,2j\_3\in\mathbb{N}$ and $2m\_1\in\mathbb{Z}$. For an nonvanishing sum we also need $j\_3\in\{|j\_1-j\_2|,|j\_1-j\_2|+1,\ldots j\_1+j\_2-1, j\_1+j\_2\}$ and $m\_1\in\{-j\_1,-j\_1+1,\ldots,j\_1-1, j\_1\}$.  

With these restrictions the sum is independent of $m\_1$, so I may set $m\_1=j\_1$. The sum also increases monotonically with increasing $j\_3$, reaching its maximal value $J\_{\rm max}$ for $j\_3=j\_1+j\_2$, hence
$$J\_{\rm max}=\sum\_{m=-j\_2}^{j\_2} \left(C^{j\_1+j\_2,j\_1+m}\_{j\_1, j\_1; j\_2, m} \right)^2=\sum\_{m=-j\_2}^{j\_2}\frac{\Gamma (2 j\_2+1) \Gamma (2 j\_1+j\_2+m+1)}{(1+2j\_2)\Gamma (2 j\_1+2 j\_2+1) \Gamma (j\_2+m+1)}$$
$$\qquad=\frac{1+2j\_1+2j\_2}{(1+2j\_1)(1+2j\_2)}.$$




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$^\ast$ Mathematica normalizes these coefficients such that $\sum\_{j\_3}J=1+2j\_2$. Here I use the normalization of the OP, where $\sum\_{j\_3}J=1$.
