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In this paper, one introduces the polynomials
R_{n}(x) and numbers
R_{n} and derives some interesting identities related to the numbers and polynomials:
R_{n} and
R_{n}(x). We also give relation between the Stirling numbers, the Bell numbers, the
R_{n} and
R_{n}(x).

Recently, many mathematicians have studied the area of the Stirling numbers, the Euler numbers and polynomials (see [

Throughout this paper, we use the following notations. By

numbers,

For

the fermionic p-adic integral on

If we take

From (1.2), we obtain

where

The classical Euler polynomials are defined by the following generating function

with the usual convention of replacing

But in this paper, Euler numbers are when

The Stirling number of the second kind

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

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

Also, let

In the special case,

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

Our primary goal of this section is to define numbers

By (1.2) and using p-adic integral on

Let

Hence, by (2.1) we get the following:

Also, Let

From (2.2) and (2.3), we define numbers and polynomials

respectively.

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

with the usual convention of replacing

From (2.6) and

Hence, we get the following;

where

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

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

From (2.5) and by the simple calculation

where

By comparing the coefficients of

theorem immediately.

Theorem 1. For

where

From (2.5), one has

Let

By comparing the coefficients of

immediately.

Theorem 2. For

where

where

Also, from (2.1) one has

By comparing the coefficients of

Theorem 3. For

Let

Left side of (1.3) is as below:

and right side of (1.3) is as below:

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

Theorem 4. For

where

By using the definition of

and the equality above is expressed as follows:

It is well known that

Hence, one has the following theorem.

Theorem 5. For

where

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

Corollary 6. For

where

It is well known that

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

The left-hand-side of (3.8) is

The right-hand-side of (3.8) is

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

Theorem 7. For

where

It is not difficult to see that

From the expression (3.11), one has

Specially, if

where

In this section, we investigate the zeros of the Bell, Euler, and

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

We plot the zeros of

Next, we plot the zeros of

Our numerical results for numbers of real and complex zeros of

We observe a remarkably regular structure of the complex roots of the Bell polynomials

Next, we calculate an approximate solution satisfying

Stacks of zeros of

Since n is the degree of the polynomial

Degree n | ||||
---|---|---|---|---|

Real zeros | Complex zeros | Real zeros | Complex zeros | |

1 | 1 | 0 | 1 | 0 |

2 | 2 | 0 | 2 | 0 |

3 | 3 | 0 | 3 | 0 |

4 | 4 | 0 | 2 | 2 |

5 | 5 | 0 | 3 | 2 |

6 | 6 | 0 | 4 | 2 |

7 | 7 | 0 | 3 | 4 |

8 | 8 | 0 | 4 | 4 |

9 | 9 | 0 | 5 | 4 |

10 | 10 | 0 | 4 | 6 |

11 | 11 | 0 | 5 | 6 |

12 | 12 | 0 | 6 | 6 |

13 | 13 | 0 | 5 | 8 |

Degree n | x |
---|---|

1 | 0 |

2 | −1, 0 |

3 | 0, −2.6180, −0.3820 |

4 | −4.491, −1.343, −0.1658 |

5 | −6.51, −2.65, −0.762, −0.076 |

6 | −8.63, −4.18, −1.70, −0.453, −0.04 |

zeros

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

Hui Young Lee,Cheon Seoung Ryoo, (2016) On Polynomials R_{n}(x) Related to the Stirling Numbers and the Bell Polynomials Associated with the p-Adic Integral on