On fractional Stieltjes constants

Abstract

We study the non-integral generalized Stieltjes constants γα(a) arising from the Laurent series expansions of fractional derivatives of the Hurwitz zeta functions ζ(α)(s,a), and we prove that if ha(s)≔ζ(s,a)−1∕(s−1)−1∕as and Cα(a)≔γα(a)−logα(a)a, then Cα(a)=(−1)−αha(α)(1),for all real α≥0, where h(α)(x) denotes the α-th Grünwald–Letnikov fractional derivative of the function h at x. This result confirms the conjecture of Kreminski (2003), originally stated in terms of the Weyl fractional derivatives.