Mellin Transforms attached to certain utomorphic integrals

Abstract

Let k be any real number with k < 2. We will consider complexvalued smooth functions f, <(f)over tilde> on H of period 1, having exponential decay at infinity (i.e. they are << e(-cy) for y = I(z) -> infinity with c> 0) and such that f |(k) W(N) = C (f) over tilde + q(g). Here |(k) is an appropriately defined Petersson slash operator in weight k, C is an element of C* is a constant and q(g)(z) := integral(i infinity)(0) g(tau)(tau - (z) over bar)(-k) d tau (z is an element of H) is a period integral attached to a holomorphic function g : H -> C such that both g and g|(2-k)W(N) have period 1, have only positive terms in their Fourier expansions and the Fourier coefficients are of polynomial growth. An arbitrary power of a non-zero complex number is defined by means of the principal branch of the complex logarithm. Under the assumption that k < 1, we will show that the Mellin transform M(f, s) (sigma >> 1) naturally attached to I has meromorphic continuation to C and we will establish an explicit formula for it (Section 2, Theorem 1). There are possible simple poles at the points s = -n where n = 0, 1, 2, ... and the residue at s = -n essentially is equal to the "n-th period" integral(infinity)(0) g(it)t(n) dt of g. Moreover, there again is a functional equation relating m(f, s) and M((f) over tilde, k - s). (C) 2011 Elsevier Inc. All rights reserved.