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A research on the new polynomials involved with the central factorial numbers, Stirling numbers and others polynomials
Journal of Inequalities and Applications volume 2014, Article number: 26 (2014)
Recently, many mathematicians have studied different kinds of the Euler, Bernoulli, and Genocchi polynomials. In this paper, we give another definition of the polynomials . We find some theorems and identities related to polynomials containing the central factorial numbers and Stirling numbers. We also derive interesting relations between the polynomials and the Euler polynomials and the Genocchi polynomials.
The Stirling numbers of the first kind are defined by 
The generating function of (1.1) is as follows:
From (1.1) and (1.2), we become aware of some properties of the Stirling numbers of the first kind, :
The generating function of (1.3) is as follows:
By using (1.3) and (1.4), we become aware of some properties of the central factorial numbers :
The Euler numbers and Euler polynomials are defined by
For a real or complex parameter α, the generalized Euler polynomials of degree n are defined by the following generating functions:
The Genocchi polynomials are defined by
In Section 2, we define polynomials . We consider the addition theorem for these polynomials. We also investigate some identities which are related to polynomials . We also try to find relations between the polynomials , the Stirling numbers , and the central factorial numbers . In Section 3, we derive some special relations of the polynomials and the Euler polynomials. We also find a link between the polynomials and the Genocchi polynomials.
2 Some properties involving a certain family of polynomials
In this section, we define the polynomials and study several theorems of the polynomials . We can see some interesting properties of the polynomials .
Definition 2.1 For , the polynomials are defined by
Note that . That is, are called the n th numbers. By using Definition 2.1, we have the addition theorem of the polynomials .
Theorem 2.2 Let and let n be non-negative integers. Then we get
Proof From Definition 2.1, we get
By comparing the coefficients of both sides, we complete the proof of Theorem 2.2. □
We also find the relation of the polynomials, , and the numbers, .
Theorem 2.3 Let and n be non-negative integers. Then we get
Proof This proof is very similar to the proof of Theorem 2.3:
By comparing the coefficients of both sides, we complete the proof of the Theorem 2.3. Of course, we can get a simple proof by substituting in Theorem 2.2. □
Theorem 2.4 Let and . For , we have
Proof From Definition 2.1, we easily see that the following equation holds true.
Let . Then we get
Differentiating with respect to t, we find
If we multiply the above equation throughout by , then we have
Then, the left-hand side gets transformed as follows in (2.1):
And the right-hand side of (2.1) gets transformed in the following form:
Comparing the coefficients of in (2.2) and (2.3), we can represent the equation as
Rearranging, we get
Thus we complete the proof. □
Remark From Theorem 2.4, we easily see the following.
If and , then
If and , then
So far, we got some properties of polynomials . From now on, we will investigate the relation of the polynomials , the Stirling numbers and the central factorial numbers.
Theorem 2.5 Let and . Then we have
and is the greatest integer not exceeding x.
Proof By using the Stirling numbers and the central factorial numbers, we express the polynomials as follows:
After some calculation, we get
where is the greatest integer not exceeding x.
Then we have
From the above equation, we have to consider odd terms and even terms by using the Cauchy product. Thus, we get generating terms by dividing the odd terms and the even terms, respectively,
The following equations represent for n even and n odd terms:
and is the greatest integer not exceeding x.
Thus, we complete the proof of Theorem 2.5. □
From Theorem 2.5, we find examples of the polynomials.
Example 2.6 Let . Then we have
Therefore, we can express as a function of x and α explicitly. For instance,
3 Some relations of the polynomials , the Euler polynomials, and the Genocchi polynomials
In this chapter, we find interesting relations between the polynomials , the Euler polynomials, and the Genocchi polynomials. In other words, the polynomials can be shown in a combined form by using Euler numbers and polynomials. We also easily see the polynomials that are represented by Genocchi numbers and polynomials.
First, we study a link between the polynomials and the Euler polynomials .
Theorem 3.1 Let , n be a non-negative integer. Then we have
Proof From Definition 2.1, we have
By comparing the coefficients of both sides, we have
Hence the proof of the Theorem 3.1 is complete. □
From Theorem 3.1, we easily get the following corollary.
Corollary 3.2 Let . Then one has
Proof Let in Definition 2.1 and Theorem 3.1, respectively. Then we easily obtain the proof of Corollary 3.2. □
Theorem 3.3 Let n, , . Then we have
Proof From the Definition 2.1 and the Euler numbers and polynomials, we can write the following equation:
Here, we can represent the above right-hand side as the following equation:
Therefore, we obtain
Secondly, we note a link between the polynomials , the Genocchi numbers, , and the polynomials .
Theorem 3.4 Let . Then we have
Proof is represented as follows:
By comparing the coefficients of both sides in the above equation, we derive
Thus we complete the proof of Theorem 3.4. □
From Theorem 3.4, we get the following corollary.
Corollary 3.5 Let , and . Then we derive
Proof Let , in Definition 2.1 and Theorem 3.4, respectively. Then we easily obtain the proof of Corollary 3.5. □
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The authors express their gratitude to the referee for his/her valuable comments. This work was supported by NRF (National Research Foundation of Korea) Grant funded by the Korean Government (NRF-2013-Fostering Core Leaders of the Future Basic Science Program).
The authors declare that they have no competing interests.
All authors contributed equally to the manuscript and read and approved the final manuscript.
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Kang, J.Y., Ryoo, C.S. A research on the new polynomials involved with the central factorial numbers, Stirling numbers and others polynomials. J Inequal Appl 2014, 26 (2014). https://doi.org/10.1186/1029-242X-2014-26
- Stirling numbers
- central factorial numbers
- Euler polynomials
- Genocchi polynomials