Abstract
For \(s,t,u \in {\mathbb{C}}\), we show rapidly (or globally) convergent series representations of the Tornheim double zeta function T(s, t, u) and (desingularized) symmetric Tornheim double zeta functions. As a corollary, we give a new proof of known results on the values of T(s, s, s) at non-positive integers and the location of the poles of T(s, s, s). Furthermore, we prove that T(s, s, s) can not be written by a polynomial in the form of \(\sum_{k=1}^j c_k \prod_{r=1}^q \zeta^{d_{kr}} (a_{kr} s + b_{kr})\), where \(a_{kr}, b_{kr}, c_k \in {\mathbb{C}}\) and \(d_{kr} \in {\mathbb{Z}}_{\ge 0}\).
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The author would like to thank the referee for a careful reading of the manuscript and valuable comments.
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The author is partially supported by JSPS grant 16K05077.
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Nakamura, T. Rapidly convergent series representations of symmetric Tornheim double zeta functions. Acta Math. Hungar. 165, 397–414 (2021). https://doi.org/10.1007/s10474-021-01189-9
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DOI: https://doi.org/10.1007/s10474-021-01189-9