On p-adic Integral Representation of q-Bernoulli Numbers Arising from Two Variable q-Bernstein Polynomials
Abstract
:1. Introduction
2. Some Integral Representations of q-Bernoulli Numbers and Polynomials
3. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
- Bayad, A.; Kim, T. Identities involving values of Bernstein q-Bernoulli, and q-Euler polynomials. Russ. J. Math. Phys. 2011, 18, 133–143. [Google Scholar] [CrossRef]
- Carlitz, L. Expansions of q-Bernoulli numbers. Duke Math. J. 1958, 25, 355–364. [Google Scholar] [CrossRef]
- Carlitz, L. q-Bernoulli and Eulerian numbers. Trans. Am. Math. Soc. 1954, 76, 332–350. [Google Scholar]
- Kim, T. Some identities on the q-integral representation of the product of several q-Bernstein-type polynomials. Abstr. Appl. Anal. 2011, 2011, 634675. [Google Scholar] [CrossRef]
- Jang, L.C.; Kim, T.; Kim, D.S.; Dolgy, D.V. On p-adic fermionic integrals of q-Bernstein polynomials associated with q-Euler numbers and polynomials. Symmetry 2018, 10, 311. [Google Scholar] [CrossRef]
- Kim, T. A note on q-Bernstein polynomials. Russ. J. Math. Phys. 2011, 18, 73–82. [Google Scholar] [CrossRef]
- Kim, T. A study on the q-Euler numbers and the fermionic q-integral of the product of several type q-Bernstein polynomials on ℤp. Adv. Stud. Contemp. Math. 2013, 23, 5–11. [Google Scholar]
- Kim, T. On p-adic q-Bernoulli numbers. J. Korean Math. Soc. 2000, 37, 21–30. [Google Scholar]
- Kim, T.; Kim, H.S. Remark on the p-adic q-Bernoulli numbers. Algerbraic number theory (Hapcheon/Saga, 1996). Adv. Stud. Contemp. Math. 1999, 1, 127–136. [Google Scholar]
- Kim, T. q-Volkenborn integration. Russ. J. Math. Phys. 2002, 9, 288–299. [Google Scholar]
- Kurt, V. Some relation between the Bernstein polynomials and second kind Bernpolli polynomials. Adv. Stud. Contemp. Math. 2013, 23, 43–48. [Google Scholar]
- Oruç, H.; Phillips, G.M. A generalization of Bernstein polynomials. Proc. Edinb. Math. Soc. 1999, 42, 403–413. [Google Scholar] [CrossRef]
- Oruç, H.; Tuncer, N. On the convergence and iterates of q-Bernstein polynomials. J. Approx. Theory 2002, 117, 301–313. [Google Scholar] [CrossRef]
- Ostrovska, S. On the q-Bernstein polynomials. Adv. Stud. Contemp. Math. 2015, 11, 193–204. [Google Scholar]
- Ostrovska, S. On the q-Bernstein polynomials of the logarithmic function in the case q > 1. Math. Slovaca 2016, 66, 73–78. [Google Scholar] [CrossRef]
- Phillips, G.M. Bernstein polynomials based on the q-integers. Ann. Numer. Math. 1997, 4, 511–518. [Google Scholar]
- Rim, S.-H.; Joung, J.; Jin, J.-H.; Lee, S.-J. A note on the weighted Carlitz’s type q-Euler numbers and q-Bernstein polynomials. Proc. Jangjeon Math. Soc. 2012, 15, 195–201. [Google Scholar]
- Kim, D.S.; Kim, T. Some p-adic integrals on ℤp associated with trigonometric functions. Russ. J. Math. Phys. 2018, 25, 300–308. [Google Scholar] [CrossRef]
- Siddiqui, M.A.; Agrawal, R.R.; Gupta, N. On a class of modified new Bernstein operators. Adv. Stud. Contemp. Math. 2014, 24, 97–107. [Google Scholar]
- Simsek, Y. On parametrization of the q-Bernstein basis functions and their applications. J. Inequal. Spec. Funct. 2017, 8, 158–169. [Google Scholar]
- Kim, T.; Kim, D.S. Identities for degenerate Bernoulli polynomials and Korobov polynomials of the first kind. Sci. China Math. 2018. [Google Scholar] [CrossRef]
- Taberski, R. Approximation properties of the integral Bernstein operators and their derivatives in some classes of locally integral functions. Funct. Approx. Comment. Math. 1992, 21, 85–96. [Google Scholar]
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Kim, D.S.; Kim, T.; Ryoo, C.S.; Yao, Y. On p-adic Integral Representation of q-Bernoulli Numbers Arising from Two Variable q-Bernstein Polynomials. Symmetry 2018, 10, 451. https://doi.org/10.3390/sym10100451
Kim DS, Kim T, Ryoo CS, Yao Y. On p-adic Integral Representation of q-Bernoulli Numbers Arising from Two Variable q-Bernstein Polynomials. Symmetry. 2018; 10(10):451. https://doi.org/10.3390/sym10100451
Chicago/Turabian StyleKim, Dae San, Taekyun Kim, Cheon Seoung Ryoo, and Yonghong Yao. 2018. "On p-adic Integral Representation of q-Bernoulli Numbers Arising from Two Variable q-Bernstein Polynomials" Symmetry 10, no. 10: 451. https://doi.org/10.3390/sym10100451