A divisor function of Wigert and higher degree forms

Let $k\in\mathbb{N}$. Wigert's divisor function $d^{\left(\frac{1}{k}\right)}(j)$ counts the number of representations of $j$ of the form $m^k+mn$ with $m\geq1 , n\geq0$. Let $\mathcal{F}_k(s)$ denote the Dirichlet series of $d^{\left(\frac{1}{k}\right)}(j)$. While $\mathcal{F}_2(s)$ is essentially a well-known special case of the Euler-Zagier double zeta function, and hence well-studied, very little is known about $\mathcal{F}_k(s)$ for $k>2$. We offer three new representations for $\mathcal{F}_k(s)$ for $k\geq2$, one of which is an analogue of the Chowla-Selberg formula as well as of a formula of Atkinson. The meromorphicity of $\mathcal{F}_k(s)$ is also discussed. The special value $\mathcal{F}_3\left(\frac{3}{2}\right)$ is expressed in terms of an infinite series of Bessel functions and a generalized divisor function.