We prove a generalization of the q-Selberg integral evaluation formula. The integrand is that multiplied by factor same form with respect to part variables. proof relies on quadratic norm formula Koornwinder polynomials. also derive generalizations Mehta's as limit cases our integral.

The notion of a tower of Rankin-Selberg integrals was introduced in [G-R]. To recall this notion, let G be a reductive group defined over a global field F . Let G denote the L group of G. Let ρ denote a finite dimensional irreducible representation of G. Given an irreducible generic cuspidal representation of G(A), we let L(π, ρ, s) denote the partial L function associated with π and ρ. Here s ...

Duke and Kowalski in [6] derive a large sieve inequality for automorphic forms on GL(n) via the Rankin-Selberg method. We give here a partial complement to this result: using some explicit geometry of fundamental regions, we prove a large sieve inequality yielding sharp results in a region distinct to that in [6]. As an application, we give a generalization to GL(n) of Duke’s multiplicity theor...

appear (the uj run over an orthonormal basis of automorphic Laplaceeigenforms), so our formula (Theorem 1) is a duality between such integrals and certain geodesic integrals of u. New integral transformations are involved depending on the Laplace-eigenvalue of u. We invert these integral transformations in Section 5, Theorem 2. We develop the formula for finite volume Fuchsian groups, so (as in...