# zbMATH — the first resource for mathematics

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
MultInt
Full Text:
##### References:
 [1] K. Alladi and M. L. Robinson, Legendre polynomials and irrationality, J. Reine Angew. Math. 318 (1980), 137 – 155. · Zbl 0425.10039 [2] 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). · Zbl 1136.11315 [3] George E. Andrews, Richard Askey, and Ranjan Roy, Special functions, Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, Cambridge, 1999. · Zbl 0920.33001 [4] R. Apéry, Irrationalité de $$\zeta(2)$$ et $$\zeta(3)$$, Astérisque 61 (1979), 11-13. · Zbl 0401.10049 [5] 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. · Zbl 1033.33002 [6] 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. [7] George D. Birkhoff and W. J. Trjitzinsky, Analytic theory of singular difference equations, Acta Math. 60 (1933), no. 1, 1 – 89. · Zbl 0006.16802 [8] H. Cohen, Accélération de la convergence de certaines récurrences linéaires, Sémin. Théor. Nombres 1980-1981, Exposé no.16, 2 pp. (1981). [9] C. Elsner, On a sequence transformation with integral coefficients for Euler’s constant, Proc. Amer. Math. Soc. 123 (1995), no. 5, 1537 – 1541. · Zbl 0828.65001 [10] Stéphane Fischler and Tanguy Rivoal, Un exposant de densité en approximation rationnelle, Int. Math. Res. Not. , posted on (2006), Art. ID 95418, 48 (French). · Zbl 1147.11039 [11] 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. · Zbl 0897.01044 [12] Masayoshi Hata, Rational approximations to \? and some other numbers, Acta Arith. 63 (1993), no. 4, 335 – 349. · Zbl 0776.11033 [13] Maxim Kontsevich and Don Zagier, Periods, Mathematics unlimited — 2001 and beyond, Springer, Berlin, 2001, pp. 771 – 808. · Zbl 1039.11002 [14] Christian Krattenthaler and Tanguy Rivoal, How can we escape Thomae’s relations?, J. Math. Soc. Japan 58 (2006), no. 1, 183 – 210. · Zbl 1092.33006 [15] Jean-Pierre Ramis, Séries divergentes et théories asymptotiques, Bull. Soc. Math. France 121 (1993), no. Panoramas et Synthèses, suppl., 74 (French). · Zbl 0830.34045 [16] T. Rivoal, Simultaneous polynomial approximations of the Lerch function, Preprint (2007), 15 pages, to appear in Canadian J. Math. [17] T. Rivoal and W. Zudilin, Diophantine properties of numbers related to Catalan’s constant, Math. Ann. 326 (2003), no. 4, 705 – 721. · Zbl 1028.11046 [18] Akalu Tefera, MultInt, a MAPLE package for multiple integration by the WZ method, J. Symbolic Comput. 34 (2002), no. 5, 329 – 353. · Zbl 1015.33013 [19] 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). · Zbl 0976.11037 [20] Herbert S. Wilf and Doron Zeilberger, An algorithmic proof theory for hypergeometric (ordinary and ”\?”) multisum/integral identities, Invent. Math. 108 (1992), no. 3, 575 – 633. · Zbl 0739.05007 [21] Jet Wimp and Doron Zeilberger, Resurrecting the asymptotics of linear recurrences, J. Math. Anal. Appl. 111 (1985), no. 1, 162 – 176. · Zbl 0579.05007 [22] Doron Zeilberger, A holonomic systems approach to special functions identities, J. Comput. Appl. Math. 32 (1990), no. 3, 321 – 368. · Zbl 0738.33001 [23] V. V. Zudilin, On third-order Apéry-type recursion for \?(5), Mat. Zametki 72 (2002), no. 5, 796 – 800 (Russian); English transl., Math. Notes 72 (2002), no. 5-6, 733 – 737. · Zbl 1041.11057
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.