On Polynomials Rn(x) Related to the Stirling Numbers and the Bell Polynomials Associated with the p-Adic Integral on

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Received 29 February 2016; accepted 5 April 2016; published 8 April 2016

1. Introduction

Recently, many mathematicians have studied the area of the Stirling numbers, the Euler numbers and polynomials (see [1] - [11] ). We studied some properties of the polynomials and numbers in com- plex field (see [12] ). In this paper, based on the Euler numbers and polynomials, we define the numbers and polynomials by using the p-adic integrals on in p-adic field. Then, we get some interesting properties and relations of the Stirling numbers, the, and the Bell numbers. It is interesting that the Euler polynomials and to be define in this paper have a different structure (see [Figure 2]). Zeros of are a symmetric structure but zeros of are not.

Throughout this paper, we use the following notations. By, we denote the ring of p-adic rational integers, denotes the field of p-adic rational numbers, denotes the completion of algebraic closure of, denotes the set of natural numbers, denotes the ring of rational integers, Q denotes the field of rational

numbers, denotes the set of complex numbers, and and denote the binomial coefficient. Let be the normalized exponential valuation of with.

For

the fermionic p-adic integral on is defined by T. Kim as below:

(cf. [5] ). (1.1)

If we take in (1.1), then we easily see that

(1.2)

From (1.2), we obtain

(1.3)

where (cf. [5] - [10] ).

The classical Euler polynomials are defined by the following generating function

(1.4)

with the usual convention of replacing by. In generally, the original Euler numbers are when

and normalizing by gives the Euler number as following:

But in this paper, Euler numbers are when. In other words, and in this paper, Euler numbers mean the Euler numbers having a generating function as below(cf. [5] - [10] ):

(1.5)

The Stirling number of the second kind is the number of partitions of n things into r non-empty sets; it is positive if and zero for other values of r (see [1] ). It satisfies the recurrence relation

The generating function of the Stirling numbers is defined as below:

(1.6)

As well known definition, the Bell polynomials are defined by Bell (1934) as below

(1.7)

Also, let be denote the Stirling numbers of the second kind. Then

(1.8)

In the special case, are called the n-th Bell numbers.

The motivation of this paper is the Euler numbers and Bell numbers’s generating function. From this idea, we induce some interesting properties related to the Stirling numbers, the Bell numbers, the Euler numbers and the.

Our aim in this paper is to define analogue Euler numbers and polynomials. We investigate some properties which are related to,. Especially, we derive the relations of the Stirling numbers and the, the.

2. An Introduction to Numbers and Polynomials

Our primary goal of this section is to define numbers and polynomials. We also find the witt’s formula for numbers and polynomials by (1.2).

By (1.2) and using p-adic integral on, we get as below:

Let.

(2.1)

Hence, by (2.1) we get the following:

(2.2)

Also, Let. By the same method (2.1), we get the following:

(2.3)

From (2.2) and (2.3), we define numbers and polynomials, as below:

(2.4)

(2.5)

respectively.

From above definition, one easily has the Witt’s formula as below:

(2.6)

(2.7)

with the usual convention of replacing by respectively. In the special case, , are called the n-th R-numbers.

From (2.6) and

Hence, we get the following;

(2.8)

where is the Euler numbers.

Also, from (2.5) and by simple calculus, one has

(2.9)

From (2.8) and (2.9), we get some polynomials as below:

3. Basic Properties for and Related to the Stirling Numbers, the Bell Numbers and the Euler Numbers

From (2.5) and by the simple calculation

(3.1)

where are the Bell polynomials.

By comparing the coefficients of on the both sides of the above equation, we get the following the

theorem immediately.

Theorem 1. For with, one has

where are the Bell polynomials.

From (2.5), one has

Let. Then from (1.2)

(3.2)

By comparing the coefficients of on the both sides of the above equation, we get the following theorem

immediately.

Theorem 2. For with, let be the stirling numbers. Then, one has

where and are the Euler polynomials and the Euler numbers respectively. And

where is the n-th Bell polynomial.

Also, from (2.1) one has

(3.3)

By comparing the coefficients of on the both sides of the above equation, we get the following theorem immediately.

Theorem 3. For with, one has

Let. Then from (1.3), we derive the following:

Left side of (1.3) is as below:

(3.4)

and right side of (1.3) is as below:

(3.5)

Hence, from (3.4) and (3.5), we get the following theorem.

Theorem 4. For with, one has

where and are the Euler polynomials and numbers respectively.

By using the definition of and simple calculation, we get the following:

and the equality above is expressed as follows:

It is well known that. By the definition and some calculation, we get the fol- lowing:

(3.6)

Hence, one has the following theorem.

Theorem 5. For with, one has

where are the Bell polynomials.

By the same method above Theorem 5, we get the corollary as follows:

Corollary 6. For with, one has

(3.7)

where are the Bell polynomials.

It is well known that is the generating function of the Euler polynomials. We substitude for

t in the generating function of the Euler polynomials as below:

(3.8)

The left-hand-side of (3.8) is

(3.9)

The right-hand-side of (3.8) is

(3.10)

By (3.9),(3.10) and comparing the coefficient of both sides, we get the following theorem.

Theorem 7. For with, one has

where and are the Euler polynomials and the Bell polynomials respectively.

It is not difficult to see that

(3.11)

From the expression (3.11), one has

Specially, if,

where are the n-th Bell polynomials.

4. Zeros of the Bell Polynomials and the Polynomials

In this section, we investigate the zeros of the Bell, Euler, and polynomials by using a computer.

From (1.7), we get some polynomials as below:

We plot the zeros of for (Figure 1). In Figure 1 (top-left), we choose. In Figure 1 (top-right), we choose. In Figure 1 (bottom-left), we choose In Figure 1 (bottom-right), we choose.

Next, we plot the zeros of for (Figure 2). In Figure 2 (left), we choose and plot of zeros of. In Figure 2 (middle), we choose and plot of zeros of In Figure 2 (right),we choose and plot of zeros of.

Our numerical results for numbers of real and complex zeros of and are displayed in Table 1.

We observe a remarkably regular structure of the complex roots of the Bell polynomials and polynomials. We hope to verify a remarkably regular structure of the complex roots of the Bell polynomials and polynomials (Table 1). Prove that the numbers of complex zeros of is

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

Stacks of zeros of for from a 3-D structure are presented (Figure 3). Next, we present stacks of zeros of for from a 3-D structure. In Figure 3 (left), stacks of zeros of for from a 3D structure are presented. In Figure 3 (middle), stacks of zeros of for from a 3D structure are presented. In Figure 3 (right), stacks of zeros of for from a 3D structure are presented .

Since n 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. Prove or disprove: has n distinct solutions. Find the numbers of complex

Figure 1. Zeros of.

Figure 2. Zeros of, and.

Figure 3. Zeros of, and.

Table 1. Numbers of real and complex zeros of and.

Table 2. Approximate solutions of B_{n}(x) = 0.

zeros of Using numerical investigation, we observed the behavior of complex roots of the Euler polynomials. By means of numerical experiments, we demonstrate a remarkably regular structure of the complex roots of the Euler polynomials (see [12] ). The theoretical prediction on the zeros of is await for further study. These figures give mathematicians an unbounded capacity to create visual mathematical investigations of the behavior of the roots of the. For more studies and results in this subject, you may see [12] - [14] .

Acknowledgements

This research was supported by Hannam University Research Fund, 2015.

NOTES

^{*}Corresponding author.

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