  For a fixed $z\in\mathbb{C}$ and a fixed $k\in\mathbb{N}$, let
$\sigma_{z}^{(k)}(n)$ denote the sum of $z$-th powers of those divisors $d$ of
$n$ whose $k$-th powers also divide $n$. This arithmetic function is a
simultaneous generalization of the well-known divisor function $\sigma_z(n)$ as
well as the divisor function $d^{(k)}(n)$ first studied by Wigert. The
Dirichlet series of $\sigma_{z}^{(k)}(n)$ does not fall under the purview of
Chandrasekharan and Narasimhan's fundamental work on Hecke's functional
equation with multiple gamma factors. Nevertheless, as we show here, an
explicit and elegant Vorono\"{\dotlessi} summation formula exists for this
function. As its corollaries, some transformations of Wigert are generalized.
The kernel $H_{z}^{(k)}(x)$ of the associated integral transform is a new
generalization of the Bessel kernel. Several properties of this kernel such as
its differential equation, asymptotic behavior and its special values are
derived. A crucial relation between $H_{z}^{(k)}(x)$ and an associated integral
$K_{z}^{(k)}(x)$ is obtained, the proof of which is deep, and employs the
uniqueness theorem of linear differential equations and the properties of
Stirling numbers of the second kind and elementary symmetric polynomials.
