Numbers generated by the reciprocal of $e^{x}-x-1$

Abstract:

In this paper we examine the polynomials ${A_n}(z)$ and the rational numbers ${A_n} = {A_n}(0)$ defined by means of \[ {e^{xz}}{x^2}{({e^x} - x - 1)^{ - 1}} = 2\sum \limits _{n = 0}^\infty {{A_n}(z){x^n}/n!} .\] We prove that the numbers ${A_n}$ are related to the Stirling numbers and associated Stirling numbers of the second kind, and we show that this relationship appears to be a logical extension of a similar relationship involving Bernoulli and Stirling numbers. Other similarities between ${A_n}$ and the Bernoulli numbers are pointed out. We also reexamine and extend previous results concerning ${A_n}$ and ${A_n}(z)$. In particular, it has been conjectured that ${A_n}$ has the same sign as $- \cos n\theta$, where $r{e^{i\theta }}$ is the zero of ${e^x} - x - 1$ with smallest absolute value. We verify this for $1 \leqslant n \leqslant 14329$ and show that if the conjecture is not true for ${A_n}$, then $|\cos n\theta | < {10^{ - (n - 1)/5}}$. We also show that ${A_n}(z)$ has no integer roots, and in the interval $[0,1]$, ${A_n}(z)$ has either two or three real roots.