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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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A Note on Bernoulli Numbers and Shintani Generalized Bernoulli Polynomials
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by Minking Eie PDF
Trans. Amer. Math. Soc. 348 (1996), 1117-1136 Request permission

Abstract:

Generalized Bernoulli polynomials were introduced by Shintani in 1976 in order to express the special values at non-positive integers of Dedekind zeta functions for totally real numbers. The coefficients of such polynomials are finite combinations of products of Bernoulli numbers which are difficult to get hold of. On the other hand, Zagier was able to get the explicit formula for the special values in cases of real quadratic number fields. In this paper, we shall improve Shintani’s formula by proving that the special values can be determined by a finite set of polynomials. This provides a convenient way to evaluate the special values of various types of Dedekind functions. Indeed, a much broader class of zeta functions considered by the author [Minking Eie, The special values at negative integers of Dirichlet series associated with polynomials of several variables, Proceedings of A. M. S. 119 (1993), 51–61] admits a similar formula for its special values. As a consequence, we are able to find infinitely many identities among Bernoulli numbers through identities among zeta functions. All these identities are difficult to prove otherwise.
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Additional Information
  • Minking Eie
  • Affiliation: Institute of Applied Mathematics, National Chung Cheng University, Ming- Hsiung, Chia-Yi 621, Taiwan, Republic of China.
  • Email: mkeie@math.ccu.edu.tw
  • Received by editor(s): January 24, 1994
  • Received by editor(s) in revised form: March 2, 1995
  • Additional Notes: This work was supported by the Department of Mathematics, National Chung Cheng University, and by the National Science Foundation of Taiwan, Republic of China.
  • © Copyright 1996 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 348 (1996), 1117-1136
  • MSC (1991): Primary 11M41
  • DOI: https://doi.org/10.1090/S0002-9947-96-01479-1
  • MathSciNet review: 1321572