A research on the new polynomials involved with the central factorial numbers, Stirling numbers and others polynomials
© Kang and Ryoo; licensee Springer. 2014
Received: 28 September 2013
Accepted: 26 December 2013
Published: 24 January 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.
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 .
Note that . That is, are called the n th numbers. By using Definition 2.1, we have the addition theorem of the polynomials .
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, .
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. □
Proof From Definition 2.1, we easily see that the following equation holds true.
Thus we complete the proof. □
Remark From Theorem 2.4, we easily see the following.
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.
and is the greatest integer not exceeding x.
where is the greatest integer not exceeding x.
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.
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 .
Hence the proof of the Theorem 3.1 is complete. □
From Theorem 3.1, we easily get the following corollary.
Proof Let in Definition 2.1 and Theorem 3.1, respectively. Then we easily obtain the proof of Corollary 3.2. □
Secondly, we note a link between the polynomials , the Genocchi numbers, , and the polynomials .
Thus we complete the proof of Theorem 3.4. □
From Theorem 3.4, we get the following corollary.
Proof Let , in Definition 2.1 and Theorem 3.4, respectively. Then we easily obtain the proof of Corollary 3.5. □
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).
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