## 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.## References

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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