Sondow's conjecture, convergents to \(e\), and \(p\)-adic analytic functions (Q2633146)

The authors study \(p\)-adic functions and methods following from the work by \textit{B. C. Berndt} et al. [Adv. Math. 248, 1298--1331 (2013; Zbl 1292.11079)] on Sondow's conjecture. This conjecture states that exactly two partial sums of the Taylor series for \(e\) (the numbers \(2\) and \(8/3\)) are also convergents for the continued fraction of \(e\). This conjecture was proved by Berndt et al. [loc. cit.]. As indicated in the abstract and in the Introduction, certain \(p\)-adic functions arising in the paper by Berndt et al. merit further attention. The authors solve an open concerning these functions: they are locally analytic of minimal radius \(1/2\) (cf. \S4). The layout of the paper is as follows. \textbf{\S1 Introduction} (2 pages) \textbf{\S2 Recurrence relations} (1 page) \textbf{\S3 Relations and generating functions} (4 pages) \textbf{\S4 Proof of the (local) analyticity of \(f_r, g_r\) and \(A\)} (\(2\frac{1}{2}\) pages) \textbf{Appendix: integral representations of locally analytic functions} (2 pages) \textbf{References} (11 items)