A Note on Bernoulli Numbers and Shintani

Generalized Bernoulli Polynomials

Author:
Minking Eie

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

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Abstract | References | Similar Articles | Additional Information

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 [4] 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.

**1.**B. C. Berndt,*Ramanujan's notebooks*, Part I and Part II, Springer-Verlag, 1985 and 1989. MR**86c:01062**; MR**90b:01039****2.**B. C. Carlson,*Special functions of applied mathematics*, Academic Press, 1977. MR**58:28707****3.**Minking Eie,*On a Dirichlet series associated with a polynomial*, Proceedings of A. M. S.**99**(1990), 583--590. MR**91m:11071****4.**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. MR**93h:11082****5.**G. van der Geer,*Hilbert modular surface*, Springer-Verlag, 1988. MR**89c:11073****6.**T. Shintani,*On evaluation of zeta functions of totally real algebraic number fields at non-positive integers*, J. Fac. Sci. University of Tokyo**23**(1976), 393--417. MR**55:266****7.**H. Rademacher,*Topics in analytic number theory*, Springer-Verlag, 1973. MR**51:358****8.**C. L. Siegel,*Lectures on advanced analytic number theory*, Tata Institute of Fundamental Research, Bombay, 1965. MR**41:6760****9.**L. C. Washington,*Introduction to cyclotomic fields*, Springer-Verlag, 1982. MR**85g:11001****10.**D. Zagier,*Valeurs des fontions zêta des corps quadratiques réels aux entiers négatifs*, Astérisque**41-42**(1977), 135--151. MR**56:316**

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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

DOI:
https://doi.org/10.1090/S0002-9947-96-01479-1

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.

Article copyright:
© Copyright 1996
American Mathematical Society