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Duality Formulas for Arakawa–Kaneko Zeta Values and Related Variants

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Abstract

In this paper, we present some new identities for multiple polylogarithm functions by using the methods of iterated integral computations of logarithm functions. Then, by applying these formulas obtained, we establish several duality formulas for Arakawa–Kaneko zeta values and Kaneko–Tsumura \(\eta \)-values. At the end of the paper, we study a variant of Kaneko–Tsumura \(\eta \)-function with r-complex variables and establish two formulas about the values of this variant; these two formulas were proved previously by Yamamoto.

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References

  1. Arakawa, T., Kaneko, M.: Multiple zeta values, poly-Bernoulli numbers, and related zeta functions. Nagoya Math. J. 153, 189–209 (1999)

    Article  MathSciNet  Google Scholar 

  2. Bayad, A., Hamahata, Y.: Arakawa–Kaneko \(L\)-functions and generalized poly-Bernoulli polynomials. J. Number Theory 131, 1020–1036 (2011)

    Article  MathSciNet  Google Scholar 

  3. Chen, K.-W.: Generalized Arakawa–Kaneko zeta functions. Integral Transforms Spec. Funct. 30(4), 282–300 (2019)

    Article  MathSciNet  Google Scholar 

  4. Coppo, M.A., Candelpergher, B.: The Arakawa–Kaneko zeta function. Ramanujan J. 22, 153–162 (2010)

    Article  MathSciNet  Google Scholar 

  5. Coppo, M.A., Candelpergher, B.: Inverse binomial series and values of Arakawa–Kaneko zeta functions. J. Number Theory 150, 98–119 (2015)

    Article  MathSciNet  Google Scholar 

  6. Hoffman, M.E.: Multiple harmonic series. Pac. J. Math. 152, 275–290 (1992)

    Article  MathSciNet  Google Scholar 

  7. Ito, T.: On analogues of Arakawa–Kaneko zeta functions of Mordell–Tornheim type. Commentarii Mathematici Universitatis Sancti Pauli (2016). arXiv:1603.04145v1

  8. Kaneko, M., Tsumura, H.: Multi-poly-Bernoulli numbers and related zeta functions. Nagoya Math. J. 232, 19–54 (2018)

    Article  MathSciNet  Google Scholar 

  9. Kaneko, M., Tsumura, H.: Zeta functions connecting multiple zeta values and poly-Bernoulli numbers. Adv. Stud. Pure Math. 84, 181–204 (2020)

    Article  Google Scholar 

  10. Kaneko, M., Yamamoto, S.: A new integral-series identity of multiple zeta values and regularizations. Sel. Math. 24, 2499–2521 (2018)

    Article  MathSciNet  Google Scholar 

  11. Kawasaki, N., Ohno, Y.: Combinatorial proofs of identities for special values of Arakawa–Kaneko multiple zeta functions. Kyushu J. Math. 72, 215–222 (2018)

    Article  MathSciNet  Google Scholar 

  12. Kuba, M.: On functions of Arakawa and Kaneko and multiple zeta values. Appl. Anal. Discrete Math. 4, 45–53 (2010)

    Article  MathSciNet  Google Scholar 

  13. Xu, C.: Multiple zeta values and Euler sums. J. Number Theory 177, 443–478 (2017)

    Article  MathSciNet  Google Scholar 

  14. Yamamoto, S.: Multiple zeta functions of Kaneko–Tsumura type and their values at positive integers. arXiv: 1607.01978v1

  15. Young, P.T.: Symmetries of Bernoulli polynomial series and Arakawa–Kaneko zeta functions. J. Number Theory 143, 142–161 (2014)

    Article  MathSciNet  Google Scholar 

  16. Young, P.T.: The \(p\)-adic Arakawa–Kaneko zeta functions and \(p\)-adic Lerch transcendent. J. Number Theory 155, 13–35 (2015)

    Article  MathSciNet  Google Scholar 

  17. Zagier, D.: Values of zeta functions and their applications. In: First European Congress of Mathematics, Volume II, vol. 120, pp. 497–512. Birkhauser, Boston (1994)

  18. Zhao, J.: Multiple Zeta Functions, Multiple Polylogarithms and Their Special Values, Series on Number Theory and Its Applications, vol. 12. World Scientific Publishing Co. Pte. Ltd., Hackensack (2016)

    Book  Google Scholar 

Download references

Acknowledgements

The author would like to thank the anonymous referee for his/her valuable comments and suggestions, and express deep gratitude to Professors Masanobu Kaneko and Jianqiang Zhao for valuable discussions. The author is supported by the Scientific Research Foundation for Scholars of Anhui Normal University and the University Natural Science Research Project of Anhui Province (Grant No. KJ2020A0057).

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  1. School of Mathematics and Statistics, Anhui Normal University, Wuhu, 241000, People’s Republic of China

    Ce Xu

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  1. Ce Xu
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Correspondence to Ce Xu.

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Communicated by Emrah Kilic.

Dedicated to Professor Masanobu Kaneko on the occasion of his 60th birthday.

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Xu, C. Duality Formulas for Arakawa–Kaneko Zeta Values and Related Variants. Bull. Malays. Math. Sci. Soc. 44, 3001–3018 (2021). https://doi.org/10.1007/s40840-021-01099-7

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  • DOI: https://doi.org/10.1007/s40840-021-01099-7

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