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Poly-p-Bernoulli polynomials and generalized Arakawa–Kaneko zeta function

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Abstract

In this paper, we first obtain several properties of poly-p-Bernoulli polynomials. In particular, we achieve some new results for poly-Bernoulli polynomials. We next define a generalization of the Arakawa–Kaneko zeta function associated with poly-p-Bernoulli polynomials, investigate some its particular values, and give asymptotic and series expansions.

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References

  1. M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover Publications, New York, 1974.

    MATH  Google Scholar 

  2. T. Arakawa, T. Ibukiyama, and M. Kaneko, Bernoulli Numbers and Zeta Functions, Springer Monogr. Math., Springer Japan, Tokyo, 2014.

    Book  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  4. T. Arakawa and M. Kaneko, On poly-Bernoulli numbers, Comment. Math. Univ. St. Pauli, 48(2):159–167, 1999.

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  6. A. Bayad and Y. Hamahata, Polylogarithms and poly-Bernoulli polynomials, Kyushu J. Math., 65(1):15–24, 2011.

    Article  MathSciNet  Google Scholar 

  7. K.N. Boyadzhiev, Evaluation of Euler–Zagier sums, Int. J. Math. Math. Sci., 27(7):407–412, 2001.

    Article  MathSciNet  Google Scholar 

  8. K.N. Boyadzhiev, A series transformation formula and related polynomials, Int. J. Math. Math. Sci., 23:3849–3866, 2005.

    Article  MathSciNet  Google Scholar 

  9. K.N. Boyadzhiev, Power series with binomial sums and asymptotic expansions, Int. J. Math. Anal., 8(28):1389–1414, 2014.

    Article  Google Scholar 

  10. A.Z. Broder, The r-Stirling numbers, Discrete Math., 49(3):241–259, 1984.

    Article  MathSciNet  Google Scholar 

  11. M. Cenkci and T. Komatsu, Poly-Bernoulli numbers and polynomialswith a q parameter, J. Number Theory, 152:38–54, 2015.

    Article  MathSciNet  Google Scholar 

  12. M. Cenkci and P.T. Young, Generalizations of poly-Bernoulli and poly-Cauchy numbers, Eur. J. Math., 1:799–828, 2015.

    Article  MathSciNet  Google Scholar 

  13. M.A. Coppo and B. Candelpergher, The Arakawa–Kaneko zeta functions, Ramanujan J., 22:153–162, 2010.

    Article  MathSciNet  Google Scholar 

  14. M.A. Coppo and B. Candelpergher, Inverse binomial series and values of Arakawa–Kaneko zeta functions, J. Number Theory, 150:98–119, 2010.

    Article  MathSciNet  Google Scholar 

  15. A. Dil and V. Kurt, Investigating geometric and exponential polynomials with Euler–Seidel matrices, J. Integer Seq., 14:11.4.6, 2011.

  16. Y. Hamahata, The Arakawa–Kaneko zeta function and poly-Bernoulli polynomials, Glas. Mat., III. Ser., 48(2):249–263, 2013.

    Article  MathSciNet  Google Scholar 

  17. Y. Hamahata and H. Masubuchi, Special multi-poly-Bernoulli numbers, J. Integer Seq., 10:07.4.1, 2007.

    MathSciNet  MATH  Google Scholar 

  18. K. Kamano, Sums of products of bernoulli numbers, including poly-Bernoulli numbers, J. Integer Seq., 13:10.5.2, 2010.

    MathSciNet  MATH  Google Scholar 

  19. M. Kaneko, Poly-Bernoulli numbers, J. Théor. Nombres Bordx., 9:221–228, 1997.

    Article  MathSciNet  Google Scholar 

  20. L. Kargın, Some formulae for products of geometric polynomialswith applications, J. Integer Seq., 20:17.4.4, 2017.

  21. L. Kargın, p-Bernoulli and geometric polynomials, Int. J. Number Theory, 14(2):595–613, 2018.

    Article  MathSciNet  Google Scholar 

  22. L. Kargın and M. Rahmani, A closed formula for the generating function of p-Bernoulli numbers, Quaest. Math., 41(7):975–983, 2018.

    Article  MathSciNet  Google Scholar 

  23. B.C. Kellner, Identities between polynomials related to Stirling and harmonic numbers, Integers, 14:A54, 2014.

    MathSciNet  MATH  Google Scholar 

  24. T. Komatsu, J.L. Ramírez, and V.F. Sirvent, Multi-poly-Bernoulli numbers and polynomials with a q parameter, Lith. Math. J., 57(4):490–505, 2017.

    Article  MathSciNet  Google Scholar 

  25. T. Komatsu and P.T. Young, Generalized Stirling numbers with poly-Bernoulli and poly-Cauchy numbers, Int. J. Number Theory, 14(5):1211–1222, 2018.

    Article  MathSciNet  Google Scholar 

  26. S.D. Lin and H.M. Srivastava, Some families of the Hurwitz–Lerch zeta functions and associated fractional derivative and other integral representations, Appl. Math. Comput., 154(3):725–733, 2004.

    Article  MathSciNet  Google Scholar 

  27. M. Rahmani, On p-Bernoulli numbers and polynomials, J. Number Theory, 157:350–366, 2015.

    Article  MathSciNet  Google Scholar 

  28. S. Roman, The Umbral Calculus, Academic Press, New York, 1984.

    MATH  Google Scholar 

  29. R. Sanchez-Peregrino, Closed formula for poly-Bernoulli numbers, Fibonacci Q., 40:362–364, 2002.

    MathSciNet  MATH  Google Scholar 

  30. Y. Sasaki, On generalized poly-Bernoulli numbers and related L-functions, J. Number Theory, 132(1):156–170,2012.

    Article  MathSciNet  Google Scholar 

  31. A. Sofo, Harmonic sums and integral representations, J. Appl. Anal., 16(2):265–277, 2010.

    Article  MathSciNet  Google Scholar 

  32. A. Sofo and H.M. Srivastava, Identities for the harmonic numbers and binomial coefficients, Ramanujan J., 25(1): 93–113, 2011.

    Article  MathSciNet  Google Scholar 

  33. W. Wang and Y. Lyu, Euler sums and Stirling sums, J. Number Theory, 185:160–193, 2018.

    Article  MathSciNet  Google Scholar 

  34. C. Xu,M. Zhang, and W. Zhu, Some evaluation of harmonic number sums, Integral Transforms Spec. Funct., 27(12): 937–955, 2016.

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  37. P.T. Young, Bernoulli and poly-Bernoulli polynomial convolutions and identities of p-adic Arakawa–Kaneko zeta functions, Int. J. Number Theory, 12(5):1295–1309, 2016.

    Article  MathSciNet  Google Scholar 

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Correspondence to Levent Kargın.

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This work was supported by Research Fund of Akdeniz University. Project Number: FBA-2018-3723.

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Kargın, L. Poly-p-Bernoulli polynomials and generalized Arakawa–Kaneko zeta function. Lith Math J 60, 29–50 (2020). https://doi.org/10.1007/s10986-019-09448-7

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  • DOI: https://doi.org/10.1007/s10986-019-09448-7

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