Mellin transforms attached to certain automorphic integrals

Let k be any real number with k<2. We will consider complex-valued smooth functions f,f˜ on H of period 1, having exponential decay at infinity (i.e. they are ≪e−cy for y=ℑ(z)→∞ with c>0) and such that f|kWN=Cf˜+qg. Here |k is an appropriately defined Petersson slash operator in weight k, C∈C⁎ is a constant andqg(z):=∫0i∞g(τ)(τ−z¯)−kdτ¯(z∈H) is a period integral attached to a holomorphic function g:H→C such that both g and g|2−kWN 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) (σ≫1) naturally attached to f 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”∫0∞g(it)tndt of g. Moreover, there again is a functional equation relating M(f,s) and M(f˜,k−s).