The Christoffel-Darboux formula applied to Bessel functions states that
$$\sum\limits_{j=0}^{+\infty}J_{j+n}(t)J_{j+m}(t)=\frac{t}{2(m-n)}\left(J_{m-1}(t)J_n(t)-J_m(t)J_{n-1}(t) \right)$$
See for instance Sum of Bessel functions
Is there a similar simplification for the following sum ?
$$\mathcal{I}=\sum\limits_{j=0}^{+\infty}J_{2j+n}(t)J_{2j+m}(t)$$
The sum is taken over all even integers. The best I could find is this series which is close http://functions.wolfram.com/Bessel-TypeFunctions/BesselJ/23/01/0016/ but there is no reference or proof of this identity.