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 theory of linear differential equations and the properties of Stirling numbers of the second kind and elementary symmetric polynomials.
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