## Rational approximations for values of derivatives of the Gamma function

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**361**(2009), 6115-6149 Request permission

## Abstract:

The arithmetic nature of Euler’s constant $\gamma$ 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 $a_n$ and $b_n$ such that $a_n/b_n$ converges sub-exponentially to $\gamma$, viewed as $-\Gamma ’(1)$, where $\Gamma$ is the usual Gamma function. Although this is not yet enough to prove that $\gamma \not \in \mathbb {Q}$, it is a major step in this direction.

In this paper, we present a different, but related, approach based on simultaneous Padé approximants to Euler’s functions, from which we construct and study a new third order recurrence that produces a sequence in $\mathbb {Q}(z)$ whose height grows like the factorial and that converges sub-exponentially to $\log (z)+\gamma$ for any complex number $z\in \mathbb {C}\setminus (-\infty ,0]$, where $\log$ is defined by its principal branch. We also show how our approach yields in theory rational approximations of numbers related to $\Gamma ^{(s)}(1)$ for any integer $s\ge 1$. In particular, we construct a sixth order recurrence which provides simultaneous rational approximations (of factorial height) converging sub-exponentially to the numbers $\gamma$ and $\Gamma ''(1)-2\Gamma ’(1)^2=\zeta (2)-\gamma ^2.$

## References

- K. Alladi and M. L. Robinson,
*Legendre polynomials and irrationality*, J. Reine Angew. Math.**318**(1980), 137–155. MR**579389** - Yves André,
*Arithmetic Gevrey series and transcendence. A survey*, J. Théor. Nombres Bordeaux**15**(2003), no. 1, 1–10 (English, with English and French summaries). Les XXIIèmes Journées Arithmetiques (Lille, 2001). MR**2018997** - George E. Andrews, Richard Askey, and Ranjan Roy,
*Special functions*, Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, Cambridge, 1999. MR**1688958**, DOI 10.1017/CBO9781107325937 - R. Apéry,
*Irrationalité de $\zeta (2)$ et $\zeta (3)$*, Astérisque**61**(1979), 11–13. - A. I. Aptekarev, A. Branquinho, and W. Van Assche,
*Multiple orthogonal polynomials for classical weights*, Trans. Amer. Math. Soc.**355**(2003), no. 10, 3887–3914. MR**1990569**, DOI 10.1090/S0002-9947-03-03330-0 - A. I. Aptekarev (editor),
*Rational approximants for Euler constant and recurrence relations*, Sovremennye Problemy Matematiki (“Current Problems in Mathematics”), vol.**9**, MIAN (Steklov Institute), Moscow, 2007. - George D. Birkhoff and W. J. Trjitzinsky,
*Analytic theory of singular difference equations*, Acta Math.**60**(1933), no. 1, 1–89. MR**1555364**, DOI 10.1007/BF02398269 - H. Cohen,
*Accélération de la convergence de certaines récurrences linéaires*, Sémin. Théor. Nombres 1980–1981, Exposé no.16, 2 pp. (1981). - C. Elsner,
*On a sequence transformation with integral coefficients for Euler’s constant*, Proc. Amer. Math. Soc.**123**(1995), no. 5, 1537–1541. MR**1233969**, DOI 10.1090/S0002-9939-1995-1233969-4 - Stéphane Fischler and Tanguy Rivoal,
*Un exposant de densité en approximation rationnelle*, Int. Math. Res. Not. , posted on (2006), Art. ID 95418, 48 (French). MR**2272100**, DOI 10.1155/IMRN/2006/95418 - G. H. Hardy,
*Divergent series*, Éditions Jacques Gabay, Sceaux, 1992. With a preface by J. E. Littlewood and a note by L. S. Bosanquet; Reprint of the revised (1963) edition. MR**1188874** - Masayoshi Hata,
*Rational approximations to $\pi$ and some other numbers*, Acta Arith.**63**(1993), no. 4, 335–349. MR**1218461**, DOI 10.4064/aa-63-4-335-349 - Maxim Kontsevich and Don Zagier,
*Periods*, Mathematics unlimited—2001 and beyond, Springer, Berlin, 2001, pp. 771–808. MR**1852188** - Christian Krattenthaler and Tanguy Rivoal,
*How can we escape Thomae’s relations?*, J. Math. Soc. Japan**58**(2006), no. 1, 183–210. MR**2204570** - Jean-Pierre Ramis,
*Séries divergentes et théories asymptotiques*, Bull. Soc. Math. France**121**(1993), no. Panoramas et Synthèses, suppl., 74 (French). MR**1272100** - T. Rivoal,
*Simultaneous polynomial approximations of the Lerch function*, Preprint (2007), 15 pages, to appear in Canadian J. Math. - T. Rivoal and W. Zudilin,
*Diophantine properties of numbers related to Catalan’s constant*, Math. Ann.**326**(2003), no. 4, 705–721. MR**2003449**, DOI 10.1007/s00208-003-0420-2 - Akalu Tefera,
*MultInt, a MAPLE package for multiple integration by the WZ method*, J. Symbolic Comput.**34**(2002), no. 5, 329–353. MR**1937465**, DOI 10.1006/jsco.2002.0561 - Michel Waldschmidt,
*Valeurs zêta multiples. Une introduction*, J. Théor. Nombres Bordeaux**12**(2000), no. 2, 581–595 (French, with English and French summaries). Colloque International de Théorie des Nombres (Talence, 1999). MR**1823204** - Herbert S. Wilf and Doron Zeilberger,
*An algorithmic proof theory for hypergeometric (ordinary and “$q$”) multisum/integral identities*, Invent. Math.**108**(1992), no. 3, 575–633. MR**1163239**, DOI 10.1007/BF02100618 - Jet Wimp and Doron Zeilberger,
*Resurrecting the asymptotics of linear recurrences*, J. Math. Anal. Appl.**111**(1985), no. 1, 162–176. MR**808671**, DOI 10.1016/0022-247X(85)90209-4 - Doron Zeilberger,
*A holonomic systems approach to special functions identities*, J. Comput. Appl. Math.**32**(1990), no. 3, 321–368. MR**1090884**, DOI 10.1016/0377-0427(90)90042-X - V. V. Zudilin,
*On third-order Apéry-type recursion for $\zeta (5)$*, Mat. Zametki**72**(2002), no. 5, 796–800 (Russian); English transl., Math. Notes**72**(2002), no. 5-6, 733–737. MR**1963141**, DOI 10.1023/A:1021473409544

## Additional Information

**Tanguy Rivoal**- Affiliation: Institut Fourier, CNRS UMR 5582, Université Grenoble 1, 100 rue des Maths, BP 74, 38402 Saint-Martin d’Hères cedex, France
- MR Author ID: 668668
- Received by editor(s): February 28, 2008
- Published electronically: June 25, 2009
- © Copyright 2009 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**361**(2009), 6115-6149 - MSC (2000): Primary 11J13; Secondary 33C45, 33F10, 39A11
- DOI: https://doi.org/10.1090/S0002-9947-09-04905-8
- MathSciNet review: 2529926