Abstract
In this chapter, we define and study a generalization of Bernoulli numbers referred to as poly-Bernoulli numbers, which is a different generalization than the generalized Bernoulli numbers introduced in Chap. 4.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
John Wilson (born on August 6, 1741 in Applethwaite, England—died on October 18, 1793 in Kendal, England).
- 2.
Harry Schultz Vandiver (born on October 21, 1882 in Philadelphia, USA, died on January 9, 1973 in Austin, USA).
- 3.
The following proof, which greatly simplifies the original proof in [10], is due to Hiroyuki Ochiai . The authors would like to thank him for providing this simple proof.
References
Arakawa, T., Kaneko, M.: Multiple zeta values, poly-Bernoulli numbers, and related zeta functions. Nagoya Math. J. 153, 189–209 (1999)
Arakawa, T., Kaneko, M.: On poly-Bernoulli numbers. Comment. Math. Univ. St. Pauli 48, 159–167 (1999)
Brewbaker, C.: Lonesum (0, 1)-matrices and poly-Bernoulli numbers of negative index. Master’s thesis, Iowa State University (2005)
Ireland, K., Rosen, M.: A Classical Introduction to Modern Number Theory, 2nd edn. Graduate Texts in Mathematics, vol. 84. Springer (1990)
Kaneko, M.: Poly-Bernoulli numbers. J. Th. Nombre Bordeaux 9, 199–206 (1997)
Kaneko, M.: Multiple zeta values. Sugaku Expositions 18(2), 221–232 (2005)
Launois, S.: Combinatorics of \(\mathcal{H}\)-primes in quantum matrices. J. Algebra 309(1), 139–167 (2007)
Nörlund, N.E.: Vorlesungen über Differenzenrechnung. Chelsea Publication, New York (1954). (First edition, Springer, Berlin, 1924)
Vandiver, H.S.: On developments in an arithmetic theory of the Bernoulli and allied numbers. Scripta Math. 25, 273–303 (1961)
Waldschmidt, M.: Valeurs zêta multiples. Une introduction. Colloque International de Théorie des Nombres (Talence, 1999). J. Théor. Nombres Bordeaux 12(2), 581–595 (2000)
Zagier, D.: Values of zeta functions and their applications, in ECM volume. Progress Math. 120, 497–512 (1994)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer Japan
About this chapter
Cite this chapter
Ibukiyama, T., Kaneko, M. (2014). Poly-Bernoulli Numbers. In: Bernoulli Numbers and Zeta Functions. Springer Monographs in Mathematics. Springer, Tokyo. https://doi.org/10.1007/978-4-431-54919-2_14
Download citation
DOI: https://doi.org/10.1007/978-4-431-54919-2_14
Published:
Publisher Name: Springer, Tokyo
Print ISBN: 978-4-431-54918-5
Online ISBN: 978-4-431-54919-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)