Dynamics of the zeros of analytic continued polynomials and differential equations associated with $q$-tangent polynomials

Ryoo, Cheon Seoung ¹ ; Hwang, Kyung Won ² ; Kim, Do Jin ³ ; Jung, Nam Soon ⁴

¹ Department of Mathematics, Hannam University, Daejeon 306-791, Republic of Korea
² Department of Mathematics, Dong-A University, Busan 604-714, Republic of Korea
³ Department of Mathematics, Kyungpook National University, Daegu, 702-701, Republic of Korea
⁴ College of Talmage Liberal Arts, Hannam University,, Daejeon 306-791, Republic of Korea

Journal of nonlinear sciences and its applications, Tome 11 (2018) no. 6, p. 785-797.

Résumé

In this paper, we study the analytic continuation $T_q(s)$ and $T_q(s,w)$ of the $q$-Tangent numbers $T_{n, q}$ and $q$-Tangent polynomials $T_{n, q}(x)$ introduced by authors. The new concept of dynamics of the zeros of analytic continued $q$-tangent polynomials is investigated observing an interesting phenomenon of `scattering' of the zeros of $T_q(s, w)$. Finally, we study linear differential equations arising from the generating functions of $q$-tangent polynomials giving explicit identities for the $q$-tangent polynomials.

Détail

DOI : 10.22436/jnsa.011.06.06

Classification : 11B68, 11S40
Keywords: Tangent numbers and polynomials, \(q\)-tangent polynomial, \(q\)-tangent Zeta function, analytic continuation, analytic continued \(q\)-tangent polynomials, zeros, differential equations

Bibliographie

[1] Ayoub, R. Euler and the zeta function, Amer. Math., Monthly, Volume 81 (1974), pp. 1067-1086

[2] Erdélyi, A.; Magnus, W.; Oberhettinger, F.; Tricomi, F. G. Higher Transcendental Functions, Vol. III., Krieger Publishing Co., Melbourne, 1981

[3] Kang, J. Y.; Lee, H. Y.; N. S. Jung Some relations of the twisted q-Genocchi numbers and polynomials with weight \(\alpha\) and weak Weight \(\beta\), Abstr. Appl. Anal., Volume 2012 (2012), pp. 1-9 | Zbl

[4] T. Kim Euler numbers and polynomials associated with zeta functions, Abstr. Appl. Anal., Volume 2008 (2008), pp. 1-11

[5] Kim, M.-S.; S. Hu On p-adic Hurwitz-type Euler Zeta functions, J. Number Theory, Volume 132 (2012), pp. 2977-3015 | DOI

[6] Kim, T.; Kim, D. S.; Ryoo, C. S.; H. I. Kwon Differential equations associated with Mahler and Sheffer-Mahler polynomials ( submitted for publication.)

[7] Kim, T.; Ryoo, C. S.; Jang, L. C.; Rim, S. H. Exploring the q-Riemann Zeta function and q-Bernoulli polynomials, Discrete Dyn. Nat. Soc., Volume 2005 (2005), pp. 171-181 | Zbl

[8] Ozden, H.; Y. Simsek A new extension of q-Euler numbers and polynomials related to their interpolation functions, Appl. Math. Lett., Volume 21 (2008), pp. 934-938 | Zbl | DOI

[9] Rim, S. H.; Park, K. H.; E. J. Moon On Genocchi numbers and polynomials, Abstr. Appl. Anal., Volume 2008 (2008), pp. 1-7

[10] Ryoo, C. S. A Note on the Tangent Numbers and Polynomials, Adv. Studies Theor. Phys., Volume 7 (2013), pp. 447-454

[11] Ryoo, C. S. On the q-Tangent Numbers and Polynomials, Appl. Math. Sci. (Ruse), Volume 7 (2013), pp. 4935-4941 | DOI

[12] Ryoo, C. S. A Numerical investigation on the zeros of the tangent polynomials, J. Appl. Math. Inform., Volume 32 (2014), pp. 315-322 | Zbl | DOI

[13] Ryoo, C. S. Analytic Continuation of Euler Polynomials and the Euler Zeta Function, Discrete Dyn. Nat. Soc., Volume 2014 (2014), pp. 1-6

[14] Ryoo, C. S. Differential equations associated with tangent numbers, J. Appl. Math. Inform., Volume 34 (2016), pp. 487-494 | DOI | Zbl

[15] Ryoo, C. S.; Kim, T.; Agarwal, R. P. A numerical investigation of the roots of q-polynomials, Int. J. Comput. Math., Volume 83 (2006), pp. 223-234 | Zbl | DOI

[16] Simsek, Y. Twisted (h, q)-Bernoulli numbers and polynomials related to twisted (h, q)-zeta function and L-function, J. Math. Anal. Appl., Volume 324 (2006), pp. 790-804 | Zbl | DOI

[17] Simsek, Y. Generating functions of the twisted Bernoulli numbers and polynomials, Adv. Stud. Contemp. Math., Volume 16 (2008), pp. 251-278