Spectral Moment Formulae for $GL(3)\times GL(2)$ $L$-functions I: The Cuspidal Case

Spectral moment formulae of various shapes have proven to be very successful in studying the statistics of central $L$-values. In this article, we establish, in a completely explicit fashion, such formulae for the family of $GL(3)\times GL(2)$ Rankin-Selberg $L$-functions using the period integral method. The Kuznetsov and the Voronoi formulae are not needed in our argument. We also prove the essential analytic properties and explicit formulae for the integral transform of our moment formulae. It is hoped that our method will provide insights into moments of $L$-functions for higher-rank groups.