On a continued fraction expansion for Euler's constant

JOURNAL OF NUMBER THEORY

卷:133

Article

Pilehrood, Kh Hessami¹  Pilehrood, T. Hessami¹ 

[1] Inst Res Fundamental Sci IPM, Sch Math, Tehran, Iran

关键词: Euler constant;    Euler-Gompertz constant;    Meijer G-function;    Rational approximation;    Second-order linear recurrence;    Continued fraction;    Zeilberger's algorithm of creative;    telescoping;    Whittaker function;    Laguerre orthogonal polynomials;   

DOI  :  10.1016/j.jnt.2012.08.016

来源: Elsevier

【 摘 要 】

Recently, A.I. Aptekarev and his collaborators found a sequence of rational approximations to Euler's constant gamma defined by a third-order homogeneous linear recurrence. In this paper, we give a new interpretation of Aptekarev's approximations in terms of Meijer G-functions and hypergeometric-type series. This approach allows us to describe a very general construction giving linear forms in 1 and gamma with rational coefficients. Using this construction we find new rational approximations to gamma generated by a second-order inhomogeneous linear recurrence with polynomial coefficients. This leads to a continued fraction (though not a simple continued fraction) for Euler's constant. It seems to be the first non-trivial continued fraction expansion convergent to Euler's constant sub-exponentially, the elements of which can be expressed as a general pattern. It is interesting to note that the same homogeneous recurrence generates a continued fraction for the Euler-Gompertz constant found by Stieltjes in 1895. (C) 2012 Elsevier Inc. All rights reserved.

【 授权许可】

Free