Higher power moments of shifted convolutions of Fourier coefficients

Let [Formula: see text] be the normalized Fourier coefficient of [Formula: see text], where [Formula: see text] is a primitive holomorphic cusp form of even integral weight [Formula: see text] for the full modular group. Let [Formula: see text] be the square-full kernel function. In this paper, we will study the asymptotic behaviour of the sums [Formula: see text] Instead of the Perron formula to deal with error term, we mainly use the individual and averaged convexity bounds of [Formula: see text]-functions in the critical strip, the functional equation of [Formula: see text]-functions, and large sieve inequalities of general [Formula: see text]-functions to deal with the error term of the asymptotic formula. Therefore we can provide a very good error term. Particularly, for [Formula: see text], we will improve the result of Lü and Wang [10], the error term from [Formula: see text] to [Formula: see text]. For [Formula: see text], we will improve the error term results of Hua [6] from [Formula: see text] to [Formula: see text].