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Rational approximations for values of derivatives of the gamma function. (English) Zbl 1236.11061
In 2007, A. I. Aptekarev and his collaborators [Rational approximations of Euler’s constant and recurrence relations. Collection of articles. Sovrem. Probl. Mat. 9. Moskva: Matematicheskiĭ Institut im. V. A. Steklova, RAN (2007; Zbl 1134.41001)] found an explicit third order recurrence with two solutions \((p_n)_{n\geq 0}\) and \((q_n)_{n\geq 0}\) with \(p_n\) and \(q_n\) in \(\mathbb Q\) such that \(p_n/q_n\) converges to Euler’s constant \(\gamma\). These approximations are not good enough to prove the irrationality of \(\gamma\), but this maybe a major step in this direction. In the paper under review, the author produces further similar results. He first approximates, for \(x>0\), \(\log x+\gamma\) by \(P_n(x)/Q_n(x)\), where \(P_n(X)\) and \(Q_n(X)\) are rational fractions in \(\mathbb Q(X)\) satisfying an explicit linear recurrence of order \(3\). The second result produces three solutions \((a_{n,1})_{n\geq 0}\), \((a_{n,2})_{n\geq 0}\) and \((b_n)_{n\geq 0}\) with \(a_{n,1}\), \(a_{n,2}\) and \(b_n\) in \(\mathbb Q\), of an explicit linear recurrence sequence of order \(6\), such that \(a_{n,1}/b_n\) converges to \(\gamma\) and \(a_{n,2}/b_n\) converges to \(\zeta(2)-\log 2\). In each case, explicit estimates for the denominators of the rational numbers are given. The proofs involve simultaneous Padé approximants to some Euler’s functions whose Taylor expansion at infinity are related with derivatives of the Euler Gamma function at the point \(1\).

MSC:
11J13 Simultaneous homogeneous approximation, linear forms
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
33F10 Symbolic computation of special functions (Gosper and Zeilberger algorithms, etc.)
39A12 Discrete version of topics in analysis
Software:
MultInt
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