RATIONAL APPROXIMATIONS FOR VALUES OF DERIVATIVES OF THE GAMMA FUNCTION

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RATIONAL APPROXIMATIONS FOR VALUES OF DERIVATIVES OF THE GAMMA FUNCTION TANGUY RIVOAL Abstract. The arithmetical nature of Euler's constant ? is still unknown and even getting good rational approximations to it is difficult. Recently, Aptekarev managed to find a third order linear recurrence with polynomial coefficients which admits two rational solutions an and bn such that an/bn converges sub-exponentially to ?, viewed as ???(1), where ? is the usual Gamma function. Although this is not yet enough to prove that ? 6? Q, it is major step in this direction. In this paper, we present a different, but related, approach based on simultaneous Pade approximants to Euler's functions, from which we constuct and study a new third order recurrence that produces a sequence in Q(z) whose height grows like the factorial and that converges sub-exponentially to log(z) + ? for any complex number z ? C \ (?∞, 0], where log is defined by its principal branch. We also show how our approach yields in theory rational approximations of numbers related to ?(s)(1) for any integer s ≥ 1. In particular, we construct a sixth order recurrence which provides simultaneous rational approximations (of factorial height) converging sub-exponentially to the numbers ? and ???(1)? 2??(1)2 = ?(2)? ?2.

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Publié par

profil-urra-2012

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RATIONAL

APPROXIMATIONS FOR VALUES OF DERIVATIVES THE GAMMA FUNCTION

TANGUY RIVOAL

Abstract.The arithmetical nature of Euler’s constantγis still unknown and even getting good rational approximations to it is diﬃcult. Recently, Aptekarev managed to ﬁnd a third order linear recurrence with polynomial coeﬃcients which admits two rational solutionsanandbnsuch thatan/bnconverges sub-exponentially toγ, viewed as−Γ0(1), where Γ is the usual Gamma function. Although this is not yet enough to prove that γ6∈Q, it is major step in this direction. Inthispaper,wepresentadiﬀerent,butrelated,approachbasedonsimultaneousPad´e approximants to Euler’s functions, from which we constuct and study a new third order recurrence that produces a sequence inQ(z) whose height grows like the factorial and that converges sub-exponentially to log(z) +γfor any complex numberz∈C\(−∞,0], where log is deﬁned by its principal branch. We also show how our approach yields in theory rational approximations of numbers related to Γ(s)(1) for any integers≥1. In particular, we construct a sixth order recurrence which provides simultaneous rational approximations (of factorial height) converging sub-exponentially to the numbersγand Γ00(1)−2Γ0(1)2=ζ(2)−γ2.

1.Introduction

The arithmetical nature of Euler’s constantγ, deﬁned as the limit of the sequence sn=Pjn=11/j−ln(n It+1), is still open. is conjectured to be a transcendental number over Q Abut even its irrationality seems currently out of reach. standard method for proving the irrationality of a classical constantαis to pull out of a hat or construct a sequence of rational approximations (an/bn)n≥0such thatan, bn∈Zand 0<|bnα−an| →0 as n→+∞ fact, even getting a sequence of rationals. Inan/bnwhose height does not grow too fast and which converges fast toγis a diﬃcult problem. It is easier to ﬁnd averaging processes of the formPnk=0ak,n(γ−sk) that converge geometrically to 0, for some well-chosen integral weights (ak,n)0≤k≤nwhich are usually products of binomial coeﬃcients (see [9]). But this does not help to study the arithmetic nature ofγ. For many classical constants, it turns out to be possible to construct “irrationality proving” rational approximationsan/bnthat satisfy a special property: both (an)n≥0and (bn)n≥0are solutions of arecurrence of ﬁnite order with polynomial coeﬃcientslinear . For eits regular continued fraction, which was found, we can use the recurrence that generates

Date: January 13, 2009. 2000Mathematics Subject Classiﬁcation.Primary 11J13; Secondary 33C45, 33F10, 39A11. Key words and phrases.ioatlanatans,rntreluocs’Ents,ximapprod´ea,saPitnoixamppor-rucerraenil rences, Birkhoﬀ–Trjitzinsky theory. 1

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