Triple correlations of Fourier coefficients of cusp forms

Abstract: We treat an unbalanced shifted convolution sum of Fourier coefficients of cusp forms. As a consequence, we obtain an upper bound for correlation of three Hecke eigenvalues of holomorphic cusp forms $\sum_{H\leq h\leq 2H}W\big(\frac{h}{H}\big)\sum_{X\leq n\leq 2X}\lambda_{1}(n-h)\lambda_{2}(n)\lambda_{3}(n+h)$, which is nontrivial provided that $H\geq X^{2/3+\varepsilon}$. The result can be viewed as a cuspidal analogue of a recent result of Blomer on triple correlations of divisor functions.