Let \( f \) be a normalized Hecke eigenform of even weight \( k \) for \( SL _2(\mathbb {Z}) \) . For a fixed integer \( \ell \ge 1 \) , we derive asymptotic formulae for generalized divisor sums involving the coefficients of an \(\ell \) -fold product \(L\) -function associated with \( f \) over sparse sequences arising from generalized Eisenstein series. We emphasize that the \(\ell \) -fold product L-function is not assumed to be automorphic; rather, our analysis relies on its Euler product structure together with available results on symmetric power L-functions. Additionally, we analyze shifted convolution sums related to these divisor sums and extend the results to the coefficients of Maass cusp forms under suitable assumptions.
