Approximations to q-logarithms and q-dilogarithms, with applications to q-zeta values期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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Approximations to q-logarithms and q-dilogarithms, with applications to q-zeta values

Authors:

W Zudilin

Institution:

(1) Department of Mechanics and Mathematics, Moscow Lomonosov State University, Moscow, Russia

Abstract:

We construct simultaneous rational approximations to q-series L₁(x₁; q) and L₁(x₂; q) and, if x = x₁ = x₂, to series L₁(x; q) and L₂(x; q), where

$\begin{gathered} L_1 (x;q) = \sum\limits_{n = 1}^\infty {\frac{{(xq)^n }}{{1 - q^n }}} = \sum\limits_{n = 1}^\infty {\frac{{xq^n }}{{1 - xq^n }}} , \hfill \\ L_2 (x;q) = \sum\limits_{n = 1}^\infty {\frac{{n(xq)^n }}{{1 - q^n }}} = \sum\limits_{n = 1}^\infty {\frac{{xq^n }}{{(1 - xq^n )^2 }}} . \hfill \\ \end{gathered}$

. Applying the construction, we obtain quantitative linear independence over ℚ of the numbers in the following collections: 1, ζ_q(1) = L₁(1; q), $\zeta _{q^2 }$ and 1, ζ_q(1), ζ_q(2) = L₂(1; q) for q = 1/p, p ε ℤ \ {0,±1}. Bibliography: 14 titles. Published in Zapiski Nauchnykh Seminarov POMI, Vol. 322, 2005, pp. 107–124.

Keywords:

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