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Let Q (x) be a positive definite integral quadratic form with the determinant D being squarefree, and r (n , Q) denote the number of representations of n by the quadratic form Q. In this paper, we apply the Hardy-Littlewood-Kloosterman circle method to derive the asymptotic formula for the shifted convolution sum of the divisor function d (n) and Fourier coefficients r (n , Q). With more efforts, our method should have a number of applications for other multiplicative functions. [ABSTRACT FROM AUTHOR]

Let λ f (n) be the normalized Fourier coefficients of a Hecke-Maass or Hecke holomorphic cusp form f for congruence group Γ 0 (N) with level N and nebentypus χ N. Let Q (x) be a positive definite integral quadratic form, and r (n , Q) denote the number of representations of n by the quadratic form Q. In this paper, we apply Jutila's circle method to derive a sharp bound for the shifted convolution sum of Fourier coefficients λ f (n) and r (n , Q). We generalize and improve previous results without the Ramanujan conjecture. [ABSTRACT FROM AUTHOR]