Difference operators for partitions under the Littlewood decomposition

Let n be a positive integer. Let (delta _3(n)) denote the difference between the number of (positive) divisors of n congruent to 1 modulo 3 and the number of those congruent to 2 modulo 3. In 2004, Farkas proved that the arithmetic convolution sum

$$begin{aligned} D_3(n):=sum _{j=1}^{n-1}delta _3(j)delta _3(n-j) end{aligned}$$

satisfies the relation

$$begin{aligned} 3D_3(n)+delta _3(n)={sum _{mathop {_{d mid n}}limits _{3 not mid d}}}d. end{aligned}$$

In this paper, we use a result about binary quadratic forms to prove a general arithmetic convolution identity which contains Farkas’ formula and two other similar known formulas as special cases. From our identity, we deduce a number of analogous new convolution formulas.