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On q-analogues of quadratic Euler sums

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Abstract

In this paper we study q-analogues of Euler sums and present a new family of identities by using the method of Jackson q-integral representations of series. We then apply it to obtain a family of identities relating quadratic Euler sums to linear sums and q-polylogarithms. Furthermore, we use certain stuffle products to evaluate several q-series with q-harmonic numbers. Some interesting new results and illustrative examples are considered. Finally, if q tends to 1, we obtain some explicit relations for the classical Euler sums.

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Acknowledgements

Funding was provided by National Natural Science Foundation of China (Grant No. 11471245).

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Correspondence to Ce Xu.

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The first author is supported by the National Natural Science Foundation of China (Grant No. 11471245) and the Natural Science Foundation of Shanghai (Grant No. 14ZR1443500). We thank the anonymous referee for suggestions which led to improvements in the exposition.

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Li, Z., Xu, C. On q-analogues of quadratic Euler sums. Period Math Hung (2020) doi:10.1007/s10998-020-00312-7

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Keywords

  • q-harmonic number
  • q-Euler sum
  • q-polylogarithm function

Mathematics Subject Classification

  • 05A30
  • 65B10
  • 33D05
  • 11M99
  • 11M06
  • 11M32