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

Author:
F. T. Howard

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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

- John Brillhart,
*On the Euler and Bernoulli polynomials*, J. Reine Angew. Math.**234**(1969), 45–64. MR**242790**, DOI https://doi.org/10.1515/crll.1969.234.45 - L. Carlitz,
*Note on irreducibility of the Bernoulli and Euler polynomials*, Duke Math. J.**19**(1952), 475–481. MR**49381** - L. Carlitz,
*Note on the numbers of Jordan and Ward*, Duke Math. J.**38**(1971), 783–790. MR**292693** - L. Carlitz,
*The Staudt-Clausen theorem*, Math. Mag.**34**(1960/61), 131–146. MR**130397**, DOI https://doi.org/10.2307/2688488 - L. Carlitz,
*Set partitions*, Fibonacci Quart.**14**(1976), no. 4, 327–342. MR**427087** - F. T. Howard,
*A sequence of numbers related to the exponential function*, Duke Math. J.**34**(1967), 599–615. MR**217035** - F. T. Howard,
*Factors and roots of the van der Pol polynomials*, Proc. Amer. Math. Soc.**53**(1975), no. 1, 1–8. MR**379347**, DOI https://doi.org/10.1090/S0002-9939-1975-0379347-9 - F. T. Howard,
*Some sequences of rational numbers related to the exponential function*, Duke Math. J.**34**(1967), 701–716. MR**217036** - F. T. Howard,
*Roots of the Euler polynomials*, Pacific J. Math.**64**(1976), no. 1, 181–191. MR**417394** - K. Inkeri,
*The real roots of Bernoulli polynomials*, Ann. Univ. Turku. Ser. A I**37**(1959), 20. MR**110835** - Dunham Jackson,
*Fourier Series and Orthogonal Polynomials*, Carus Monograph Series, no. 6, Mathematical Association of America, Oberlin, Ohio, 1941. MR**0005912**
C. JORDAN, - Konrad Knopp,
*Infinite sequences and series*, Dover Publications, Inc., New York, 1956. Translated by Frederick Bagemihl. MR**0079110** - Percy A. MacMahon,
*Combinatory analysis*, Chelsea Publishing Co., New York, 1960. Two volumes (bound as one). MR**0141605**
N. E. NÖRLUND, - John Riordan,
*An introduction to combinatorial analysis*, Wiley Publications in Mathematical Statistics, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1958. MR**0096594**
E. C. TITCHMARSH,

*Calculus of Finite Differences*, Hungarian Agent Eggenberger Book-Shop, Budapest, 1939; Chelsea, New York, 1950. MR

**1**, 74.

*Vorlesungen über Differenzrechnung*, Springer-Verlag, Berlin, 1924.

*The Theory of Functions*, 2nd ed., Oxford, London, 1939.

Retrieve articles in *Mathematics of Computation*
with MSC:
10A40,
05A17

Retrieve articles in all journals with MSC: 10A40, 05A17

Additional Information

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