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Article Dans Une Revue Journal of Number Theory Année : 2015

Inverse binomial series and values of Arakawa-Kaneko zeta functions

Résumé

In this article, we present a variety of evaluations of series of polylogarithmic nature. More precisely, we express the special values at positive integers of two classes of zeta functions of Arakawa-Kaneko-type by means of certain inverse binomial series involving harmonic sums which appeared fifteen years ago in physics in relation with the Feynman diagrams. In some cases, these series may be explicitly evaluated in terms of zeta values and other related numbers. Incidentally, this connection allows us to deduce new identities for the constant $C= \sum_{n\geq 1} \frac{1}{(2n)^3}(1+\frac13 + \dots + \frac{1}{2n-1})$ considered by S. Ramanujan in his notebooks.
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Dates et versions

hal-00995770 , version 1 (26-05-2014)
hal-00995770 , version 2 (01-09-2014)
hal-00995770 , version 3 (10-09-2014)
hal-00995770 , version 4 (22-09-2014)
hal-00995770 , version 5 (12-12-2014)

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  • HAL Id : hal-00995770 , version 5

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Marc-Antoine Coppo, Bernard Candelpergher. Inverse binomial series and values of Arakawa-Kaneko zeta functions. Journal of Number Theory, 2015, 150, pp.98-119. ⟨hal-00995770v5⟩
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