On the rational approximation of the sum of the reciprocals of the Fermat numbers

Title: On the rational approximation of the sum of the reciprocals of the Fermat numbers
Creator: Coons, Michael
Relation: The Ramanujan Journal Vol. 30, Issue 1, p. 39-65
Publisher Link: http://dx.doi.org/10.1007/s11139-012-9410-x
Publisher: Springer New York
Resource Type: journal article
Date: 2013
Description: Let G(z):=∑n⩾0z2n(1−z2n)−1 denote the generating function of the ruler function, and F(z):=∑n⩾z2n(1+z2n)−1; note that the special value F(1/2) is the sum of the reciprocals of the Fermat numbers Fn:=22n+1. The functions F(z)and G(z)as well as their special values have been studied by Mahler, Golomb, Schwarz, and Duverney; it is known that the numbers F(α) and G(α) are transcendental for all algebraic numbers α which satisfy 0<α<1. For a sequence u, denote the Hankel matrix Hpn(u):=(u(p+i+j−2))1⩽i,j⩽n. Let alpha be a real number. The irrationality exponent μ(α) is defined as the supremum of the set of real numbers μ such that the inequality |α−p/q|
Subject: irrationality exponents; Padé approximants; Hankel determinants; Fermat numbers
Identifier: http://hdl.handle.net/1959.13/1063504
Identifier: uon:17309
Identifier: ISSN:1382-4090
Language: eng
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