Some Properties and Identities of Bernoulli and Euler Polynomials Associated with p-adic Integral on Zp

and Applied Analysis 3 2. Some Identities of Bernoulli and Euler Polynomials By 1.4 , we get Ek ( x y ) k ∑ j 0 ( k j ) yk−jEj x , for ∈ Z . 2.1 From 2.1 , we note that Ek ( x y ) k ∑ j 0 ( k j ) yk−jEj x y k ∑ j 1 k j ( k − 1 j − 1 ) yk−jEj x . 2.2 Thus, we have k−1 ∑ j 0 ( k − 1 j ) yk−1−j Ej 1 x j 1 Ek ( x y ) − y k . 2.3 Replacing k by k 1 in 2.3 , we obtain the following proposition. Proposition 2.1. For k ∈ Z , one has k ∑ j 0 ( k j ) yk−j Ej 1 x j 1 Ek 1 ( x y ) − y 1 k 1 . 2.4 Let us replace y by −y in Proposition 2.1. Then we have k ∑ j 0 ( k j ) −1 k−jyk−j Ej 1 x j 1 Ek 1 ( x − y − −1 k yk 1 k 1 . 2.5