Bulletin of the
Korean Mathematical Society BKMS

In this paper, we consider several convolution sums, namely, $\mathcal{A}_i(m,n;$ $N)$ $(i=1, 2, 3, 4)$, $\mathcal{B}_j(m,n;N)$ $(j=1, 2, 3)$, and $\mathcal{C}_k(m,n;N)$ $(k=1, 2, 3, \ldots$, $12)$, and establish certain identities involving their finite products. Then we extend these types of product convolution identities to products involving Faulhaber sums. As an application, an identity involving the Weierstrass $\wp$-function, its derivative and certain linear combination of Eisenstein series is established.

Keywords: sum of divisor functions, convolution sums, Faulhaber sums, Eisenstein series, elliptic function

MSC numbers: Primary 33E05, 11A25, 11B65; Secondary 11Y35, 11Y70