On certain kernel functions and shifted convolution sums of Hecke eigenvalues

Let j 2 be a given integer. Let f be a normalized primitive holomorphic cusp form of even integral weight for the full modular group =SL (2, Z). Denote by ₒₘ₌^₉f (n) the nth normalized coefficient of the Dirichlet expansion of the jth symmetric power L-function L (s, sym^jf). In this paper, we are interested in the behavior of the shifted convolution sum involving ₒₘ₌^₉f (n) with a weight function to be the k-full kernel function for any fixed integer k 2.