Note on the Kuznetsov sum formula

In the theory of automorphic L-functions, the Kuznetsov trace formula is one of the highlights. There are a few different statements of the formula. The most comprehensive one for a Fuchsian group of the first kind is given in Iwaniec's book. In Motohashi's book, the case of the full modular group is treated and their results look different. In the companion paper by Ma and Agarwal, it is shown on the basis of modular relation principle that the statement of Motohashi is another version of Theorem 2.4 and that it has different outlook lacking the Neumann series. We slightly generalized the method of Selberg adopted by Motohashi of equating the two different expressions for the inner product of two Poincare series to deduce Theorem 2. 4 below? which is the reversed form according to Iwaniec but it is the form stated as the Kuznetsov trace formula in most of the literature. In this note we shall elaborate the proof in the Ma and Agarwal's paper using more familiar special functions and give proofs of intermediate formulas involving Bessel functions. This makes the situation more transparent surrouding the Kuznetsov trace formula and make it more accessible.