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Transactions of the American Mathematical Society

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Rational approximations to the dilogarithm

Author: Masayoshi Hata
Journal: Trans. Amer. Math. Soc. 336 (1993), 363-387
MSC: Primary 11J82; Secondary 11J72
MathSciNet review: 1147401
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Abstract: The irrationality proof of the values of the dilogarithmic function ${L_2}(z)$ at rational points $z = 1/k$ for every integer $k \in ( - \infty , - 5] \cup [7,\infty )$ is given. To show this we develop the method of Padé-type approximations using Legendre-type polynomials, which also derives good irrationality measures of ${L_2}(1/k)$. Moreover, the linear independence over ${\mathbf {Q}}$ of the numbers $1$, $\log (1 - 1/k)$, and ${L_2}(1/k)$ is also obtained for each integer $k \in ( - \infty , - 5] \cup [7,\infty )$ .

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Keywords: Dilogarithm, irrationality measure, Padé approximation
Article copyright: © Copyright 1993 American Mathematical Society