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Kim, Min-Soo; Kim, Taekyun; Son, Jin-Woo

Multivariate \(p\)-adic fermionic \(q\)-integral on \(\mathbb Z_{p}\) and related multiple zeta-type functions. (English) Zbl 1195.11156

Abstr. Appl. Anal. 2008, Article ID 304539, 13 p. (2008).
Summary: In [Abstr. Appl. Anal. 2008, Article ID 498173, 7 p. (2008; Zbl 1195.11029)], L.-C. Jang and C.-S. Ryoo constructed generating functions of the multiple twisted Carlitz’s type \(q\)-Bernoulli polynomials and obtained the distribution relation for them. They also raised the following problem: “are there analytic multiple twisted Carlitz’s type \(q\)-zeta functions which interpolate multiple twisted Carlitz’s type \(q\)-Euler (Bernoulli) polynomials?” The aim of this paper is to give a partial answer to this problem. Furthermore we derive some interesting identities related to twisted \(q\)-extension of Euler polynomials and multiple twisted Carlitz’s type \(q\)-Euler polynomials.

Cited in 2 Documents

MSC:

11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)
11B65 Binomial coefficients; factorials; \(q\)-identities
11B68 Bernoulli and Euler numbers and polynomials
33D15 Basic hypergeometric functions in one variable, \({}_r\phi_s\)

Keywords:

identities; twisted \(q\)-extension of Euler polynomials; multiple twisted Carlitz’s type \(q\)-Euler polynomials

Citations:

Zbl 1195.11029
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Full Text: DOI
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References:

[1] T. Kim, “On the q-extension of Euler and Genocchi numbers,” Journal of Mathematical Analysis and Applications, vol. 326, no. 2, pp. 1458-1465, 2007. · Zbl 1112.11012 · doi:10.1016/j.jmaa.2006.03.037
[2] T. Kim, “On p-adic interpolating function for q-Euler numbers and its derivatives,” Journal of Mathematical Analysis and Applications, vol. 339, no. 1, pp. 598-608, 2008. · Zbl 1160.11013 · doi:10.1016/j.jmaa.2007.07.027
[3] T. Kim, “On the multiple q-Genocchi and Euler numbers,” to appear in Russian Journal of Mathematical Physics. · Zbl 1192.11011
[4] T. Kim, M.-S. Kim, L.-C. Jang, and S.-H. Rim, “New q-Euler numbers and polynomials associated with p-adic q-integrals,” Advanced Studies in Contemporary Mathematics, vol. 15, no. 2, pp. 243-252, 2007. · Zbl 1217.11117
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[7] H. Ozden, I. N. Cangul, and Y. Simsek, “Remarks on sum of products of (h,q)-twisted Euler polynomials and numbers,” Journal of Inequalities and Applications, vol. 2008, Article ID 816129, 8 pages, 2008. · Zbl 1140.11314 · doi:10.1155/2008/816129
[8] T. Kim, L.-C. Jang, and C.-S. Ryoo, “Note on q-extensions of Euler numbers and polynomials of higher order,” Journal of Inequalities and Applications, vol. 2008, Article ID 371295, 9 pages, 2008. · Zbl 1195.11030 · doi:10.1155/2008/371295
[9] Y. Simsek, “q-analogue of twisted l-series and q-twisted Euler numbers,” Journal of Number Theory, vol. 110, no. 2, pp. 267-278, 2005. · Zbl 1114.11019 · doi:10.1016/j.jnt.2004.07.003
[10] L.-C. Jang and C.-S. Ryoo, “A note on the multiple twisted Carlitz’s type q-Bernoulli polynomials,” Abstract and Applied Analysis, vol. 2008, Article ID 498173, 7 pages, 2008. · Zbl 1195.11029 · doi:10.1155/2008/498173
[11] L.-C. Jang, “Multiple twisted q-Euler numbers and polynomials associated with p-adic q-integrals,” Advances in Difference Equations, vol. 2008, Article ID 738603, 11 pages, 2008. · Zbl 1140.11311 · doi:10.1155/2008/738603
[12] Y. Yamasaki, “On q-analogues of the Barnes multiple zeta functions,” Tokyo Journal of Mathematics, vol. 29, no. 2, pp. 413-427, 2006. · Zbl 1192.11060 · doi:10.3836/tjm/1170348176
[13] M. Wakayama and Y. Yamasaki, “Integral representations of q-analogues of the Hurwitz zeta function,” Monatshefte für Mathematik, vol. 149, no. 2, pp. 141-154, 2006. · Zbl 1110.11029 · doi:10.1007/s00605-005-0369-1
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