## Abstract and Applied Analysis

### On the Symmetries of the $q$-Bernoulli Polynomials

Taekyun Kim

#### Abstract

Kupershmidt and Tuenter have introduced reflection symmetries for the $q$-Bernoulli numbers and the Bernoulli polynomials in (2005), (2001), respectively. However, they have not dealt with congruence properties for these numbers entirely. Kupershmidt gave a quantization of the reflection symmetry for the classical Bernoulli polynomials. Tuenter derived a symmetry of power sum polynomials and the classical Bernoulli numbers. In this paper, we study the new symmetries of the $q$-Bernoulli numbers and polynomials, which are different from Kupershmidt's and Tuenter's results. By using our symmetries for the $q$-Bernoulli polynomials, we can obtain some interesting relationships between $q$-Bernoulli numbers and polynomials.

#### Article information

Source
Abstr. Appl. Anal., Volume 2008 (2008), Article ID 914367, 7 pages.

Dates
First available in Project Euclid: 10 February 2009

https://projecteuclid.org/euclid.aaa/1234298991

Digital Object Identifier
doi:10.1155/2008/914367

Mathematical Reviews number (MathSciNet)
MR2448390

Zentralblatt MATH identifier
1217.11022

#### Citation

Kim, Taekyun. On the Symmetries of the $q$ -Bernoulli Polynomials. Abstr. Appl. Anal. 2008 (2008), Article ID 914367, 7 pages. doi:10.1155/2008/914367. https://projecteuclid.org/euclid.aaa/1234298991

#### References

• M. Cenkci, M. Can, and V. Kurt, $p$-adic interpolation functions and Kummer-type congruences for $q$-twisted and $q$-generalized twisted Euler numbers,'' Advanced Studies in Contemporary Mathematics, vol. 9, no. 2, pp. 203--216, 2004.
• 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.
• T. Kim, $q$-Bernoulli numbers and polynomials associated with Gaussian binomial coefficients,'' Russian Journal of Mathematical Physics, vol. 15, no. 1, pp. 51--57, 2008.
• T. Kim, $q$-extension of the Euler formula and trigonometric functions,'' Russian Journal of Mathematical Physics, vol. 14, no. 3, pp. 275--278, 2007.
• T. Kim, J. Y. Choi, and J. Y. Sug, Extended $q$-Euler numbers and polynomials associated with fermionic $p$-adic $q$-integral on $\mathbbZ_p$,'' Russian Journal of Mathematical Physics, vol. 14, no. 2, pp. 160--163, 2007.
• T. Kim, Multiple $p$-adic $L$-function,'' Russian Journal of Mathematical Physics, vol. 13, no. 2, pp. 151--157, 2006.
• H. M. Srivastava, T. Kim, and Y. Simsek, $q$-Bernoulli numbers and polynomials associated with multiple $q$-zeta functions and basic $L$-series,'' Russian Journal of Mathematical Physics, vol. 12, no. 2, pp. 241--268, 2005.
• T. Kim, Power series and asymptotic series associated with the $q$-analog of the two-variable $p$-adic $L$-function,'' Russian Journal of Mathematical Physics, vol. 12, no. 2, pp. 186--196, 2005.
• T. Kim, Analytic continuation of multiple $q$-zeta functions and their values at negative integers,'' Russian Journal of Mathematical Physics, vol. 11, no. 1, pp. 71--76, 2004.
• T. Kim, Non-Archimedean $q$-integrals associated with multiple Changhee $q$-Bernoulli polynomials,'' Russian Journal of Mathematical Physics, vol. 10, no. 1, pp. 91--98, 2003.
• T. Kim, On Euler-Barnes multiple zeta functions,'' Russian Journal of Mathematical Physics, vol. 10, no. 3, pp. 261--267, 2003.
• T. Kim, $q$-Volkenborn integration,'' Russian Journal of Mathematical Physics, vol. 9, no. 3, pp. 288--299, 2002.
• T. Kim, A note on $p$-adic $q$-integral on $\mathbbZ_p$ associated with $q$-Euler numbers,'' Advanced Studies in Contemporary Mathematics, vol. 15, no. 2, pp. 133--137, 2007.
• B. A. Kupershmidt, Reflection symmetries of $q$-Bernoulli polynomials,'' Journal of Nonlinear Mathematical Physics, vol. 12, supplement 1, pp. 412--422, 2005.
• H. Ozden, Y. Simsek, S.-H. Rim, and I. N. Cangul, A note on $p$-adic $q$-Euler measure,'' Advanced Studies in Contemporary Mathematics, vol. 14, no. 2, pp. 233--239, 2007.
• M. Schork, Ward's calculus of sequences'', $q$-calculus and the limit $q\rightarrow-1$,'' Advanced Studies in Contemporary Mathematics, vol. 13, no. 2, pp. 131--141, 2006.
• 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, Theorems on twisted $L$-function and twisted Bernoulli numbers,'' Advanced Studies in Contemporary Mathematics, vol. 11, no. 2, pp. 205--218, 2005.
• 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.
• H. J. H. Tuenter, A symmetry of power sum polynomials and Bernoulli numbers,'' The American Mathematical Monthly, vol. 108, no. 3, pp. 258--261, 2001.
• S.-L. Yang, An identity of symmetry for the Bernoulli polynomials,'' Discrete Mathematics, vol. 308, no. 4, pp. 550--554, 2008.
• P. T. Young, Degenerate Bernoulli polynomials, generalized factorial sums, and their applications,'' Journal of Number Theory, vol. 128, no. 4, pp. 738--758, 2008. \endthebibliography