Journal of the Mathematical Society of Japan

On an integral representation of special values of the zeta function at odd integers

Takashi ITO

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Abstract

An integral representation of the p -series of odd p is shown; n = 1 1 n 2 p + 1 = ( - 1 ) p ( 2 π ) 2 p ( 2 p ) ! 0 1 B 2 p ( t ) log ( sin π t ) d t ( p = 1 , 2 , ) , where B 2 p ( t ) is a Bernoulli polynomial of degree 2 p . As a consequence of this we have n = 1 1 n 2 p + 1 = ( - 1 ) p ( 2 π ) 2 p ( 2 p ) ! 2 k = 0 p 2 p 2 k B 2 p - 2 k 1 2 b 2 k , where b 2 k = 0 1 2 t 2 k log ( cos π t ) d t , k = 0 , 1 , , p .

Article information

Source
J. Math. Soc. Japan, Volume 58, Number 3 (2006), 681-691.

Dates
First available in Project Euclid: 23 August 2006

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1156342033

Digital Object Identifier
doi:10.2969/jmsj/1156342033

Mathematical Reviews number (MathSciNet)
MR2254406

Zentralblatt MATH identifier
1102.11014

Subjects
Primary: 11B68: Bernoulli and Euler numbers and polynomials
Secondary: 42A85: Convolution, factorization

Keywords
Bernoulli polynomials convolutions Fourier series

Citation

ITO, Takashi. On an integral representation of special values of the zeta function at odd integers. J. Math. Soc. Japan 58 (2006), no. 3, 681--691. doi:10.2969/jmsj/1156342033. https://projecteuclid.org/euclid.jmsj/1156342033


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References

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