Arithmetical properties of hypergeometric Bernoulli numbers (Q505044)

In this paper, the authors prove two conjectures for arithmetic properties of the denominators of the reduce fraction of the hypergeometric Bernoulli numbers. Remember that the hypergeometric Bernoulli numbers \(B_n(N)\) can be defined by the following exponential generating function \[ \sum_{n=0}^\infty B_n(N)\frac{z^n}{n!}=\frac{z^N/N!}{e^z-T_N(z)}, \] where \[ T_N(z)=\sum_{n=0}^N\frac{z^n}{n!} \] In particular, the author prove that the number of odd terms at the beginning of the sequence of denominators of the sequence \(B_n(N)\) is given by \(2^{\nu_2(N)}\), where \(\nu_2(N)\) is the highest power of 2 that divides \(N\). This conjecture was made by \textit{A. Byrnes} et al. in [Int. J. Number Theory 10, No. 7, 1761--1782 (2014; Zbl 1372.11021)].