Geodesic
Approximations to $q$-logarithms and $q$-dilogarithms, with applications to $q$-zeta values
Zapiski Nauchnykh Seminarov POMI, Proceedings on number theory, Tome 322 (2005), pp. 107-124 
W. Zudilin. Approximations to $q$-logarithms and $q$-dilogarithms, with applications to $q$-zeta values. Zapiski Nauchnykh Seminarov POMI, Proceedings on number theory, Tome 322 (2005), pp. 107-124. http://geodesic.mathdoc.fr/item/ZNSL_2005_322_a7/
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     title = {Approximations to $q$-logarithms and $q$-dilogarithms, with applications to $q$-zeta values},
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     url = {http://geodesic.mathdoc.fr/item/ZNSL_2005_322_a7/}
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

We construct simultaneous rational approximations to the $q$-series $L_1(x_1;q)$ and $L_1(x_2;q)$, and, if $x=x_1=x_2$, to the series $L_1(x;q)$ and $L_2(x;q)$, where \begin{gather*} L_1(x;q)=\sum_{n=1}^\infty\frac{(xq)^n}{1-q^n}=\sum_{n=1}^\infty\frac{xq^n}{1-xq^n}, \\ L_2(x;q)=\sum_{n=1}^\infty\frac{n(xq)^n}{1-q^n}=\sum_{n=1}^\infty\frac{xq^n}{(1-xq^n)^2}. \end{gather*} Applying the construction, we obtain quantitative linear independence over $\mathbb Q$ of the numbers in the following collections: $1$, $\zeta_q(1)=L_1(1;q)$, $\zeta_{q^2}(1)$, and $1$, $\zeta_q(1)$, $\zeta_q(2)=L_2(1;q)$ for $q=1/p$, $p\in\mathbb Z\setminus\{0,\pm1\}$.

[1] W. Van Assche, “Little $q$-Legendre polynomials and irrationality of certain Lambert series”, Ramanujan J., 5:3 (2001), 295–310 | DOI | MR | Zbl

[2] J.-P. Bézivin, “Indépendence linéaire des valeurs des solutions transcendantes de certaines équations fonctionelles”, Manuscripta Math., 61 (1988), 103–129 | DOI | MR | Zbl

[3] P. Borwein, “On the irrationality of $\sum(1/(q^n+r))$”, J. Number Theory, 37 (1991), 253–259 | DOI | MR | Zbl

[4] P. Bundschuh, K. Väänänen, “Arithmetical investigations of a certain infinite product”, Compositio Math., 91 (1994), 175–199 | MR | Zbl

[5] P. Bundschuh, K. Väänänen, “Linear independence of $q$-analogues of certain classical constants”, Results in Math., 2005 (to appear) | MR

[6] P. Bundschuh, W. Zudilin, “Rational approximations to a $q$-analogue of $\pi$ and some other $q$-series”, A 70th birthday conference in honour of Wolfgang M. Schmidt (November 2003, Vienna), Springer-Verlag, Berlin, 2005 (to appear) | MR

[7] P. Erdős, “On arithmetical properties of Lambert series”, J. Indiana Math. Soc., 12 (1948), 63–66 | MR | Zbl

[8] G. Gasper, M. Rahman, Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, 35, Cambridge Univ. Press, Cambridge, 1990 | MR | Zbl

[9] M. Hata, “Rational approximations to $\pi$ and some other numbers”, Acta Arith., 63:4 (1993), 335–349 | MR | Zbl

[10] T. Matalo-aho, K. Väänänen, W. Zudilin, “New irrationality measures for $q$-logarithms”, Math. Comput., 2004, submitted

[11] K. Postelmans, W. Van Assche, Irrationality of $\zeta_q(1)$ and $\zeta_q(2)$, Manuscript, 2004

[12] W. Zudilin, “On the irrationality measure for a $q$-analogue of $\zeta(2)$”, Mat. Sb., 193:8 (2002), 49–70 | MR | Zbl

[13] W. Zudilin, “Diophantine problems for $q$-zeta values”, Mat. Zametki, 72:6 (2002), 936–940 | MR | Zbl

[14] W. Zudilin, “Heine's basic transform and a permutation group for $q$-harmonic series”, Acta Arith., 111:2 (2004), 153–164 | DOI | MR | Zbl