Certain series attached to an even number of elliptic modular forms (Q597130)

Let \(f_j(z)\) and \(g_j(z)\) (\(j=1,2,\dots,n\)) be holomorphic modular forms for \(\text{SL}_2(\mathbb{Z})\) such that \(f_j(z)g_j(z)\) is a cusp form. Here, the author defines a certain series \({\mathcal D}(s; f_1,\dots,f_n;g_1,\dots,g_n)\) whose terms are expressed by the Fourier coefficients of the above modular forms and the Mellin transform of the product of modified Bessel functions. He proves the meromorphic continuation and the functional equations of the above series via a Rankin-Selberg-type integral involving a pullback of a certain Eisenstein series for the Siegel modular group \(\text{Sp}_n (\mathbb{Z})\) of degree \(n\). Note that this Eisenstein series is not for the Siegel parabolic subgroup but for a Jacobi parabolic subgroup of \(\text{Sp}_n(\mathbb{Z})\).