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Mat. Sb., 2002, Volume 193, Number 8, Pages 49–70 (Mi msb674)  

This article is cited in 14 scientific papers (total in 14 papers)

On the irrationality measure for a $q$-analogue of $\zeta(2)$

W. V. Zudilin

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: A Liouville-type estimate is proved for the irrationality measure of the quantities
$$ \zeta_q(2) =\sum_{n=1}^\infty\frac{q^n}{(1-q^n)^2} $$
with $q^{-1}\in\mathbb Z\setminus\{0,\pm1\}$. The proof is based on the application of a $q$-analogue of the arithmetic method developed by Chudnovsky, Rukhadze, and Hata and of the transformation group for hypergeometric series–the group-structure approach introduced by Rhin and Viola.

DOI: https://doi.org/10.4213/sm674

Full text: PDF file (352 kB)
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English version:
Sbornik: Mathematics, 2002, 193:8, 1151–1172

Bibliographic databases:

UDC: 511.3
MSC: Primary 11J72, 11J82; Secondary 33D15
Received: 08.11.2001

Citation: W. V. Zudilin, “On the irrationality measure for a $q$-analogue of $\zeta(2)$”, Mat. Sb., 193:8 (2002), 49–70; Sb. Math., 193:8 (2002), 1151–1172

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. W. V. Zudilin, “Diophantine Problems for $q$-Zeta Values”, Math. Notes, 72:6 (2002), 858–862  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    2. Zudilin W., “An Apery-like difference equation for Catalan's constant”, Electron. J. Combin., 10:1 (2003), 14, 10 pp.  crossref  mathscinet  zmath  isi
    3. Zudilin W., “Heine's basic transform and a permutation group for $q$-harmonic series”, Acta Arith., 111:2 (2004), 153–164  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    4. J. Math. Sci. (N. Y.), 137:2 (2006), 4673–4683  mathnet  crossref  mathscinet  zmath  elib
    5. Bradley D.M., “Multiple $q$-zeta values”, J. Algebra, 283:2 (2005), 752–798  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    6. Krattenthaler C., Rivoal T., Zudilin W., “Séries hypergéométriques basiques, $q$-analogues des valeurs de la fonction zêta et séries d'Eisenstein [Basic hypergeometric series, $q$-analogues of the values of zeta functions and Eisenstein series]”, J. Inst. Math. Jussieu, 5:1 (2006), 53–79  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    7. Bundschuh P., “Linear independence of values of a certain Lambert series”, Results Math., 51:1-2 (2007), 29–42  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    8. Postelmans K., “Irrationality of $\zeta_q(1)$ and $\zeta_q(2)$”, J. Number Theory, 126:1 (2007), 119–154  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    9. Smet C., Van Assche W., “Irrationality proof of a $q$-extension of $\zeta(2)$ using little $q$-Jacobi polynomials”, Acta Arith., 138:2 (2009), 165–178  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus  scopus
    10. Koelink E., Van Assche W., “Leonhard Euler and a $q$-analogue of the logarithm”, Proc. Amer. Math. Soc., 137:5 (2009), 1663–1676  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    11. Merilä V., “On arithmetical properties of certain $q$-series”, Results Math., 53:1-2 (2009), 129–151  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    12. Fischler S., Zudilin W., “A refinement of Nesterenko's linear independence criterion with applications to zeta values”, Mathematische Annalen, 347:4 (2010), 739–763  crossref  mathscinet  zmath  isi  scopus  scopus
    13. W. Zudilin, “Arithmetic hypergeometric series”, Russian Math. Surveys, 66:2 (2011), 369–420  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    14. Warnaar S.O., Zudilin W., “A q-rious positivity”, Aequationes Mathematicae, 81:1–2 (2011), 177–183  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
  • Математический сборник - 1992–2005 Sbornik: Mathematics (from 1967)
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