Voronoï summation formula for the generalized divisor function σz(k)(n)

Abstract

For a fixed z∈C and a fixed k∈N, let σz(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 σz(n) as well as the divisor function d(k)(n) first studied by Wigert. The Dirichlet series of σz(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 Voronoï summation formula exists for this function. As its corollaries, some transformations of Wigert are generalized. The kernel Hz(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 Hz(k)(x) and an associated integral Kz(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. © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2026.