Just solve for the sum in the addition formula (31) of Neumann 1867, p. 65 (also in Watson, p. 128): $$ J_0\left(2r\sqrt{\frac{1-\cos\theta}2}\right) = J_0(r)^2 + 2\sum_{n=1}^\infty J_n(r)^2\cos(n\theta). $$
Francois Ziegler
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