Recently, some interesting and new identities are introduced in (Hwang et al., Communicated). From these identities, we derive some new and interesting integral formulae for the Bernoulli polynomials.

1. Introduction

As is well known, the Bernoulli polynomials are defined by generating functions as follows:

(1.1)

(see [1–11]). In the special case, x = 0, B_n(0) = B_n are called the nth Bernoulli numbers. The Euler polynomials are also defined by

(1.2)

with the usual convention about replacing Eⁿ(x) by E_n(x) (see [1–11]). From (1.1) and (1.2), we can easily derive the following equation:

(1.3)

By (1.1) and (1.3), we get

(1.4)

From (1.1), we have

(1.5)

Thus, by (1.5), we get

(1.6)

It is known that E_n(0) = E_n are called the nth Euler numbers (see [7]). The Euler polynomials are also given by

(1.7)

(see [6]). From (1.7), we can derive the following equation:

(1.8)

By the definition of Bernoulli and Euler numbers, we get the following recurrence formulae:

(1.9)

where δ_n,k is the kronecker symbol (see [5]). From (1.6), (1.8), and (1.9), we note that

(1.10)

where n ∈ ℤ₊. The following identity is known in [5]:

(1.11)

From the identities of Bernoulli polynomials, we derive some new and interesting integral formulae of an arithmetical nature on the Bernoulli polynomials.

2. Integral Formulae of Bernoulli Polynomials

From (1.1) and (1.2), we note that

(2.1)

Therefore, by (1.2) and (2.1), we obtain the following theorem.

Theorem 2.1. For n ∈ ℤ₊, one has

(2.2)

Let us take the definite integral from 0 to 1 on both sides of (1.4): for n ≥ 2,

(2.3)

By (2.3), we get

(2.4)

Therefore, by (2.4), we obtain the following theorem.

Theorem 2.2. For n ∈ ℕ, with n ≥ 2, one has

(2.5)

Let us take k = m, a = 0, and b = −2 in (1.11). Then we have

(2.6)

It is easy to show that

(2.7)

Let us consider the integral from 0 to 1 in (2.6):

(2.8)

By (2.6) and (2.8), we get

(2.9)

Therefore, by (2.9), we obtain the following theorem.

Theorem 2.3. For m ∈ ℕ, one has

(2.10)

Lemma 2.4. Let a, b ∈ ℤ. For m, k ∈ ℤ₊, one has

(2.11)

(see [5]).

Let us take k = m, a = 1, b = −2 in Lemma 2.4. Then we have

(2.12)

Taking integral from 0 to 1 in (2.12), we get

(2.13)

It is easy to show that

(2.14)

Thus, by (2.13) and (2.14), we get

(2.15)

Therefore, by (2.2) and (2.15), we obtain the following theorem.

Theorem 2.5. For m ∈ ℤ₊, one has

(2.16)

3. p-Adic Integral on ℤ_p Associated with Bernoulli and Euler Numbers

Let p be a fixed odd prime number. Throughout this section, ℤ_p, ℚ_p, and ℂ_p will denote the ring of p-adic integers, the field of p-adic rational numbers, and the completion of algebraic closure of ℚ_p, respectively. Let ν_p be the normalized exponential valuation of ℂ_p with

. Let UD(ℤ_p) be the space of uniformly differentiable functions on ℤ_p. For f ∈ UD(ℤ_p), the bosonic p-adic integral on ℤ_p is defined by

(3.1)

(see [8]). Thus, by (3.1), we get

(3.2)

where f₁(x) = f(x + 1), and f^′(0) = df(x)/dx|_x=0. Let us take f(y) = e^t(x+y). Then we have

(3.3)

From (3.3), we have

(3.4)

From (1.2), we can derive the following integral equation:

(3.5)

Thus, from (3.4) and (3.5), we get

(3.6)

From (3.6), we have

(3.7)

The fermionic p-adic integral on ℤ_p is defined by Kim as follows [6, 7]:

(3.8)

Let f₁(x) = f(x + 1). Then we have

(3.9)

Continuing this process, we obtain the following equation:

(3.10)

Thus, by (3.10), we have

(3.11)

Let us take f(y) = e^t(x+y). By (3.9), we get

(3.12)

From (3.2), we have the Witt′s formula for the nth Euler polynomials and numbers as follows:

(3.13)

By (3.11) and (3.13), we get

(3.14)

Let us consider the following p-adic integral on ℤ_p:

(3.15)

From (1.4) and (3.15), we have

(3.16)

Therefore, by (3.15) and (3.16), we obtain the following theorem.

Theorem 3.1. For n ∈ ℤ₊, one has

(3.17)

Now, we set

(3.18)

By (1.4), we get

(3.19)

Therefore, by (3.18) and (3.19), we obtain the following theorem.

Theorem 3.2. For n ∈ ℤ₊, one has

(3.20)

Let us consider the following integral on ℤ_p:

(3.21)

From (2.2), we have

(3.22)

Therefore, by (3.21) and (3.22), we obtain the following theorem.

Theorem 3.3. For n ∈ ℤ₊, one has

(3.23)

Now, we set

(3.24)

By (2.2), we get

(3.25)

Therefore, by (3.24) and (3.25), we obtain the following corollary.

Corollary 3.4. For n ∈ ℤ₊, we have

(3.26)

Let us assume that a, b, c, d ∈ ℤ. From Lemma 2.4 and (3.13), we note that

(3.27)

By (3.27), we get

(3.28)

Thus, by (3.28) and (3.13), we obtain the following lemma (see [5]).

Lemma 3.5. Let a, b, c, d ∈ ℤ. For m, k ∈ ℤ₊, one has

(3.29)

Let us consider the formula in Lemma 3.5 with d = c − 1. Then we have

(3.30)

Taking

on both sides of (3.30),

(3.31)

By the same method, we get

(3.32)

Therefore, by (3.31) and (3.32), we obtain the following proposition.

Proposition 3.6. Let a, b, c ∈ ℤ. Then one has

(3.33)

Replacing c by c + 1, we have

(3.34)

From (3.4) and (3.7), we derive some identity for the first term of the LHS of (3.34).

The first term of the LHS of (3.34)

(3.35)

where

(3.36)

The second term of the LHS of (3.34)

(3.37)

Therefore, by (3.34), (3.35), and (3.37), we obtain the following theorem.

Theorem 3.7. Let a, b, c ∈ ℤ with c ≥ 1. Then one has

(3.38)

where

(3.39)

Remark 3.8. Here, we note that

(3.40)

Acknowledgment

The first author was supported by National Research Foundation of Korea Grant funded by the Korean Government 2011-0002486.

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Integral Formulae of Bernoulli Polynomials

Abstract

1. Introduction

2. Integral Formulae of Bernoulli Polynomials

3. p-Adic Integral on ℤ_p Associated with Bernoulli and Euler Numbers

Acknowledgment

References

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Integral Formulae of Bernoulli Polynomials

Abstract

1. Introduction

2. Integral Formulae of Bernoulli Polynomials

3. p-Adic Integral on ℤp Associated with Bernoulli and Euler Numbers

Acknowledgment

References

Citing Literature

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3. p-Adic Integral on ℤ_p Associated with Bernoulli and Euler Numbers