Symmetry <i>p</i> -adic invariant integral on ℤ _<i>p</i> for Bernoulli and Euler polynomials

Abstract The main purpose of this paper is to investigate several further interesting properties of symmetry for the p-adic invariant integrals on ℤ p . From these symmetry, we can derive many interesting recurrence identities for Bernoulli and Euler polynomials. Finally we introduce the new concept of symmetry of fermionic p-adic invariant integral on ℤ p . By using this symmetry of fermionic p-adic invariant integral on ℤ p , we will give some relations of symmetry between the power sum polynomials and Euler numbers. The relation between the q-Bernoulli polynomials and q-Dedekind type sums which discussed in Y. Simsek (q-Dedekind type sums related to q-zeta function and basic L-series, J. Math. Anal. Appl. 318 (2006), pp. 333–351) can be also derived by using the properties of symmetry of fermionic p-adic integral on ℤ p . Keywords: q-Bernoulli numbers q-Volkenborn integrals q-Euler numbers q-stirling numbers2000 Ams Classification number: 11B6811S80 Acknowledgement This paper is supported by Jangjeon Research Institute for Mathematical Science (JRIMS-10R-2001).