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Rational approximations of values of the Gamma function on rationals. (Approximations rationnelles des valeurs de la fonction Gamma aux rationnels.) (French) Zbl 1206.11095
Let \(x>0\) be a real number and \(\alpha\) a complex number with real part \(>-1\). The author produces explicit linear recurrences of order \(3\) \[ C_3u_{n+3}+C_2u_{n+2}+C_1u_{n+1}+C_0u_n=0, \] with coefficients \(C_0\), \(C_1\), \(C_2\), \(C_3\) which are polynomial in \(n\), \(\alpha\) and \(x\) of total degrees \(\leq 16\), having solutions \((P_n(x,\alpha))_{n\geq 0}\) and \((Q_n(x,\alpha))_{n\geq 0}\), sequences of polynomials in \(\mathbb Q[\alpha,x]\), for which the sequence \(\bigl(P_n(x,\alpha)/Q_n(x,\alpha)\bigr)_{n\geq 0}\) converges quickly towards \(\Gamma(1+\alpha)/x^\alpha\). In the special case \(x=1\) and \(1+\alpha=a/b\in{\mathbb Q}_{>0}\), this yields new rational approximations to \(\Gamma(a/b)\).
The proof rests on the methods of the previous work by the author [Trans. Am. Math. Soc. 361, No. 11, 6115–6149 (2009; Zbl 1236.11061)] involving the sequence of polynomials \[ A_{n,\alpha}(x)=\frac{1}{n!^2} e^x \bigl( x^{n-\alpha} (e^{-x} x^{n+\alpha})^{(n)}\bigr)^{(n)} \] previously introduced by A. I. Aptekarev, A. Branquinho and W. Van Assche [Trans. Am. Math. Soc. 355, No. 10, 3887–3914 (2003; Zbl 1033.33002)]. These polynomials \(A_{n,\alpha}\) are denominators of Padé simultaneous approximants at infinity to the functions \(\mathcal F_0\) and \(\mathcal F_\alpha\), where, for \(z\in\mathbb C\setminus{\mathbb R}_{\leq 0}\) and \(\alpha>-1\), \[ \mathcal F_\alpha(z)=\int_0^\infty \frac{t^\alpha e^{-t}}{z-t} dt. \]

11J91 Transcendence theory of other special functions
33B15 Gamma, beta and polygamma functions
Full Text: DOI
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