Functiones et Approximatio Commentarii Mathematici

Symmetric q-Bernoulli numbers and polynomials

Hédi Elmonser

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In this work we are interested by giving a new $q$-analogue of Bernoulli numbers and polynomials which are symmetric under the interchange $q\leftrightarrow q^{-1}$ and deduce some important relations of them. Also, we deduce a $q$-analogue of the Euler-Maclaurin formulas.

Article information

Funct. Approx. Comment. Math., Volume 56, Number 2 (2017), 181-193.

First available in Project Euclid: 27 January 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 33D05: $q$-gamma functions, $q$-beta functions and integrals

q-Bernoulli symmetric


Elmonser, Hédi. Symmetric q-Bernoulli numbers and polynomials. Funct. Approx. Comment. Math. 56 (2017), no. 2, 181--193. doi:10.7169/facm/1603.

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  • G.E. Andrews, R. Askey and R. Roy, Special Functions, Cambridge University Press. Cambridge (1999).
  • K. Brahim and Y. Sidomou, On Some Symmetric q-Special Functions, Le Mathematiche LXVIII (2013)-Fasc.II, 107–122.
  • L. Carlitz, q-Bernoulli numbers and polynomials, Duke Math. J. 15 (1948), 987–100.
  • G. Dattoli and A. Torre, Symmetric q-Bessel functions, Le Mathematiche 51 (1996), no. 1, 153–167.
  • G. Gasper and M. Rahman, Basic Hypergeometric Series, 2nd Edition, (2004), Encyclopedia of Mathematics and Its Applications, 96, Cambridge University Press, Cambridge.
  • A.S. Hegazi and M. Mansour, A Note on q-Bernoulli numbers and polynomials, Journal of Nonlinear Mathematical Physics 13 (2006), no. 1, 9–18.
  • F.H. Jackson, On a $q$-Definite Integrals, Quarterly Journal of Pure and Applied Mathematics 41 (1910), 193–203.
  • V.G. Kac and P. Cheung, Quantum Calculus, Universitext, Springer-Verlag, New York, (2002).
  • T. Kim, Non Archimeden q-integrals associated with multiple Changhee q-Bernoulli polynomials, Russ. J. Math. Phys. 10 (2003), 91–98.
  • T. Kim and S.H. Rim, Generalized Carlitz's q-Bernoulli Numbers in the p-adic number field, Adv. Stud. Contemp Math. 2 (2000), 9–19.
  • T. Kim, L.C. Jang, S.H. Rim, and H.K. Pak, On the twisted q-Zeta functions and q-Bernoulli polynomials, Far East J. Applied Math. 13 (2003), 13–21.
  • H.T. Koelink and T.H. Koornwinder, $q$-Special Functions, a Tutorial, in Deformation Theory and Quantum Groups with Applications to Mathematical Physics, Contemp. Math. 134, Editors: M. Gerstenhaber and J. Stasheff, J. Amer. Math. Soc., Providence, (1992), 141–142.
  • R. Koelink and R.F. Swarttouw, The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue, Report 94-05, Technical University Delft (1994).
  • T.H. Koornwinder, Special functions and q-commuting variables, Fields Institue Communications 14 (1997), 131–166.
  • B.A. Kupershmidt, Reflection symmetries of q-Bernoulli polynomials, J. Nonlinear Math. Phys. 12 (2005), 412–422.
  • D.S. McAnally, q-exponential and q-gamma functions I.q-exponential functions, J. Math. Phys. 36 (1995), no. 1, 546–573.
  • D.S. McAnally, q-exponential and q-gamma functions II.q-gamma functions, J. Math. Phys. 36 (1995), no. 1, 574–595.
  • E.D. Rainville, Special functions, The Macmillan company, (1965).