Glasnik Matematicki, Vol. 48, No. 2 (2013), 249-263.

THE ARAKAWA-KANEKO ZETA FUNCTION AND POLY-BERNOULLI POLYNOMIALS

Yoshinori Hamahata

Institute for Teaching and Learning, Ritsumeikan University, 1-1-1 Nojihigashi, Kusatsu, Shiga 525-8577, Japan
e-mail: hamahata@fc.ritsumei.ac.jp


Abstract.   The purpose of this paper is to introduce a generalization of the Arakawa-Kaneko zeta function and investigate their special values at negative integers. The special values are written as the sums of products of Bernoulli and poly-Bernoulli polynomials. We establish the basic properties for this zeta function and their special values.

2010 Mathematics Subject Classification.   11B68, 11M32.

Key words and phrases.   Arakawa-Kaneko zeta function, Bernoulli numbers and polynomials, poly-Bernoulli numbers and polynomials.


Full text (PDF) (free access)

DOI: 10.3336/gm.48.2.03


References:

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

  2. T. Arakawa and M. Kaneko, On poly-Bernoulli numbers, Comment. Math. Univ. St. Paul. 48 (1999), 159-167.
    MathSciNet    

  3. A. Bayad and Y. Hamahata, Polylogarithms and poly-Bernoulli polynomials, Kyushu J. Math. 65 (2011), 15-24.
    MathSciNet     CrossRef

  4. K.-W. Chen, Sums of products of generalized Bernoulli polynomials, Pacific J. Math. 208 (2003), 39-52.
    MathSciNet     CrossRef

  5. M.-A. Coppo and B. Candelpergher, The Arakawa-Kaneko zeta function, Ramanujan J. 22 (2010), 153-162.
    MathSciNet     CrossRef

  6. K. Dilcher, Sums of products of Bernoulli numbers, J. Number Theory 60 (1996), 23-41.
    MathSciNet     CrossRef

  7. K. Kamano, Sums of products of Bernoulli numbers, including poly-Bernoulli numbers, J. Integer Seq. 13 (2010), Article 10.5.2.
    MathSciNet    

  8. M. Kaneko, Poly-Bernoulli numbers, J. Théor. Nombres Bordeaux 9 (1997), 221-228.
    CrossRef

Glasnik Matematicki Home Page