Symmetryp-adic invariant integral on ℤpfor Bernoulli and Euler polynomials

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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 .

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