Abstract
We investigate some formulae for the product of two Bernoulli and Euler polynomials arising from the Euler and Bernoulli basis polynomials.
1. Introduction
As is well known, the Bernoulli polynomials are defined by the generating function as follows: (see [1–21]), with the usual convention about replacing by . In the special case, , are called the th Bernoulli numbers. The Euler polynomials are also defined by the generating function as follows: (see [6–11]), with the usual convention about replacing by . In the special case, , are called the th Euler numbers. From (1.1) and (1.2), we can derive the following recurrence relations for the Bernoulli and Euler numbers: where is the Kronecker symbol. By (1.1) and (1.2), we get From (1.4), we can derive By (1.4) and (1.5), we get It is easy to show that Thus, we have (see [11–18]). By the definition of the Euler polynomials, we get From (1.9), we have (see [1–18]). By (1.8) and (1.10), we see that the set and are the basis for the space of polynomials of degree less than or equal to with coefficients in (see [1–21]).
From , let . Then, we note that Let us assume that . Then, we have
Continuing this process, we get Let for . Then, we have Assume that . Then, we get
Continuing this process, we get By (1.16), we get From the properties of the Bernoulli and Euler basis for the space of the polynomials of degree less than or equal to with coefficients in , we derive some identities for the product of two Bernoulli and Euler polynomials.
2. Some Identities for the Bernoulli and Euler Numbers
Let us consider the polynomial , with . Then, we have From the properties of the Bernoulli basis for the space of polynomials of degree less than or equal to with coefficients in , is given by Thus, by (2.2), we get By (2.1) and (2.2), we get Therefore, by (2.3) and (2.4), we obtain the following theorem.
Theorem 2.1. For , one has
From the properties of the Euler basis for the space of polynomials of degree less than or equal to with coefficients in , is given by By (2.1) and (2.6), we get Therefore, by (2.6) and (2.7), we obtain the following theorem.
Theorem 2.2. For , one has
Let us take polynomial with . Then, we have By the basis set for the space of polynomials of degree less than or equal to with coefficients in , we see that is given by From (2.10), we note that Note that where is the beta function.
From (2.11) and (2.12), we have For , by (2.9) and (2.10), we get Therefore, by (2.13) and (2.14), we obtain the following theorem.
Theorem 2.3. For , one has
From the Euler basis for the space of polynomials of degree less than or equal to with coefficients in , we note that can be written as follows: Thus, we have Therefore, by (2.16) and (2.17), we obtain the following theorem.
Theorem 2.4. For , one has
Let us consider the polynomial . Then, we have It is easy to show that For , we have Therefore, by (2.19) and (2.21), we obtain the following theorem.
Theorem 2.5. For , one has
Remark 2.6. If , by the same method, we get
Let us consider the polynomial . Then, we have It is easy to show that
For , we have Therefore, by (2.24) and (2.26), we obtain the following theorem.
Theorem 2.7. For , one has
Let us consider the polynomial . By the same method, we obtain the following identity: Let us take . Then, the th derivative of is given by where Note that where .
By the properties of the Bernoulli basis for the space of polynomials of degree less than or equal to with coefficients in , is given by Thus, by (2.32), we get
From (2.29), (2.31), and (2.32), we note that From (2.30), we have Therefore, by (2.32), (2.34), and (2.35), we obtain the following theorem.
Theorem 2.8. For , one has
We assume that . For , we have Finally,
By the same method, we obtain the following identity: Let us take . Then, for , we have where .
Note that By the properties of the Bernoulli basis for the space of polynomials of degree less than or equal to with coefficients in , can be written as Thus, by (2.42), we get
It is easy to show that For , one has Therefore, by (2.42), (2.44), and (2.45), we obtain the following theorem.
Theorem 2.9. For , one has
We may assume that . Then, we note that
Thus, we have
By the same method, we obtain the following identity: