Abstract
We give some new formulae for product of two Genocchi polynomials including Euler polynomials and Bernoulli polynomials. Moreover, we derive some applications for Genocchi polynomials to study a matrix formulation.
1. Introduction
The history of Genocchi numbers can be traced back to Italian mathematician Angelo Genocchi (1817–1889). From Genocchi to the present time, Genocchi numbers have been extensively studied in many different context in such branches of Mathematics as, for instance, elementary number theory, complex analytic number theory, homotopy theory (stable homotopy groups of spheres), differential topology (differential structures on spheres), theory of modular forms (Eisenstein series), -adic analytic number theory (-adic -functions), and quantum physics (quantum groups). The works of Genocchi numbers and their combinatorial relations have received much attention [1–11]. For showing the value of this type of numbers and polynomials, we list some of their applications.
In the complex plane, the Genocchi numbers, named after Angelo Genocchi, are a sequence of integers that are defined by the exponential generating function: with the usual convention about replacing by , is used. When we multiply with in the left-hand side of (1), then we have where are called Genocchi polynomials. It follows from (2) that , , , , , , , , and for (for details, see [7–9]).
Differentiating both sides of (1), with respect to , then we have the following:
On account of (1) and (3), we can easily derive the following:
By (1), we get
Thanks to (4) and (5), we acquire the following equation (6):
It is not difficult to see that
By expression of (7), then we have (see [1–25]).
Let be the -dimensional vector space over . Probably, is the most natural basis for . From this, we note that is also good basis for space .
In [14], Kim et al. introduced the following integrals: where and are called Bernoulli polynomials and Euler polynomials, respectively. Also, they are defined by the following generating series: with and , symbolically. By (10), then we have
Here and are called Bernoulli numbers and Euler numbers, respectively. Additionally, the Bernoulli and Euler numbers and polynomials have the following identities: (for details, see [6, 11, 13–15, 17, 19]). By (11), we have the following recurrence relations of Euler and Bernoulli numbers, as follows: where is the Kronecker’s symbol defined by
In the complex plane, we can write the following:
By (15), we have by comparing coefficients on the both sides of the above equality, then we have (see [6]).
Via (17), our results in the present paper can be extended to Euler polynomials at the special value .
Recent works including multiplication formulas for products of Bernoulli and Euler polynomials [17], the product of two Eulerian polynomials [18], sums of products of generalized Bernoulli polynomials [20], explicit formulas for computing Euler polynomials in terms of the second kind Stirling numbers [21], explicit formulas for computing Bernoulli numbers of the second kind and Stirling numbers of the first kind [22], some identities and an explicit formula for Bernoulli and Stirling numbers [23], some identities for the product of two Bernoulli and Euler polynomials [13], some formulae for the product of two Bernoulli and Euler polynomials [14], Bernoulli basis and the product of several Bernoulli polynomials [15], some formulae of products of the Apostol-Bernoulli and Apostol-Euler Polynomials [24], and the modified -Euler numbers of higher order with weight [25] have been extensively investigated.
By the same motivation of the above knowledge, we write this paper. We give some interesting properties which are derived from the basis of Genocchi. From our methods, we obtain not only new but also interesting identities including Bernoulli and Euler polynomials. Also, by using (17), we derive our results in terms of Euler polynomials.
2. On the Genocchi Numbers and Polynomials
In this section, we introduce the following integral equation: for ,
By (18), becomes
Thus, we have the following recurrence formulas: by continuing with the above recurrence relation; then we derive that
Now also, we develop the following for sequel of this paper:
Let us now introduce the polynomial
Taking th derivative of the above equality, then we have
Theorem 1. The following equality holds true:
Proof. On account of the properties of the Genocchi basis for the space of polynomials of degree less than or equal to with coefficients in , then can be written as follows:
Therefore, by (26), we obtain
From expression of (24), we get
Substituting (27) and (28) into (26), we arrive at the desired result.
By using (17) and Theorem 1, we get the following corollary, which has been stated in terms of Euler polynomials.
Corollary 2. For any , then we have
Theorem 3. The following nice identity is true.
Proof. Let us now consider the polynomial in terms of Euler polynomials as follows: In [14], Kim et al. gave the coefficients by utilizing from the definition of Bernoulli polynomials. Now also, we give the coefficients by using the definition of Genocchi polynomials, as follows: After the above applications, we complete the proof of the theorem.
By employing (17) and Theorem 3, we have the following corollary, which is the sum of products of two Euler polynomials.
Corollary 4. For each , one has
We now discover the following theorem, which will be an interesting and worthwhile theorem for studying in analytic numbers theory.
Theorem 5. The following equality holds:
Proof. It is proved by using the following polynomial :
It is not difficult to indicate the following:
Then, we see that, for ,
By (35) and (37), we arrive at the desired result.
Theorem 6. The following identity is true.
Proof. Now also, let us take the polynomial in terms of Bernoulli polynomials as
By using the above identity, we develop as follows:
By (36), we compute coefficients, as follows:
When we substituted (40) and (41) into (39), the proof of theorem will be completed.
By using (17) and Theorem 6, we procure the following corollary.
Corollary 7. For any , one has
In [6], it is well known that
For in (43), we have the following:
By comparing the equations of (38) and (44), then we readily derive the following corollary.
Corollary 8. Consider that
Theorem 9. The following equality holds true.
Proof. To prove this theorem, we introduce the following polynomial : Then, we derive th derivative of that is given by where We want to note that where are called Harmonic numbers, which are defined by With the properties of Genocchi basis for the space of polynomials of degree less than or equal to with coefficients in , is introduced by By expression of (52), we obtain that As a result, In [14], it is well known that By (48), (52), and (55), we arrive at the desired result.
From (17) and Theorem 9, we acquire the following.
Corollary 10. The following identity holds:
3. Further Remarks
Let be the space of polynomials of a degree less than or equal to . In this final section, we will give the matrix formulation of Genocchi polynomials. Let us now consider the polynomial as a linear combination of Genocchi basis polynomials with
We can write the above as a product of two variables:
From expression of (58), we consider the following equation: where are the coefficients of the power basis that are used to determine the respective Genocchi polynomials. We now list a few Genocchi polynomials as follows:
In the quadratic case (), the matrix representation is
In the cubic case (), the matrix representation is
Remark 11. Throughout this paper, many considerations for Genocchi polynomials seem to be useful to study in a matrix formulation.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
The authors would like to thank an anonymous referee for his/her valuable comments and suggestions which have improved the presentation of the paper.