The first and second derivatives of the q-Rationals

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.