Derivation of Identities Involving Bernoulli and Euler Numbers

Let p be a fixed odd prime. Throughout this paper, Zp,Qp,Cp will, respectively, denote the ring of p-adic integers, the field of p-adic rational numbers, and the completion of algebraic closure of Qp. The p-adic absolute value | |p on Cp is normalized so that |p|p 1/p. Let Z>0 be the set of natural numbers and Z≥0 Z>0 ∪ {0}. As is well known, the Bernoulli polynomials Bn x are defined by the generating function as follows:


Introduction and Preliminaries
Let p be a fixed odd prime. Throughout this paper, Z p , Q p , C p will, respectively, denote the ring of p-adic integers, the field of p-adic rational numbers, and the completion of algebraic closure of Q p . The p-adic absolute value | | p on C p is normalized so that |p| p 1/p. Let Z >0 be the set of natural numbers and Z ≥0 Z >0 ∪ {0}.
As is well known, the Bernoulli polynomials B n x are defined by the generating function as follows: In the special case, x 0, B n 0 B n is referred to as the nth Bernoulli number. That is, the generating function of Bernoulli numbers is given by  where δ k,n is the Kronecker symbol. By 1.1 and 1.2 , we easily get the following: Let UD Z p be the space of uniformly differentiable C p -valued functions on Z p . For f ∈ UD Z p , the bosonic p-adic integral on Z p is defined by cf. 12 . Then it is easy to see that where f 1 x f x 1 and f 0 df x /dx| x 0 . By 1.6 , we have the following:  Especially, for x 1 and y 1, Therefore, from 1.9 , 1.10 , and 1.12 , we can derive the following relation. For n ∈ Z ≥0 , −1 n B n −1 B n 2 n B n 1 n B n δ 1,n n −1 n B n .

1.13
Let f y x y n 1 . By 1.6 , we have the following: x y n 1 dμ y n 1 x n , for n ∈ Z ≥0 . 1.14 By 1.8 and 1.14 , we have the following: Thus, by 1.11 and 1.15 , we have the following identity.
x n 1 n 1 n l 0 n 1 l B l x , for n ∈ Z ≥0 .

1.16
As is well known, the Euler polynomials E n x are defined by the generating function as follows: with the usual convention of replacing E x n by E n x . In the special case, x 0, E n 0 E n is referred to as the nth Euler number. That is, the generating function of Euler numbers is given by Let C Z p be the space of continuous C p -valued functions on Z p . For f ∈ C Z p , the fermionic p-adic integral on Z p is defined by Kim as follows: cf. 9 . Then it is easy to see that where f 1 x f x 1 . By 1.22 , we have the following: From 1.23 , we can derive the Witt's formula for the n-th Euler polynomial as follows: By 1.17 , we have the following: Thus, from 1.19 , 1.20 , and 1.25 , we have the following: By 1.20 , we have the following: Especially, for x 1 and y 1, International Journal of Mathematics and Mathematical Sciences 5 Therefore, from 1.25 , 1.26 , and 1.28 , we can derive the following relations. For n ∈ Z ≥0 , Let f y x y n . By 1.22 , we have the following: By 1.24 and 1.30 , we have the following: Thus, by 1.27 and 1.31 , we get the following identity.
x n 1 2 The Bernstein polynomials are defined by B k,n x n k x k 1 − x n−k , for k, n ∈ Z ≥0 , 1.33 with 0 ≤ k ≤ n cf. 14 .
By the definition of B k,n x , we note that In this paper, we derive some new and interesting identities involving Bernoulli and Euler numbers from well-known polynomial identities. Here, we note that our results are "complementary" to those in 6 , in the sense that we take a fermionic p-adic integral where a bosonic p-adic integral is taken and vice versa, and we use the identity involving Euler polynomials in 1.32 where that involving Bernoulli polynomials in 1.16 is used and vice versa. Finally, we report that there have been a lot of research activities on this direction of research, namely, on derivation of identities involving Bernoulli and Euler numbers and polynomials by exploiting bosonic and fermionic p-adic integrals cf. 6-8 .

Identities Involving Bernoulli Numbers
Taking the bosonic p-adic integral on both sides of 1.16 , we have the following:

2.1
Therefore, we obtain the following theorem.
Theorem 2.1. Let m ∈ Z ≥0 . Then on has the following: Let us apply 1.9 to the bosonic p-adic integral of 1.16 .
Z p

2.3
Then, we can express 2.3 in three different ways.

2.4
Thus, we have the following theorem.

Theorem 2.2.
Let m ∈ Z ≥0 . Then one has the following: Corollary 2.3. Let m be an integer ≥ 2. Then one has the following: Especially, for an odd integer m with m ≥ 3, we obtain the following corollary.
2.8 By 1.10 , 2.8 can be written as So, we get the following theorem.
Theorem 2.5. Let m ∈ Z ≥0 . Then one has the following: By 1.10 , 2.8 can also be written as Thus, we have the following theorem.
Theorem 2.6. Let m ∈ Z ≥0 . Then one has the following: International Journal of Mathematics and Mathematical Sciences 9

Identities Involving Euler Numbers
Taking the fermionic p-adic integral on both sides of 1.32 , we have the following:

3.1
So, we obtain the following theorem. Z p

3.3
Then, we can express 3.3 in two different ways.
International Journal of Mathematics and Mathematical Sciences By 1.29 , 3.3 can be written as

3.4
Thus, we get the following theorem.