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Rational approximations for the quotient of gamma values. (English) Zbl 1217.41016
The authors continue the study of quantities arising in rational approximation to Euler’s constant and values of the Gamma function started by A. I. Aptekarev, T. Rivoal and D. N. Tulyakov [Rational approximations of Euler’s constant and recurrence relations. Sovrem. Probl. Mat. 9. Moskva: Matematicheskiĭ Institut im. V. A. Steklova, RAN. 82 p. (2007; Zbl 1134.41001), J. Number Theory 130, No. 4, 944–955 (2010; Zbl 1206.11095) and Math. Notes 85, No. 5, 746–750 (2009); translation from Mat. Zametki 85, No. 5, 782–787 (2009; Zbl 1205.41011)].
The main result is
Theorem 1. Let \(a_1,a_2,b\in{\mathbb Q}\), \(a_1-a_2 \not\in \mathbb Z\), \(a_1,a_2>-1\), \(b>0\). For \(n=0,1,\dots\) define a sequence of rational numbers
\[ q_n(a_1,a_2,b) = \sum_{k=0}^n \binom{n+a_1-a_2}{k} \binom{n+a+2-a_2}{n-}(a_1+1)_{n+k}b^{n-k}. \]
Then
\[ \mu_{a_2-a_1}^n \mu_{a_2}^{2n} q_n(a_1,a_2,b) \in n! \mathbb Z[b], \]
with
\[ \mu_a^n = (\operatorname{den}a)^n \cdot \prod_{p|\operatorname{den}a} p^{[\frac{n}{p-1}]} \]
(\(\operatorname{den}a\) is the denominator of the rational number \(a\) when it is written in its simplest terms).
Moreover, the following asymptotic formulae hold
\[ \begin{aligned} q_n(a_1,a_2,b) \frac{\Gamma(a_2+1)/b^{a_2}}{\Gamma(a_1+1)b^{a_1}} - q_n(a_2,a_1,b) &= (2n)! \frac{e^{-2\sqrt{2bn}}}{n^{1/4-(a_1+a_2)/2}} \left(c_1+{\mathcal O}(n^{-1/2})\right),\\ q_n(a_1,a_2,b) &= (2n)!\frac{e^{\sqrt{2bn}}}{n^{1/4-(a_1+a_2)/2}}\left(c_2+{\mathcal O}(n^{-1/2})\right) \quad\text{as }n\rightarrow\infty, \end{aligned} \]
with
\[ \begin{aligned} c_1 &=\frac{2\sin{(\pi (a_2-a_1))}\Gamma(a_2+1)}{b^{1/4+(a_1-a_2)/2}e^{3b/8}\sqrt{\pi}\Gamma(a_1+1)},\\ c_2 &=\frac{2^{(a_1+a_2)2-3/4}}{b^{1/4+(a_1-a_2)/2}e^{3b/8}\sqrt{\pi}\Gamma(a_2+1)}.\end{aligned} \]

MSC:
41A20 Approximation by rational functions
33B15 Gamma, beta and polygamma functions
11J04 Homogeneous approximation to one number
11J72 Irrationality; linear independence over a field
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
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[1] (), (in Russian)
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[10] Rivoal, T., Approximations rationnelles des valeurs de la fonction gamma aux rationneels, J. number theory, 130, 944-955, (2010) · Zbl 1206.11095
[11] Tulyakov, D.N., A system of recurrence relations for rational approximations of the Euler constant, Math. notes, 85, 5, 746-750, (2009) · Zbl 1205.41011
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