For \(q>0\), \(0<\delta <1/2\) and \(x>0\), let \[ D_{q-1} (x,\delta)= {\textstyle {1\over {\Gamma(q)}}} \sum_{m\leq x} (x-m)^{q-1} \sigma_{-\delta} (m), \] where as usual for a real \(a\), \(\sigma_ a(m)= \sum_{d\mid m} d^ a\). Moreover, let \[ \Delta_{-\delta} (x)= D_ 0(x,\delta)+ {\textstyle {1\over 2}} \zeta(\delta)- x\zeta (1+\delta)- {{x^{1-\delta} \zeta(1-\delta)} \over {1-\delta}}. \] The author derives Voronoi-type expansions for \(D_{q-1} (x,\delta)\) and \(\Delta_{- \delta} (x)\) in terms of Bessel functions. As a corollary he proves that for every \(\varepsilon>0\) we have \(\Delta_{-\delta} (x)= O(x^{1/3- (\delta/6) +\varepsilon})\).