## Taiwanese Journal of Mathematics

### A NOTE ON THE GENERALIZED HIGHER-ORDER $q$-BERNOULLI NUMBERS AND POLYNOMIALS WITH WEIGHT $\alpha$

#### Abstract

In this paper we give some interesting equation of $p$-adic $q$-integrals on $\mathbb Z_p$. From those $p$-adic $q$-integrals, we present a systemic study of some families of extended Carlitz $q$-Bernoulli numbers and polynomials with weight $\alpha$ in $p$-adic number field.

#### Article information

Source
Taiwanese J. Math., Volume 17, Number 3 (2013), 785-800.

Dates
First available in Project Euclid: 10 July 2017

https://projecteuclid.org/euclid.twjm/1499705983

Digital Object Identifier
doi:10.11650/tjm.17.2013.2396

Mathematical Reviews number (MathSciNet)
MR3072261

Zentralblatt MATH identifier
1282.11013

#### Citation

Lee, H. Y.; Ryoo, C. S. A NOTE ON THE GENERALIZED HIGHER-ORDER $q$-BERNOULLI NUMBERS AND POLYNOMIALS WITH WEIGHT $\alpha$. Taiwanese J. Math. 17 (2013), no. 3, 785--800. doi:10.11650/tjm.17.2013.2396. https://projecteuclid.org/euclid.twjm/1499705983

#### References

• L. Carlitz, Expansions of $q$-Bernoulli numbers, Duke Mathematical Journal, 25 (1958), 355-364.
• L. Carlitz, $q$-Bernoulli numbers and polynomials, Duke Mathematical Journal, 15 (1948), 987-1000.
• M. Cenkci and V. Kurt, Congruences for generalized $q$-Bernoulli polynomials, J. Inequl. Appl, Art ID 270723, 2008, pp. 19.
• T. Kim, J. Choi, B. Lee and C. S. Ryoo, Some Identities on the Generalized $q$-Bernoulli Numbers and Polynomials Associated with $q$-Volkenborn Integrals, Journal of Inequalities and Applications, Vol. 2010, Article ID 575240, 17 pages.
• T. kim, On the weighted $q$-Bernoulli numbers and polynomials, Advanced Studies in Contemporary Mathematics, 21(2) (2011), 207-215.
• T. Kim, S. H. Lee, D. V. Dolgy and C. S. Ryoo, A Note on the Generalized $q$-Bernoulli Measures with Weight $\alpha$, Abstract and Applied Analysis, Vol. 2011, Article ID 867217, 9 pages.
• T. Kim and J. Choi, On the $q$-Bernoulli Numbers and Polynomials with Weight $\alpha$, Abstract and Applied Analysis, Vol. 2011, Article ID 392025, 14 pages.
• T. Kim, An analogue of Bernoulli numbers and their applications, Rep. Fac. Sci. Engrg. Saga Univ. Math., 22 (1994), 21-26.
• T. Kim, $q$-Volkenborn integration, Russ. J. Math. Phys., 9 (2002), 288-299.
• T. Kim, On the weighted $q$-Bernoulli numbers and polynomials, Advanced Studies in Contemporary Mathematics, 21 (2011), 207-215.
• T. Kim, J. Choi and Y. H. Kim, $q$-Bernstein polynomials associated with $q$-Stirling numbers and Carlitzs $q$-Bernoulli numbers, Abstract and Applied Analysis, Article ID 150975, 2010, pp. 11.
• H. Ozden, I. N. Cangul and Y. Simsek, Remarks on $q$-Bernoulli numbers associated with Daehee numbers, Advanced Studies in Contemporary Mathematics, 18 (2009), 41-48.
• H. Sirivastave, T. Kim and Y. Simsek, $q$-Bernoulli numbers and polynomials associated with multiple $q$-zeta functions and basic L-series, Russ. J. Math. Phys., 12(2) (2005), 201-228.
• Y. Simsek, Generating functions of the twisted Bernoulli numbers and polynomials associated with their interpolation functions, Advanced Studies in Contemporary Mathematics, 16 (2008), 251-278.
• Y. Simesk, V. Kurt and D. Kim, New approach to the complete sum of products of the twisted $(h, q)$-Bernoulli numbers and polynomials, J. Nonlinear Math. Phys., 14 (2007), 44-56.
• Y. Simsek, Theorem on twisted $L$-function and twisted Bernoulli numbers, Advan. Stud. Contemp. Math., 12 (2006), 237-246.