Journal of Applied Mathematics

Volume 2013 (2013), Article ID 158130, 10 pages

http://dx.doi.org/10.1155/2013/158130

Research Article

A Research on a Certain Family of Numbers and Polynomials Related to Stirling Numbers, Central Factorial Numbers, and Euler Numbers

J. Y. Kang and C. S. Ryoo

Department of Mathematics, Hannam University, Daejeon 306-791, Republic of Korea

Received 28 June 2013; Accepted 23 August 2013

Academic Editor: Ch. Tsitouras

Copyright © 2013 J. Y. Kang and C. S. Ryoo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Recently, many mathematicians have studied different kinds of the Euler, Bernoulli, and Genocchi numbers and polynomials. In this paper, we give another definition of polynomials . We observe an interesting phenomenon of “scattering” of the zeros of the polynomials in complex plane. We find out some identities and properties related to polynomials . Finally, we also derive interesting relations between polynomials , Stirling numbers, central factorial numbers, and Euler numbers.

1. Introduction

Recently, many mathematicians have studied in the areas of the Bernoulli numbers and polynomials, Euler numbers and polynomials, Genocchi numbers and polynomials, Stirling numbers, and central factorial numbers (see [1–17]). These numbers and polynomials possess many interesting properties and are arising in many areas of mathematics and physics. In this paper, we give another definition of polynomials . We obtain some interesting identities and properties related to polynomials . In order to study the polynomials , we must understand the structure of the polynomials . Therefore, using computer, a realistic study for the polynomials is very interesting. It is the aim of this paper to observe an interesting phenomenon of “scattering” of the zeros of the polynomials in complex plane.

The Stirling numbers of the first kind define that

The generating function of (1) is as follows:

From (1) and (2), we are aware of some properties of the Stirling numbers of the first kind as follows (see [1, 2, 15]):

with

We usually define the central factorial numbers as the following expansion formula (see [1, 7]):

The generating function of (5) is as follows:

By using (5) and (6), we are aware of some properties of the central factorial numbers as follows:

with

The Euler numbers are defined by the generating function:

We introduce the Euler polynomials as follows:

Zhi-Hong Sun introduces the sequence similar to Euler numbers as follows:

where is the greatest integer not exceeding (see [4, 5]).

The outline of this paper is as follows. In Section 2, this paper is to define polynomials . In Section 3, we describe the beautiful zeros of the polynomials using a numerical investigation. We investigate the roots of the polynomials . Also we carried out computer experiments for demonstrating a remarkably regular structure of the complex roots of the polynomials . In Section 4, we derive some special relations of polynomials and Euler numbers.

2. Some Properties Involving a Certain Family of Numbers and Polynomials

In this section, we introduce the polynomials and investigate some interesting properties and identities which are related to polynomials . We also try to find relations between polynomials , Stirling numbers , and central factorial numbers .

Definition 1. For or , the polynomials are defined by

From Definition 1, we have Theorem 2.

Theorem 2. Let and . Then one obtains

where is the greatest integer not exceeding and .

Proof. By Definition 1, (2), and (6), we have

From now on, we have to consider odd terms and even terms by using Cauchy product. So, we get to generating terms by dividing the odd terms and the even terms, respectively,

The following equation is the generating even terms:

where is the greatest integer not exceeding .

We also derive the generating odd terms:

where is the greatest integer not exceeding .

Thus, we complete the proof of Theorem 2.

Example 3. Let . Then we can know the following polynomials:

Corollary 4. Let and . Then one has

Proof. Corollary 4 shows identities of and . The proof of Corollary 4 is contained in Theorem 2.

Setting in Corollary 4, we easily see the following corollary.

Corollary 5. Relation of   and central factorial numbers is shown.

Let and . Then one has

Proof. Consider the following:

We find out the following equation from the above equation by substituting :

We also can find out by using similar method as above:

If we substitute , then

Let and and

where is the greatest integer not exceeding .

Then we have

By using and , we obtain Theorem 6.

Theorem 6. Let . Then one gets

Proof. From Definition 1, one easily obtains the following equation:

We also easily get the following equation by substituting :

Thus,

We have to consider odd terms and even terms in (26) and the previous equations, respectively.

Odd terms are in the following form:

and even terms are similar to the way of the proof process of the odd terms. Therefore, we omit the proof process of the even terms.

Theorem 7. Let and . Then one has

Proof. From Definition 1, one easily obtains the following equation:

We also easily get the following by using differential, namely,

Thus,

Then, right-hand side is in the following form:

and left-hand side is in the following form:

By using comparing coefficients of in the previous equations, we can represent the equation; that is,

By simple calculation, we get

Therefore, we proved that

3. Zeros of the Polynomials

In this section, we investigate the reflection symmetry of the zeros of the polynomials .

We investigate the beautiful zeros of the polynomials by using a computer. We plot the zeros of the polynomials for and (Figure 1). In Figure 1(a), we choose . In Figure 1(b), we choose . In Figure 1(c), we choose . In Figure 1(d), we choose .

(a)

(b)

(c)

(d)

Figure 1: Zeros of for .

Throughout the numerical experiments, we can finally conclude that polynomials have no reflection symmetry analytic complex functions. However, we observe that has reflection symmetry (see Figures 1, 2, and 3). The obvious corollary is that the zeros of will also inherit these symmetries:

where denotes complex conjugation (see Figure 1).

Figure 2: Real zeros of for .

Figure 3: Stacks of zeros of for .

Plots of real zeros of for structure are presented (Figure 2).

Our numerical results for approximate solutions of real zeros of the are displayed (Tables 1 and 2).

Table 1: Numbers of real and complex zeros of .

Degree
Real zeros
Complex zeros

1
1
0

2
2
0

3
3
0

4
4
0

5
5
0

6
4
2

7
5
2

8
6
2

9
5
4

10
6
4

30
20
10

Table 2: Approximate solutions of .

1
0

2
0, 2

3
0, 0, 6

4
−0.5346, 0, 1.73111, 10.8035

5
−0.6759, 0, 0, 4.60224, 16.0736

6
0, 1.64591338, 8.2083255, 21.6537433

7
0, 0, 4.0761207, 12.3245041, 27.4570253

8
−0.877412, 0, 1.6058053, 7.1033690, 16.8165987, 33.4298423

9
−0.968751, 0, 0, 3.8044567, 10.5908618, 21.598401, 39.536481

10
0, 1.5827926, 6.4946365, 14.442719, 26.611139, 45.751858

We observe a remarkably regular structure of the complex roots of the polynomials . We hope to verify a remarkably regular structure of the complex roots of the polynomials (Table 1).

Stacks of zeros of for from a 3-D structure are presented (Figure 3).

Next, we calculated an approximate solution satisfying . The results are given in Table 2.

Finally, we will consider the more general problems. Find the numbers of complex zeros of . Since is the degree of the polynomial , the number of real zeros lying on the real plane is then , where denotes complex zeros. See Table 1 for tabulated values of and . We plot the zeros of , respectively (Figures 1–3). These figures give mathematicians an unbounded capacity to create visual mathematical investigations of the behavior of the roots of the . Moreover, it is possible to create new mathematical ideas and analyze them in ways that generally are not possible by hand. The authors have no doubt that investigation along this line will lead to a new approach employing numerical methods in the field of research of to appear in mathematics and physics.

4. Some Relations of Polynomials and Euler Numbers

In this section, we find out interesting relations between polynomials and Euler numbers. We usually use the values of polynomials and Euler polynomials.

Theorem 8. Let . Then one has

with the usual convention of replacing by .

Proof. By setting in Definition 1 and using Euler numbers, then we have

Hence,

We also obtain the above equation by taking and using Euler numbers:

By using Cauchy product in the above equation, we have

By comparing the coefficient of both sides of and some calculations, we have the following equation:

with the usual convention of replacing by .

Therefore, we consummated the proof of Theorem 8.

From (26), we can suppose that

where

Therefore, we have Theorem 9.

Theorem 9. Let . Then one has

Proof. From Definition 1, we can differentiate the -times as follows:

Here we required that

Hence, we derive that

By comparing the coefficient of both sides of , we have the following equation:

From (48), we easily see that

Therefore, we clear off the proof of Theorem 9.

Theorem 10. Let . Then one has

Proof. This proof can be proved by the similar method of Theorem 9.

We also can derive relation polynomials and Euler numbers as follows.

Theorem 11. Let and . Then one derives

Proof. By the proof of Theorem 9, we can differentiate polynomials as follows:

By integrating from to , we deduce that

We already knew from the proof process of Theorem 9 that

By using the above equation, we can derive that

Therefore, we completely demonstrated the proof of Theorem 11.

Acknowledgments

The authors express their gratitude to the referee for his/her valuable comments. This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean Government (NRF-2013-Fostering Core Leaders of the Future Basic Science Program).

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