## Abstract and Applied Analysis

### A Note on the Multiple Twisted Carlitz's Type $q$-Bernoulli Polynomials

#### Abstract

We give the twisted Carlitz's type $q$-Bernoulli polynomials and numbers associated with $p$-adic $q$-inetgrals and discuss their properties. Furthermore, we define the multiple twisted Carlitz's type $q$-Bernoulli polynomials and numbers and obtain the distribution relation for them.

#### Article information

Source
Abstr. Appl. Anal., Volume 2008 (2008), Article ID 498173, 7 pages.

Dates
First available in Project Euclid: 9 September 2008

https://projecteuclid.org/euclid.aaa/1220969157

Digital Object Identifier
doi:10.1155/2008/498173

Mathematical Reviews number (MathSciNet)
MR2393122

Zentralblatt MATH identifier
1195.11029

#### Citation

Jang, Lee-Chae; Ryoo, Cheon-Seoung. A Note on the Multiple Twisted Carlitz's Type $q$ -Bernoulli Polynomials. Abstr. Appl. Anal. 2008 (2008), Article ID 498173, 7 pages. doi:10.1155/2008/498173. https://projecteuclid.org/euclid.aaa/1220969157

#### References

• L. Carlitz, “$q$-Bernoulli numbers and polynomials,” Duke Mathematical Journal, vol. 15, no. 4, pp. 987–1000, 1948.
• L.-C. Jang, “On a $q$-analogue of the $p$-adic generalized twisted $L$-functions and $p$-adic $q$-integrals,” Journal of the Korean Mathematical Society, vol. 44, no. 1, pp. 1–10, 2007.
• L.-C. Jang, T. Kim, and D.-W. Park, “Kummer congruence for the Bernoulli numbers of higher order,” Applied Mathematics and Computation, vol. 151, no. 2, pp. 589–593, 2004.
• T. Kim, “$q$-Volkenborn integration,” Russian Journal of Mathematical Physics, vol. 9, no. 3, pp. 288–299, 2002.
• T. Kim, “On a $q$-analogue of the $p$-adic log gamma functions and related integrals,” Journal of Number Theory, vol. 76, no. 2, pp. 320–329, 1999.
• T. Kim, “Some formulae for the $q$-Bernoulli and Euler polynomials of higher order,” Journal of Mathematical Analysis and Applications, vol. 273, no. 1, pp. 236–242, 2002.
• T. Kim, “On $p$-adic $q$-$L$-functions and sums of powers,” Discrete Mathematics, vol. 252, no. 1–3, pp. 179–187, 2002.
• T. Kim, “An invariant $p$-adic $q$-integrals on $\mathbbZ_p$,” Applied Mathematics Letters, vol. 21, pp. 105–108, 2008.
• T. Kim and H. S. Kim, “Remark on $p$-adic $q$-Bernoulli numbers,” Advanced Studies in Contemporary Mathematics, vol. 1, pp. 127–136, 1999.
• T. Kim and J.-S. Cho, “A note on multiple Dirichlet's $q$-$L$-function,” Advanced Studies in Contemporary Mathematics, vol. 11, no. 1, pp. 57–60, 2005.
• T. Kim, “Sums of products of $q$-Bernoulli numbers,” Archiv der Mathematik, vol. 76, no. 3, pp. 190–195, 2001.
• M. Cenkci, Y. Simsek, and V. Kurt, “Further remarks on multiple $p$-adic $q$-$L$-function of two variables,” Advanced Studies in Contemporary Mathematics, vol. 14, no. 1, pp. 49–68, 2007.
• Y. Simsek, “Twisted $(h,q)$-Bernoulli numbers and polynomials related to twisted $(h,q)$-zeta function and $L$-function,” Journal of Mathematical Analysis and Applications, vol. 324, no. 2, pp. 790–804, 2006.
• Y. Simsek, “The behavior of the twisted $p$-adic $(h,q)$-$L$-functions at $s=0$,” Journal of the Korean Mathematical Society, vol. 44, no. 4, pp. 915–929, 2007.
• 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, “On $p$-adic twisted $q$-$L$-functions related to generalized twisted Bernoulli numbers,” Russian Journal of Mathematical Physics, vol. 13, no. 3, pp. 340–348, 2006.
• Y. Simsek, “Twisted čommentPlease update the information of this reference, if possible. $(h,q)$-$L$-functions,” to appear in Journal of Nonlinear Mathematical Physics.
• Y. Simsek, “$p$-adic Dedekind čommentPlease update the information of this reference, if possible. and Hardy-Berndt type sum related to Volkenborn integral on $\mathbbZ_p$,” to appear in Journal of Nonlinear Mathematical Physics.
• H. Ozden, I. N. Cangul, and Y. Simsek, “Remarks on sum čommentWe updated the information of this reference. Please check. of products of $(h,q)$-twisted Euler polynomials and numbers,” Journal of Inequalities and Applications, vol. 2008, Article ID 816129, 8 pages, 2008.
• H. Ozden, I. N. Cangul, and Y. Simsek, “Multivariate interpolation čommentWe updated the information of this reference. Please check. functions of higher-order $q$-Euler numbers and their applications,” Abstract and Applied Analysis, vol. 2008, Article ID 390857, 16 pages, 2008.