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

Author:

Howard F. T.

Abstract

In this paper we examine the polynomials A n ( z ) {A_n}(z) and the rational numbers A n = A n ( 0 ) {A_n} = {A_n}(0) defined by means of \[ e x z x 2 ( e x x 1 ) 1 = 2 n = 0 A n ( z ) x n / n ! . {e^{xz}}{x^2}{({e^x} - x - 1)^{ - 1}} = 2\sum \limits _{n = 0}^\infty {{A_n}(z){x^n}/n!} . \] We prove that the numbers A n {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 {A_n} and the Bernoulli numbers are pointed out. We also reexamine and extend previous results concerning A n {A_n} and A n ( z ) {A_n}(z) . In particular, it has been conjectured that A n {A_n} has the same sign as cos n θ - \cos n\theta , where r e i θ r{e^{i\theta }} is the zero of e x x 1 {e^x} - x - 1 with smallest absolute value. We verify this for 1 n 14329 1 \leqslant n \leqslant 14329 and show that if the conjecture is not true for A n {A_n} , then | cos n θ | > 10 ( n 1 ) / 5 |\cos n\theta | > {10^{ - (n - 1)/5}} . We also show that A n ( z ) {A_n}(z) has no integer roots, and in the interval [ 0 , 1 ] [0,1] , A n ( z ) {A_n}(z) has either two or three real roots.

Publisher

American Mathematical Society (AMS)

Subject

Applied Mathematics,Computational Mathematics,Algebra and Number Theory

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