Sorry if this is a vague question. I remember from my younger days that before proving his trace formula, Kuznetsov had a pretty result on orthogonality of Bessel functions. The formulas that I am going to write are WRONG, but the whole point is that I would like to remember the correct one (or a pointer to a reference).

If $J_{\nu}$ denotes the $\nu$th $J$-Bessel function, then if $n$ and $m$ are positive integers, then $<J_n,J_m>=0$ when $n\ne m$ for the scalar product $\int_0^\infty f(x)g(x)dx/x$ (I know this is wrong, let me continue). Thus, as for Fourier series, one can think of expanding reasonable functions into linear combinations of $J_n$ by computing the scalar product to get the coefficients. This does not work because the $J_n$ are not complete, contrary to the $e^{inx}$. Thus, in addition, Kuznetsov adds the "continuous spectrum" functions which are $J_{i\nu}-J_{-i\nu}$ for $\nu\in\Bbb R$, and shows that these are orthonormal, orthogonal to the $J_n$, and now form a complete set. I know that this is very weak form of the Selberg trace formula, but could someone give me the correct formula ?