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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Mathematics of Computation is 1.98.

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Numbers generated by the reciprocal of $e^{x}-x-1$
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by F. T. Howard PDF
Math. Comp. 31 (1977), 581-598 Request permission

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.
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Additional Information
  • © Copyright 1977 American Mathematical Society
  • Journal: Math. Comp. 31 (1977), 581-598
  • MSC: Primary 10A40; Secondary 05A17
  • DOI: https://doi.org/10.1090/S0025-5718-1977-0439741-4
  • MathSciNet review: 0439741