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.

Cite this paper

Lee, H. and Ryoo, C. (2016) On Polynomials*R*_{n}(x) Related to the Stirling Numbers and the Bell Polynomials Associated with the *p*-Adic Integral on . *Open Journal of Discrete Mathematics*, **6**, 89-98. doi: 10.4236/ojdm.2016.62009.

Lee, H. and Ryoo, C. (2016) On Polynomials

References

[1] Rennie, B.C. and Dobson, A.J. (1969) On Stirling Numbers of the Second Kind. Journal of Combinatorial Theory, 7, 116-121.

http://dx.doi.org/10.1016/S0021-9800(69)80045-1

[2] Abbas, M. and Bouroubi, S. (2005) On New Identities for Bells Polynomials. Discrete Mathematics, 293, 5-10.

http://dx.doi.org/10.1016/j.disc.2004.08.023

[3] Yang, S.L. (2008) Some Identities Involving the Binomial Sequences. Discrete Mathematics, 308, 51-58.

http://dx.doi.org/10.1016/j.disc.2007.03.040

[4] Xu, A.-M. and Cen, Z.-D. (2014) Some Identities Involving Exponential Functions and Stirling Numbers and Applications. Journal of Computational and Applied Mathematics, 260, 201-207.

http://dx.doi.org/10.1016/j.cam.2013.09.077

[5] Kim, T. (2002) q-Volkenborn Integration. Russ. Journal of Mathematical Physics, 9, 288-299.

[6] Kim, T. (2006) On the Euler Numbers and Polynomials Associated with p-Adic q-Integral on Zp at q = 1. Russian Journal of Mathematical Physics, in Press.

[7] Kim, T. (2007) q-Euler Numbers and Polynomials Associated with p-Adic q-Integrals. Journal of Nonlinear Mathematical Physics, 14, 15-27.

http://dx.doi.org/10.2991/jnmp.2007.14.1.3

[8] Kim, T., Choi, J., Kim, Y-.H. and Ryoo, C.S. (2011) A Note on the Weighted p-Adic q-Euler Measure on . Advanced Studies in Contemporary Mathematics, 21, 35-40.

[9] Kupershmidt, B.A. (2005) Reflection Symmetries of q-Bernoulli Polynomials. Journal of Nonlinear Mathematical Physics, 12, 412-422.

http://dx.doi.org/10.2991/jnmp.2005.12.s1.34

[10] Ryoo, C.S., Kim, T. and Jang, L.-C. (2007) Some Relationships between the Analogs of Euler Numbers and Polynomials. Journal of Inequalities and Applications, 2007, Article ID: 86052.

[11] Ryoo, C.S. (2011) A Note on the Weighted q-Euler Numbers and Polynomials. Advanced Studies in Contemporary Mathematics, 21, 47-54.

[12] Lee, H.Y. and Ryoo, C.S. (2014) On New Polynomials Related to Bell Numbers and Stirling Numbers. Advanced Studies in Theoretical Physics, 8, 763-769.

[13] Ryoo, C.S. and Kang, J.Y. (2015) A Numerical Investigation on the Structure of the Zeros of Euler Polynomials. Discrete Dynamics in Nature and Society, 2015, Article ID: 174173.

[14] Ryoo, C.S. (2014) Analytic Continuation of Euler Polynomials and the Euler Zeta Function. Discrete Dynamics in Nature and Society, 2014, Article ID: 568129.

[1] Rennie, B.C. and Dobson, A.J. (1969) On Stirling Numbers of the Second Kind. Journal of Combinatorial Theory, 7, 116-121.

http://dx.doi.org/10.1016/S0021-9800(69)80045-1

[2] Abbas, M. and Bouroubi, S. (2005) On New Identities for Bells Polynomials. Discrete Mathematics, 293, 5-10.

http://dx.doi.org/10.1016/j.disc.2004.08.023

[3] Yang, S.L. (2008) Some Identities Involving the Binomial Sequences. Discrete Mathematics, 308, 51-58.

http://dx.doi.org/10.1016/j.disc.2007.03.040

[4] Xu, A.-M. and Cen, Z.-D. (2014) Some Identities Involving Exponential Functions and Stirling Numbers and Applications. Journal of Computational and Applied Mathematics, 260, 201-207.

http://dx.doi.org/10.1016/j.cam.2013.09.077

[5] Kim, T. (2002) q-Volkenborn Integration. Russ. Journal of Mathematical Physics, 9, 288-299.

[6] Kim, T. (2006) On the Euler Numbers and Polynomials Associated with p-Adic q-Integral on Zp at q = 1. Russian Journal of Mathematical Physics, in Press.

[7] Kim, T. (2007) q-Euler Numbers and Polynomials Associated with p-Adic q-Integrals. Journal of Nonlinear Mathematical Physics, 14, 15-27.

http://dx.doi.org/10.2991/jnmp.2007.14.1.3

[8] Kim, T., Choi, J., Kim, Y-.H. and Ryoo, C.S. (2011) A Note on the Weighted p-Adic q-Euler Measure on . Advanced Studies in Contemporary Mathematics, 21, 35-40.

[9] Kupershmidt, B.A. (2005) Reflection Symmetries of q-Bernoulli Polynomials. Journal of Nonlinear Mathematical Physics, 12, 412-422.

http://dx.doi.org/10.2991/jnmp.2005.12.s1.34

[10] Ryoo, C.S., Kim, T. and Jang, L.-C. (2007) Some Relationships between the Analogs of Euler Numbers and Polynomials. Journal of Inequalities and Applications, 2007, Article ID: 86052.

[11] Ryoo, C.S. (2011) A Note on the Weighted q-Euler Numbers and Polynomials. Advanced Studies in Contemporary Mathematics, 21, 47-54.

[12] Lee, H.Y. and Ryoo, C.S. (2014) On New Polynomials Related to Bell Numbers and Stirling Numbers. Advanced Studies in Theoretical Physics, 8, 763-769.

[13] Ryoo, C.S. and Kang, J.Y. (2015) A Numerical Investigation on the Structure of the Zeros of Euler Polynomials. Discrete Dynamics in Nature and Society, 2015, Article ID: 174173.

[14] Ryoo, C.S. (2014) Analytic Continuation of Euler Polynomials and the Euler Zeta Function. Discrete Dynamics in Nature and Society, 2014, Article ID: 568129.