A variation of $q$-Wolstenholme's theorem

Abstract

We investigate a variation of $q$-Wolstenholme's theorem, which extends the $q$-analogue of Wolstenholme's theorem due to Shi and Pan [Amer. Math. Monthly 114 (2007), 529--531]. The proof makes use of the Ramanujan sum and higher order logarithmic derivatives of cyclotomic polynomials.