Certain series attached to an even number of elliptic modular forms

For j=1,…,n let fj(z) and gj(z) be holomorphic modular forms for SL2(Z) such that fj(z)gj(z) is a cusp form. We define a series whose terms are expressed by the Fourier coefficients of the above modular forms and the Mellin transform of the product of modified Bessel functions. We prove the meromorphic continuation and the functional equations of the series via a Rankin–Selberg-type integral involving a pullback of a certain Eisenstein series for the Siegel modular group of degree n.