Advertisement

Nonlinear Identities for Bernoulli and Euler Polynomials

  • Karl DilcherEmail author
Conference paper
  • 48 Downloads
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 313)

Abstract

It is shown that a certain nonlinear expression for Bernoulli polynomials, related to higher-order convolutions, can be evaluated as a product of simple linear polynomials with integer coefficients. The proof involves higher-order Bernoulli polynomials. A similar result for Euler polynomials is also obtained, and identities for Bernoulli and Euler numbers follow as special cases.

Keywords

Bernoulli polynomials Bernoulli numbers Euler polynomials Euler numbers convolution identities 

References

  1. 1.
    Agoh, T., Dilcher, K.: Higher-order convolutions for Bernoulli and Euler polynomials. J. Math. Anal. Appl. 419(2), 1235–1247 (2014)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bailey, D.H., Borwein, D., Borwein, J.M.: On Eulerian log-gamma integrals and Tornheim-Witten zeta functions. Ramanujan J. 36(1–2), 43–68 (2015)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bailey, D.H., Borwein, J.M., Crandall, R.: Computation and theory of extended Mordell-Tornheim-Witten sums. Math. Comput. 83(288), 1795–1821 (2014)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bailey, D.H., Borwein, J.M.: Computation and theory of Mordell-Tornheim-Witten sums II. J. Approx. Theory 197, 115–140 (2015)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bailey, D.H., Borwein, J.M.: Computation and experimental evaluation of Mordell-Tornheim-Witten sum derivatives. Exp. Math. 27(3), 370–376 (2018)Google Scholar
  6. 6.
    Borwein, J.M.: Hilbert’s inequality and Witten’s zeta-function. Am. Math. Mon. 115(2), 125–137 (2008)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Borwein, J.M., Dilcher, K.: Derivatives and fast evaluation of the Tornheim zeta function. Ramanujan J. 45(2), 413–432 (2018)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Dilcher, K.: Sums of products of Bernoulli numbers. J. Number Theory 60(1), 23–41 (1996)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Dilcher, K., Tomkins, H.: Derivatives and special values of higher-order Tornheim zeta functions. In preparationGoogle Scholar
  10. 10.
    Dilcher, K., Vignat, C.: General convolution identities for Bernoulli and Euler polynomials. J. Math. Anal. Appl. 435(2), 1478–1498 (2016)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Gould, H.W.: Combinatorial Identities, revised edition, Gould Publications, West Virginia, Morgantown (1972)Google Scholar
  12. 12.
    Huang, I.-C., Huang, S.-Y.: Bernoulli numbers and polynomials via residues. J. Number Theory 76(2) 178–193 (1999)Google Scholar
  13. 13.
    Matiyasevich, Y.: Identities with Bernoulli numbers. http://logic.pdmi.ras.ru/~yumat/personaljournal/identitybernoulli/bernulli.htm
  14. 14.
    Miki, H.: A relation between Bernoulli numbers. J. Number Theory 10(3), 297–302 (1978)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Milne-Thomson, L.M.: The Calculus of Finite Differences. Macmillan, London (1951)zbMATHGoogle Scholar
  16. 16.
    Petojević, A.: New sums of products of Bernoulli numbers. Integr. Transform. Spec. Funct. 19(1–2), 105–114 (2008)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Riordan, J.: Combinatorial Identities. Wiley, New York (1968)zbMATHGoogle Scholar
  18. 18.
    Tomkins, H.: An exploration of multiple zeta functions. Honours Thesis, Dalhousie University (2016)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsDalhousie UniversityHalifaxCanada

Personalised recommendations