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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 $$\alpha>-1$$, $\mathcal F_\alpha(z)=\int_0^\infty \frac{t^\alpha e^{-t}}{z-t} dt.$

MSC:
 11J91 Transcendence theory of other special functions 33B15 Gamma, beta and polygamma functions
MultInt
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References:
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