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Some Relations Between the Riemann Zeta Function and the Generalized Bernoulli Polynomials of Level $m$

Year 2019, Volume: 2 Issue: 4, 188 - 201, 26.12.2019
https://doi.org/10.32323/ujma.602178

Abstract

The main purpose of this paper is to show some relations between the Riemann zeta function and the generalized Bernoulli polynomials of level $m$. Our approach is based on the use of Fourier expansions for the periodic generalized Bernoulli functions of level $m$, as well as  quadrature formulae of Euler-Maclaurin type. Some illustrative examples involving such relations are also given.

Supporting Institution

Decanato de Investigación y Desarrollo, Universidad Simón Bolívar

Project Number

DID-USB (S1-IC-CB-004-17)

References

  • [1] T. M. Apostol, Another elementary proof of Euler’s formula for z (2n), AM. Math. Monthly, 80 (1973), 425-431.
  • [2] T. M. Apostol, An elementary view of Euler’s summation formula, AM. Math. Monthly, 106 (1999), 409-418.
  • [3] T. Arakawa, T. Ibukiyama, M. Kaneko, Bernoulli numbers and Zeta Functions, Springer Monographs in Mathematics, Springer, New York, 2014.
  • [4] R. Ayoub, Euler and the zeta function, AM. Math. Monthly, 81 (1974), 1067-1086.
  • [5] R. Baker, An Introduction to Riemann’s Life, His Mathematics and His Work on the Zeta Function, H. Montgomery, A. Nikeghbali, M. Th. Rassias (editors), Exploring the Riemann Zeta Function: 190 years from Riemann’s Birth, Springer International Publishing AG, Switzerland, 2017, pp. 1-12.
  • [6] B. C. Berndt, A. Straub, Ramanujan’s Formula for z (2n+1), H. Montgomery, A. Nikeghbali, M. Th. Rassias (editors), Exploring the Riemann Zeta Function: 190 years from Riemann’s Birth, Springer International Publishing AG, Switzerland, 2017, pp. 13-14.
  • [7] O. Ciaurri, L. M. Navas, F. J. Ruiz, J. L. Varona, A simple computation of z (2k) by using Bernoulli polynomials and a telescoping series, AM. Math. Monthly, 122 (2015), 444-451.
  • [8] P. J. Davis, P. Rabinowitz, Methods of Numerical Integration, Academic Press Inc., 1984.
  • [9] E. De Amo, M. D´ıaz-Carrillo, J. Fern´andez-S´anchez, Another proof of Euler’s formula for z (2k), Proc. Amer. Math. Soc., 139 (2011), 1441-1444.
  • [10] G. B. Folland, Fourier Analysis and Its Applications, Brooks/Cole Publishing Co., 1992.
  • [11] V. Lampret, The Euler-Maclaurin and Taylor formulas: Twin, elementary derivations, Math. Mag., 74(2) (2001), 109-122.
  • [12] A. Hassen, H. D. Nguyen, Hypergeometric Bernoulli polynomials and Appell sequences, Int. J. Number Theory, 4(5) (2008), 767-774.
  • [13] P. Hernandez-Llanos, Y. Quintana, A. Urieles, About extensions of generalized Apostol-type polynomials, Results Math., 68 (2015), 203-225.
  • [14] F. T. Howard, Some sequences of rational numbers related to the exponential function, Duke Math. J., 34 (1967), 701-716.
  • [15] P. Natalini, A. Bernardini, A generalization of the Bernoulli polynomials, J. Appl. Math., 2003(3) (2003), 155-163.
  • [16] N. E. Nørlund, Vorlesungen ¨uber Differenzenrechnung, Springer-Verlag, Berlin, 1924, (reprinted 1954), (in German).
  • [17] G. M. Phillips, Interpolation and Approximation by Polynomials, Springer-Verlag, New York, 2003.
  • [18] Y. Quintana, W. Ramirez, A. Urieles, On an operational matrix method based on generalized Bernoulli polynomials of level m, Calcolo, 55(3) (2018), 29 pages.
  • [19] Y. Quintana, A. Urieles, Quadrature formulae of Euler-Maclaurin type based on generalized Euler polynomials of level m, Bull. Comput. Appl. Math., 6(2) (2018), 43-64.
  • [20] H. M. Srivastava, H. L. Manocha, A Treatise on Generating Functions, Ellis Horwood Ltd., West Sussex, England, 1984.
  • [21] H. M. Srivastava, J. Choi, Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier, London, 2012.
  • [22] H. M. Srivastava, M. Garg, S. Choudhary, A new generalization of the Bernoulli and related polynomials, Russ. J. Math. Phys., 17(2) (2010), 251-261.
  • [23] R. D. Stuart, Introduction to Fourier Analysis, Methuen & Co. Ltd., London, 1961.
Year 2019, Volume: 2 Issue: 4, 188 - 201, 26.12.2019
https://doi.org/10.32323/ujma.602178

Abstract

Project Number

DID-USB (S1-IC-CB-004-17)

References

  • [1] T. M. Apostol, Another elementary proof of Euler’s formula for z (2n), AM. Math. Monthly, 80 (1973), 425-431.
  • [2] T. M. Apostol, An elementary view of Euler’s summation formula, AM. Math. Monthly, 106 (1999), 409-418.
  • [3] T. Arakawa, T. Ibukiyama, M. Kaneko, Bernoulli numbers and Zeta Functions, Springer Monographs in Mathematics, Springer, New York, 2014.
  • [4] R. Ayoub, Euler and the zeta function, AM. Math. Monthly, 81 (1974), 1067-1086.
  • [5] R. Baker, An Introduction to Riemann’s Life, His Mathematics and His Work on the Zeta Function, H. Montgomery, A. Nikeghbali, M. Th. Rassias (editors), Exploring the Riemann Zeta Function: 190 years from Riemann’s Birth, Springer International Publishing AG, Switzerland, 2017, pp. 1-12.
  • [6] B. C. Berndt, A. Straub, Ramanujan’s Formula for z (2n+1), H. Montgomery, A. Nikeghbali, M. Th. Rassias (editors), Exploring the Riemann Zeta Function: 190 years from Riemann’s Birth, Springer International Publishing AG, Switzerland, 2017, pp. 13-14.
  • [7] O. Ciaurri, L. M. Navas, F. J. Ruiz, J. L. Varona, A simple computation of z (2k) by using Bernoulli polynomials and a telescoping series, AM. Math. Monthly, 122 (2015), 444-451.
  • [8] P. J. Davis, P. Rabinowitz, Methods of Numerical Integration, Academic Press Inc., 1984.
  • [9] E. De Amo, M. D´ıaz-Carrillo, J. Fern´andez-S´anchez, Another proof of Euler’s formula for z (2k), Proc. Amer. Math. Soc., 139 (2011), 1441-1444.
  • [10] G. B. Folland, Fourier Analysis and Its Applications, Brooks/Cole Publishing Co., 1992.
  • [11] V. Lampret, The Euler-Maclaurin and Taylor formulas: Twin, elementary derivations, Math. Mag., 74(2) (2001), 109-122.
  • [12] A. Hassen, H. D. Nguyen, Hypergeometric Bernoulli polynomials and Appell sequences, Int. J. Number Theory, 4(5) (2008), 767-774.
  • [13] P. Hernandez-Llanos, Y. Quintana, A. Urieles, About extensions of generalized Apostol-type polynomials, Results Math., 68 (2015), 203-225.
  • [14] F. T. Howard, Some sequences of rational numbers related to the exponential function, Duke Math. J., 34 (1967), 701-716.
  • [15] P. Natalini, A. Bernardini, A generalization of the Bernoulli polynomials, J. Appl. Math., 2003(3) (2003), 155-163.
  • [16] N. E. Nørlund, Vorlesungen ¨uber Differenzenrechnung, Springer-Verlag, Berlin, 1924, (reprinted 1954), (in German).
  • [17] G. M. Phillips, Interpolation and Approximation by Polynomials, Springer-Verlag, New York, 2003.
  • [18] Y. Quintana, W. Ramirez, A. Urieles, On an operational matrix method based on generalized Bernoulli polynomials of level m, Calcolo, 55(3) (2018), 29 pages.
  • [19] Y. Quintana, A. Urieles, Quadrature formulae of Euler-Maclaurin type based on generalized Euler polynomials of level m, Bull. Comput. Appl. Math., 6(2) (2018), 43-64.
  • [20] H. M. Srivastava, H. L. Manocha, A Treatise on Generating Functions, Ellis Horwood Ltd., West Sussex, England, 1984.
  • [21] H. M. Srivastava, J. Choi, Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier, London, 2012.
  • [22] H. M. Srivastava, M. Garg, S. Choudhary, A new generalization of the Bernoulli and related polynomials, Russ. J. Math. Phys., 17(2) (2010), 251-261.
  • [23] R. D. Stuart, Introduction to Fourier Analysis, Methuen & Co. Ltd., London, 1961.
There are 23 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Yamilet Quintana 0000-0003-1053-0892

Héctor Torres-guzmán This is me 0000-0002-9101-7355

Project Number DID-USB (S1-IC-CB-004-17)
Publication Date December 26, 2019
Submission Date August 5, 2019
Acceptance Date October 14, 2019
Published in Issue Year 2019 Volume: 2 Issue: 4

Cite

APA Quintana, Y., & Torres-guzmán, H. (2019). Some Relations Between the Riemann Zeta Function and the Generalized Bernoulli Polynomials of Level $m$. Universal Journal of Mathematics and Applications, 2(4), 188-201. https://doi.org/10.32323/ujma.602178
AMA Quintana Y, Torres-guzmán H. Some Relations Between the Riemann Zeta Function and the Generalized Bernoulli Polynomials of Level $m$. Univ. J. Math. Appl. December 2019;2(4):188-201. doi:10.32323/ujma.602178
Chicago Quintana, Yamilet, and Héctor Torres-guzmán. “Some Relations Between the Riemann Zeta Function and the Generalized Bernoulli Polynomials of Level $m$”. Universal Journal of Mathematics and Applications 2, no. 4 (December 2019): 188-201. https://doi.org/10.32323/ujma.602178.
EndNote Quintana Y, Torres-guzmán H (December 1, 2019) Some Relations Between the Riemann Zeta Function and the Generalized Bernoulli Polynomials of Level $m$. Universal Journal of Mathematics and Applications 2 4 188–201.
IEEE Y. Quintana and H. Torres-guzmán, “Some Relations Between the Riemann Zeta Function and the Generalized Bernoulli Polynomials of Level $m$”, Univ. J. Math. Appl., vol. 2, no. 4, pp. 188–201, 2019, doi: 10.32323/ujma.602178.
ISNAD Quintana, Yamilet - Torres-guzmán, Héctor. “Some Relations Between the Riemann Zeta Function and the Generalized Bernoulli Polynomials of Level $m$”. Universal Journal of Mathematics and Applications 2/4 (December 2019), 188-201. https://doi.org/10.32323/ujma.602178.
JAMA Quintana Y, Torres-guzmán H. Some Relations Between the Riemann Zeta Function and the Generalized Bernoulli Polynomials of Level $m$. Univ. J. Math. Appl. 2019;2:188–201.
MLA Quintana, Yamilet and Héctor Torres-guzmán. “Some Relations Between the Riemann Zeta Function and the Generalized Bernoulli Polynomials of Level $m$”. Universal Journal of Mathematics and Applications, vol. 2, no. 4, 2019, pp. 188-01, doi:10.32323/ujma.602178.
Vancouver Quintana Y, Torres-guzmán H. Some Relations Between the Riemann Zeta Function and the Generalized Bernoulli Polynomials of Level $m$. Univ. J. Math. Appl. 2019;2(4):188-201.

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