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Uspekhi Mat. Nauk, 2002, Volume 57, Issue 3(345), Pages 99–134 (Mi umn512)  

This article is cited in 16 scientific papers (total in 16 papers)

Cyclic graphs and Apéry's theorem

V. N. Sorokin

M. V. Lomonosov Moscow State University

Abstract: This is a survey of results about the behaviour of Hermite–Padé approximants for graphs of Markov functions, and a survey of interpolation problems leading to Apéry's result about the irrationality of the value $\zeta(3)$ of the Riemann zeta function. The first example is given of a cyclic graph for which the Hermite–Padé problem leads to Apéry's theorem. Explicit formulae for solutions are obtained, namely, Rodrigues' formulae and integral representations. The asymptotic behaviour of the approximants is studied, and recurrence formulae are found.

DOI: https://doi.org/10.4213/rm512

Full text: PDF file (419 kB)
References: PDF file   HTML file

English version:
Russian Mathematical Surveys, 2002, 57:3, 535–571

Bibliographic databases:

UDC: 517.53
MSC: Primary 11M06, 11J72, 41A21, 05C90; Secondary 11J82, 14G10
Received: 15.03.2001

Citation: V. N. Sorokin, “Cyclic graphs and Apéry's theorem”, Uspekhi Mat. Nauk, 57:3(345) (2002), 99–134; Russian Math. Surveys, 57:3 (2002), 535–571

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. I. Aptekarev, V. G. Lysov, “Systems of Markov functions generated by graphs and the asymptotics of their Hermite-Padé approximants”, Sb. Math., 201:2 (2010), 183–234  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    2. W. Zudilin, “Arithmetic hypergeometric series”, Russian Math. Surveys, 66:2 (2011), 369–420  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    3. A. I. Aptekarev, A. Kuijlaars, “Hermite–Padé approximations and multiple orthogonal polynomial ensembles”, Russian Math. Surveys, 66:6 (2011), 1133–1199  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    4. Beckermann B., Kalyagin V., Matos A.C., Wielonsky F., “Equilibrium Problems for Vector Potentials with Semidefinite Interaction Matrices and Constrained Masses”, Constr. Approx., 37:1 (2013), 101–134  crossref  mathscinet  zmath  isi  scopus  scopus
    5. A. P. Starovoitov, “Approksimatsii Ermita–Pade dlya sistemy funktsii Mittag-Lefflera”, PFMT, 2013, no. 1(14), 81–87  mathnet
    6. A. P. Starovoitov, “On asymptotic form of the Hermite–Pade approximations for a system of Mittag-Leffler functions”, Russian Math. (Iz. VUZ), 58:9 (2014), 49–56  mathnet  crossref
    7. Khodabakhsh Hessami Pilehrood, Tatiana Hessami Pilehrood, “On
      $$q$$
      q -analogues of two-one formulas for multiple harmonic sums and multiple zeta star values”, Monatsh Math, 2014  crossref  mathscinet  isi  scopus
    8. A. P. Starovoitov, E. P. Kechko, “O lokalizatsii nulei approksimatsii Ermita–Pade eksponentsialnykh funktsii”, PFMT, 2015, no. 3(24), 84–89  mathnet
    9. S. P. Suetin, “Distribution of the zeros of Padé polynomials and analytic continuation”, Russian Math. Surveys, 70:5 (2015), 901–951  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    10. A. P. Starovoitov, E. P. Kechko, “Upper Bounds for the Moduli of Zeros of Hermite–Padé Approximations for a Set of Exponential Functions”, Math. Notes, 99:3 (2016), 417–425  mathnet  crossref  crossref  mathscinet  isi  elib
    11. A. V. Astafieva, A. P. Starovoitov, “Hermite-Padé approximation of exponential functions”, Sb. Math., 207:6 (2016), 769–791  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    12. A. P. Starovoitov, G. N. Kazimirov, M. V. Sidortsov, “Asimptotika approksimatsii Ermita–Pade eksponentsialnykh funktsii s kompleksnymi mnozhitelyami v pokazatelyakh eksponent”, PFMT, 2016, no. 2(27), 61–67  mathnet
    13. M. V. Sidortsov, N. A. Starovoitova, A. P. Starovoitov, “Ob asimptotike approksimatsii Ermita–Pade vtorogo roda dlya eksponentsialnykh funktsii s kompleksnymi mnozhitelyami v pokazatelyakh eksponent”, PFMT, 2017, no. 1(30), 73–77  mathnet
    14. A. P. Starovoitov, “Asymptotics of Diagonal Hermite–Padé Polynomials for the Collection of Exponential Functions”, Math. Notes, 102:2 (2017), 277–288  mathnet  crossref  crossref  mathscinet  isi  elib
    15. V. G. Lysov, “Ob approksimatsiyakh Ermita–Pade dlya proizvedeniya dvukh logarifmov”, Preprinty IPM im. M. V. Keldysha, 2017, 141, 24 pp.  mathnet  crossref
    16. V. G. Lysov, “O diofantovykh priblizheniyakh proizvedeniya logarifmov”, Preprinty IPM im. M. V. Keldysha, 2018, 158, 20 pp.  mathnet  crossref
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