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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
DOI: https://doi.org/10.1090/S0025-5718-1977-0439741-4
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.

References [Enhancements On Off] (What's this?)

  • [1] J. BRILLHART, "On the Euler and Bernoulli polynomials," J. Reine Angew. Math., v. 234, 1969, pp. 45-64. MR 39 #4117. MR 0242790 (39:4117)
  • [2] L. CARLITZ, "Note on irreducibility of the Bernoulli and Euler polynomials," Duke Math. J., v. 19, 1952, pp. 475-481. MR 14, 163. MR 0049381 (14:163h)
  • [3] L. CARLITZ, "Note on the numbers of Jordan and Ward," Duke Math. J., v. 38, 1971, pp. 783-790. MR 45 #1776. MR 0292693 (45:1776)
  • [4] L. CARLITZ, "The Staudt-Clausen theorem," Math. Mag., v. 34, 1960/61, pp. 131-146. MR 24 #A258. MR 0130397 (24:A258)
  • [5] L. CARLITZ, "Set partitions," Fibonacci Quart. (To appear.) MR 0427087 (55:123)
  • [6] F. T. HOWARD, "A sequence of numbers related to the exponential function," Duke Math. J., v. 34, 1967, pp. 599-616. MR 36 #130. MR 0217035 (36:130)
  • [7] F. T. HOWARD, "Factors and roots of the van der Pol polynomials," Proc. Amer. Math. Soc., v. 53, 1975, pp. 1-8. MR 52 #252. MR 0379347 (52:252)
  • [8] F. T. HOWARD, "Some sequences of rational numbers related to the exponential function," Duke Math. J., v. 34, 1967, pp. 701-716. MR 36 #131. MR 0217036 (36:131)
  • [9] F. T. HOWARD, "Roots of the Euler polynomials," Pacific J. Math., v. 64, 1976, pp. 181-191. MR 0417394 (54:5444)
  • [10] K. INKERI, "The real roots of Bernoulli polynomials," Ann. Univ. Turku. Ser. A I, v. 37, 1959, pp. 3-20. MR 22 #1703. MR 0110835 (22:1703)
  • [11] D. JACKSON, Fourier Series and Orthogonal Polynomials, Carus Monograph Ser., no. 6, Math. Assoc. of America, Oberlin, Ohio, 1941. MR 3, 230. MR 0005912 (3:230f)
  • [12] C. JORDAN, Calculus of Finite Differences, Hungarian Agent Eggenberger Book-Shop, Budapest, 1939; Chelsea, New York, 1950. MR 1, 74.
  • [13] K. KNOPP, Infinite Sequences and Series, Dover, New York, 1956. MR 18, 30. MR 0079110 (18:30c)
  • [14] P. A. MacMAHON, Combinatory Analysis, Chelsea, New York, 1960. MR 25 #5003. MR 0141605 (25:5003)
  • [15] N. E. NÖRLUND, Vorlesungen über Differenzrechnung, Springer-Verlag, Berlin, 1924.
  • [16] J. RIORDAN, An Introduction to Combinatorial Analysis, Chapman & Hall, London; Wiley, New York, 1958. MR 20 #3077. MR 0096594 (20:3077)
  • [17] E. C. TITCHMARSH, The Theory of Functions, 2nd ed., Oxford, London, 1939.

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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1977-0439741-4
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

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