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 Γ=SL(2,Z). Denote by λ_symʲf(n) the nth normalized coefficient of the Dirichlet expansion of the jth symmetric power L-function L(s,symʲf). In this paper, we are interested in the behavior of the shifted convolution sum involving λ_symʲf(n) with a weight function to be the k-full kernel function for any fixed integer k≥ 2.