Sondow’s conjecture, convergents to e, and p-adic analytic functions

Abstract

In their study of diophantine approximation of the exponential function in connection with Sondow’s Conjecture, Berndt et al. (Adv Math 348:1298–1331, 2013) have constructed certain p-adic functions arising from the sequence of convergents to the continued fraction of e. We solve an open problem posed in [2], more precisely we show that those p-adic functions are locally analytic (of minimal radius 1 / 2). We leave open the question of the existence of nontrivial zeros (i.e. zeros that are not forced by the functional equations) for these functions.