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 ?



Kuznetsov trace formula, orthogonality of Bessel functions
The Bessel functions $J\_\ell$ for $\ell\geq 1$ odd are pairwise orthogonal on the positive axis with respect to the measure $dx/x$. They correspond to the holomorphic spectrum (of various even weights $\ell+1$) of $L^2(\Gamma\backslash H)$. The orthogonal complement of the span of these $J\_\ell$'s is continuously (and orthogonally) spanned by the functions $J\_{2it}-J\_{-2it}$ with $t>0$. This corresponds to the (weight zero and tempered) Maass and Eisenstein spectrum of $L^2(\Gamma\backslash H)$ (of various Laplace eigenvalues $1/4+t^2$). For more details, I recommend Sections 9.3-9.4 in Iwaniec: Introduction to the spectral theory of automorphic forms.


The Bruggeman-Kuznetsov formula is not a weak form of the Selberg trace formula, in fact in many situations it is a more refined (or more suitable) tool than the Selberg trace formula. It can be interpreted as a relative trace formula, see e.g. [the paper of Knightly and Li](http://doi.org/10.4064/aa122-3-5) in Acta Arithmetica.


