On a general divisor problem associated to coefficients of Rankin–Selberg L-functions over a sparse set of sequences of positive integers

Let f and g be two distinct normalized primitive holomorphic cusp forms of even integral weights \(k_{1}\) $k_1$ and \(k_{2}\) $k_2$ for the full modular group \(\Gamma =SL(2,{\mathbb {Z}})\) $\mathrm{\Gamma }=SL(2,\mathbb{Z})$ , respectively. In this paper, we establish the asymptotic formula for the higher power moments of general divisor problem associated to the coefficients of Rankin–Selberg L-function \(L(f\times g,s)\) $L(f\times g,s)$ , supported at the sequence of positive integers represented by primitive integral positive definite reduced binary quadratic forms of a fixed discriminant \(D<0\) $D<0$ . By analogy, we also investigate the similar problem for the coefficients of a certain Rankin–Selberg L-function.