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Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



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

Author: F. T. Howard
Journal: Math. Comp. 31 (1977), 581-598
MSC: Primary 10A40; Secondary 05A17
MathSciNet review: 0439741
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Abstract: In this paper we examine the polynomials $ {A_n}(z)$ and the rational numbers $ {A_n} = {A_n}(0)$ defined by means of

$\displaystyle {e^{xz}}{x^2}{({e^x} - x - 1)^{ - 1}} = 2\sum\limits_{n = 0}^\infty {{A_n}(z){x^n}/n!} .$

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 $ \vert\cos n\theta \vert < {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.

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Keywords: Bernoulli number and polynomial, Stirling numbers of the second kind, associated Stirling numbers of the second kind, Eisenstein's irreducibility criterion, set partition, composition, Staudt-Clausen theorem
Article copyright: © Copyright 1977 American Mathematical Society