Journal of the
Korean Mathematical Society JKMS

The study of the identities of symmetry for the Bernoulli polynomials arises from the study of Gauss's multiplication formula for the gamma function. There are many works in this direction. In the sense of $p$-adic analysis, the $q$-Bernoulli polynomials are natural extensions of the Bernoulli and Apostol-Bernoulli polynomials (see the introduction of this paper). By using the $N$-fold iterated Volkenborn integral, we derive serval identities of symmetry related to the $q$-extension power sums and the higher order $q$-Bernoulli polynomials. Many previous results are special cases of the results presented in this paper, including Tuenter's classical results on the symmetry relation between the power sum polynomials and the Bernoulli numbers in [A symmetry of power sum polynomials and Bernoulli numbers, Amer. Math. Monthly {108} (2001), no. 3, 258--261] and D. S. Kim's eight basic identities of symmetry in three variables related to the $q$-analogue power sums and the $q$-Bernoulli polynomials in [Identities of symmetry for $q$-Bernoulli polynomials, Comput. Math. Appl. {60} (2010), no. 8, 2350--2359].

Keywords: $p$-adic analysis, higher order $q$-Bernoulli polynomials, power sums, Volkenborn integral, $q$-extension of the power sums, identities of symmetry

MSC numbers: 11B68, 11S80