## Journal of Applied Mathematics

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

#### 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 ${\stackrel{̃}{U}}_{n}\left(x\right)$. We observe an interesting phenomenon of “scattering” of the zeros of the polynomials ${\stackrel{̃}{U}}_{n}\left(x\right)$ in complex plane. We find out some identities and properties related to polynomials ${\stackrel{̃}{U}}_{n}\left(x\right)$. Finally, we also derive interesting relations between polynomials ${\stackrel{̃}{U}}_{n}\left(x\right)$, Stirling numbers, central factorial numbers, and Euler numbers.

#### Article information

Source
J. Appl. Math., Volume 2013 (2013), Article ID 158130, 10 pages.

Dates
First available in Project Euclid: 14 March 2014

https://projecteuclid.org/euclid.jam/1394808152

Digital Object Identifier
doi:10.1155/2013/158130

Mathematical Reviews number (MathSciNet)
MR3115274

Zentralblatt MATH identifier
06950539

#### Citation

Kang, J. Y.; Ryoo, C. S. A Research on a Certain Family of Numbers and Polynomials Related to Stirling Numbers, Central Factorial Numbers, and Euler Numbers. J. Appl. Math. 2013 (2013), Article ID 158130, 10 pages. doi:10.1155/2013/158130. https://projecteuclid.org/euclid.jam/1394808152

#### References

• M. Açikgöz, D. Erdal, and S. Araci, “A new approach to q-Bernoulli numbers and q-Bernoulli polynomials related to q-Bernstein polynomials,” Advances in Difference Equations, vol. 2010, Article ID 951764, 9 pages, 2010.
• A. Bayad, “Modular properties of elliptic Bernoulli and Euler functions,” Advanced Studies in Contemporary Mathematics, vol. 20, no. 3, pp. 389–401, 2010.
• N. K. Govil and V. Gupta, “Convergence of q-Meyer-König-Zeller-Durrmeyer operators,” Advanced Studies in Contemporary Mathematics, vol. 19, no. 1, pp. 97–108, 2009.
• M.-S. Kim and S. Hu, “On p-adic Hurwitz-type Euler zeta functions,” Journal of Number Theory, vol. 132, no. 12, pp. 2977–3015, 2012.
• G. Liu, “The \emphD numbers and the central factorial numbers,” Publicationes Mathematicae Debrecen, vol. 79, no. 1-2, pp. 41–53, 2011.
• H. Ozden and Y. Simsek, “A new extension of q-Euler numbers and polynomials related to their interpolation functions,” Applied Mathematics Letters, vol. 21, no. 9, pp. 934–939, 2008.
• S.-H. Rim, K. H. Park, and E. J. Moon, “On Genocchi numbers and polynomials,” Abstract and Applied Analysis, vol. 2008, Article ID 898471, 7 pages, 2008.
• S.-H. Rim, J.-H. Jin, E.-J. Moon, and S.-J. Lee, “Some identities on the q-Genocchi polynomials of higher-order and q-Stirling numbers by the fermionic p-adic integral on ${\mathbb{Z}}_{p}$,” International Journal of Mathematics and Mathematical Sciences, vol. 2010, Article ID 860280, 14 pages, 2010.
• C. S. Ryoo, “A numerical computation on the structure of the roots of q-extension of Genocchi polynomials,” Applied Mathematics Letters, vol. 21, no. 4, pp. 348–354, 2008.
• Y. Simsek, V. Kurt, and D. Kim, “New approach to the complete sum of products of the twisted (h,q)-Bernoulli numbers and polynomials,” Journal of Nonlinear Mathematical Physics, vol. 14, no. 1, pp. 44–56, 2007.
• Y. Simsek, “Generating functions of the twisted Bernoulli numbers and polynomials associated with their interpolation functions,” Advanced Studies in Contemporary Mathematics, vol. 16, no. 2, pp. 251–278, 2008.
• Y. Simsek, “Twisted (h,q)-Bernoulli numbers and polynomials related to twisted (h,q)-zeta function and \emphL-function,” Journal of Mathematical Analysis and Applications, vol. 324, no. 2, pp. 790–804, 2006.
• Z.-H. Sun, “Congruences for sequences similar to Euler numbers,” Journal of Number Theory, vol. 132, no. 4, pp. 675–700, 2012.
• H. M. Srivastava, B. Kurt, and Y. Simsek, “Some families of Genocchi type polynomials and their interpolation functions,” Integral Transforms and Special Functions, vol. 23, no. 12, pp. 919–938, 2012.
• Y. Simsek, “Generating functions for generalized Stirling type numbers, Array type polynomials, Eulerian type polynomials and their applications,” Fixed Point Theory and Applications, vol. 2013, article 87, 2013.
• Y. Simsek, A. Bayad, and V. Lokesha, “q-Bernstein polynomials related to q-Frobenius-Euler polynomials, l-functions, and q-Stirling numbers,” Mathematical Methods in the Applied Sciences, vol. 35, no. 8, pp. 877–884, 2012.
• R. Dere, Y. Simsek, and H. M. Srivastava, “A unified presentation of three families of generalized Apostol type polynomials based upon the theory of the umbral calculus and the umbral algebra,” Journal of Number Theory, vol. 133, no. 10, pp. 3245–3263, 2013.