The first and second derivatives of the q-Rationals

Abstract

The q-rationals, introduced by Valentin Ovsienko and Sophie Morier Genoud, are an extension of Gauss' q-integers. Like the q-integers, the q-rationals reduce to their non-quantized values at $q=1$. In this paper, I prove closed-form expressions for the first and second derivatives of the q-rationals at this point. My expressions are written in terms of the q-rationals' non-quantized values; both feature Thomae's function, whereas my expression for the second derivative additionally features a generalized form of the Dedekind sum.