×

Some identities of symmetry for the generalized Bernoulli numbers and polynomials. (English) Zbl 1246.11054

Summary: By the properties of \(p\)-adic invariant integral on \(\mathbb Z_p\), we establish various identities concerning the generalized Bernoulli numbers and polynomials. From the symmetric properties of \(p\)-adic invariant integral on \(\mathbb Z_p\), we give some interesting relationship between the power sums and the generalized Bernoulli polynomials.

MSC:

11B68 Bernoulli and Euler numbers and polynomials
11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] T. Kim, “q-Volkenborn integration,” Russian Journal of Mathematical Physics, vol. 9, no. 3, pp. 288-299, 2002. · Zbl 1092.11045
[2] L. Carlitz, “q-Bernoulli numbers and polynomials,” Duke Mathematical Journal, vol. 15, pp. 987-1000, 1948. · Zbl 0032.00304
[3] 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. · Zbl 1143.11043
[4] M. Cenkci, Y. Simsek, and V. Kurt, “Multiple two-variable p-adic q-L-function and its behavior at s=0,” Russian Journal of Mathematical Physics, vol. 15, no. 4, pp. 447-459, 2008. · Zbl 1192.11079
[5] T. Ernst, “Examples of a q-umbral calculus,” Advanced Studies in Contemporary Mathematics, vol. 16, no. 1, pp. 1-22, 2008. · Zbl 1151.33009
[6] A. S. Hegazi and M. Mansour, “A note on q-Bernoulli numbers and polynomials,” Journal of Nonlinear Mathematical Physics, vol. 13, no. 1, pp. 9-18, 2006. · Zbl 1109.33024
[7] 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. · Zbl 1072.11090
[8] 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. · Zbl 1190.11049
[9] T. Kim, “Multiple p-adic L-function,” Russian Journal of Mathematical Physics, vol. 13, no. 2, pp. 151-157, 2006. · Zbl 1140.11352
[10] T. Kim, “q-Euler numbers and polynomials associated with p-adic q-integrals,” Journal of Nonlinear Mathematical Physics, vol. 14, no. 1, pp. 15-27, 2007. · Zbl 1159.11049
[11] T. Kim, “A note on p-adic q-integral on \Bbb Zp associated with q-Euler numbers,” Advanced Studies in Contemporary Mathematics, vol. 15, no. 2, pp. 133-137, 2007. · Zbl 1132.11369
[12] 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. · Zbl 1196.11040
[13] T. Kim, “On the symmetry of the q-Bernoulli polynomials,” Abstract and Applied Analysis, vol. 2008, Article ID 914367, 7 pages, 2008. · Zbl 1217.11022
[14] T. Kim, “Symmetry p-adic invariant integral on \Bbb Zp for Bernoulli and Euler polynomials,” Journal of Difference Equations and Applications, vol. 14, no. 12, pp. 1267-1277, 2008. · Zbl 1229.11152
[15] T. Kim, “Note on q-Genocchi numbers and polynomials,” Advanced Studies in Contemporary Mathematics, vol. 17, no. 1, pp. 9-15, 2008. · Zbl 1246.11047
[16] T. Kim, “Symmetry of power sum polynomials and multivariate fermionic p-adic invariant integral on \Bbb Zp,” Russian Journal of Mathematical Physics, vol. 16, no. 1, pp. 93-96, 2009. · Zbl 1200.11089
[17] Y.-H. Kim, W. Kim, and L.-C. Jang, “On the q-extension of Apostol-Euler numbers and polynomials,” Abstract and Applied Analysis, vol. 2008, Article ID 296159, 10 pages, 2008. · Zbl 1247.11028
[18] B. A. Kupershmidt, “Reflection symmetries of q-Bernoulli polynomials,” Journal of Nonlinear Mathematical Physics, vol. 12, pp. 412-422, 2005. · Zbl 1362.33021
[19] 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. · Zbl 1143.11008
[20] K. H. Park and Y.-H. Kim, “On some arithmetical properties of the Genocchi numbers and polynomials,” Advances in Difference Equations, vol. 2008, Article ID 195049, 14 pages, 2008. · Zbl 1193.11114
[21] M. Schork, “A representation of the q-fermionic commutation relations and the limit q=1,” Russian Journal of Mathematical Physics, vol. 12, no. 3, pp. 394-399, 2005. · Zbl 1200.81090
[22] Y. Simsek, “Theorems on twisted L-function and twisted Bernoulli numbers,” Advanced Studies in Contemporary Mathematics, vol. 11, no. 2, pp. 205-218, 2005. · Zbl 1178.11058
[23] 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. · Zbl 1163.11312
[24] Y. Simsek, “Complete sums of (h,q)-extension of the Euler polynomials and numbers,” http://arxiv.org/abs/0707.2849. · Zbl 1223.11027
[25] Y.-H. Kim and K.-W. Hwang, “Symmetry of power sum and twisted Bernoulli polynomials,” Advanced Studies in Contemporary Mathematics, vol. 18, no. 2, pp. 127-133, 2009. · Zbl 1218.11023
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.