On a continued fraction expansion for Euler?s constant

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• 作者：Kh. Hessami Pilehrood¹ ; ^{hessamik@gmail.com} ; T. Hessami Pilehrood ; ¹ ; ^{hessamit@gmail.com}
• 关键词：Euler constant ; Euler&ndash ; Gompertz constant ; Meijer G-function ; Rational approximation ; Second-order linear recurrence ; Continued fraction ; Zeilberger?s algorithm of creative telescoping ; Whittaker function ; Laguerre orthogonal polynomials
• 刊名：Journal of Number Theory
• 出版年：2013
• 出版时间：February, 2013
• 年：2013
• 卷：133
• 期：2
• 页码：769-786
• 全文大小：235 K

文摘

Recently, A.I. Aptekarev and his collaborators found a sequence of rational approximations to Euler?s constant ¦Ã 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 ¦Ã with rational coefficients. Using this construction we find new rational approximations to ¦Ã 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.