Rocky Mountain Journal of Mathematics

Results on values of Barnes polynomials

Abdelmejid Bayad and Taekyun Kim

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Abstract

In this paper, we investigate rationality of the Barnes numbers, and we find explicit good bounds for their denominators. In addition, we give Fourier expansion of Barnes polynomials and, from this study, we connect generalized Barnes numbers to values of Dirichlet $L$-function at non-negative integers.

Article information

Source
Rocky Mountain J. Math., Volume 43, Number 6 (2013), 1857-1869.

Dates
First available in Project Euclid: 25 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1393336659

Digital Object Identifier
doi:10.1216/RMJ-2013-43-6-1857

Mathematical Reviews number (MathSciNet)
MR3178446

Zentralblatt MATH identifier
1345.11013

Subjects
Primary: 11B68: Bernoulli and Euler numbers and polynomials 11B83: Special sequences and polynomials 11B99: None of the above, but in this section 11M32: Multiple Dirichlet series and zeta functions and multizeta values

Keywords
Bernoulli numbers and polynomials Barnes numbers and polynomials Von Staudt theorem Dirichlet L -function. The present research has been conducted by a Research Grant of Kwangwoon University in 2011

Citation

Bayad, Abdelmejid; Kim, Taekyun. Results on values of Barnes polynomials. Rocky Mountain J. Math. 43 (2013), no. 6, 1857--1869. doi:10.1216/RMJ-2013-43-6-1857. https://projecteuclid.org/euclid.rmjm/1393336659


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References

  • T. Agoh and K. Dilcher, Higher-order recurrences for Bernoulli numbers, J. Num. Theor. 129 (2009), 1837-1847.
  • A. Bayad and Y. Simsek, Dedekind sums involving Jacobi modular forms and special values of Barnes zeta functions, Annal. Inst. Fourier 61 (2011).
  • –––, Identities on values at negative integers of twisted Barnes zeta functions, preprint.
  • K. Dilcher and L. Louise, Arithmetic properties of Bernoulli-Padé numbers and polynomials, J. Num. Theor. 92 (2002), 330-347.
  • E. Friedman and S. Ruijsenaars, Shintani-Barnes zeta and gamma functions, Adv. Math. 187 (2004), 362-395.
  • K. Ireland and M. Rosen, A classical introduction to modern number theory, Springer, New York, 1982.
  • K. Katayama, Barne's multiple function and Apostol's generalized Dedekind sum, Tokyo J. Math. 27 2004, 57-74.
  • T. Kim, Non-Archimedean $q$-integrals associated with multiple Changhee $q$-Bernoulli polynomials, Russ. J. Math. Phys. 10 (2003), 91-98.
  • –––, On a $p$-adic interpolation function for the $q$-extension of the generalized Bernoulli polynomials and its derivative, Discr. Math. 309 (2009), 1593-1602.
  • –––, On Euler-Barnes multiple zeta functions, Russ. J. Math. Phys. 10 (2003), 261-267.
  • L. Kronecker, Zur Theorie der elliptischen Modulfunktionen 4 (1929), pages 347-495.
  • L.M. Navas, F.J. Ruiz and J.L. Varona, The Möbius inversion formula for Fourier series applied to Bernoulli and Euler polynomials, J. Approx. Theor. 163 (2011), 22-40.
  • K. Ota, On Kummer-type congruences for derivatives of Barnes multiple Bernoulli polynomials, J. Num. Theor. 92 (2002), 1-36.
  • S. Ruijsenaars, On Barnes' multiple zeta and gamma functions, Adv. Math. 156 (2000), 107-132.
  • Y. Simsek, Generating functions of the twisted Bernoulli numbers and polynomials associated with their interpolation functions, Adv. Stud. Contemp. Math. 16 (2008), 251-278.
  • M. Spreafico, On the Barnes double zeta and Gamma functions, J. Num. Theor. 129 (2009), 2035-2063.
  • L.C. Washington, Introduction to cyclotomic fields, Springer, New York, 1982.
  • A. Weil, Elliptic functions according to Eisenstein and Kronecker, Ergeb. Math. 88, Springer-Verlag, Berlin, 1976. \noindentstyle