Abstract
In this paper, we investigate some properties of q-Bernoulli polynomials arising from q-umbral calculus. We find a formula for expressing any polynomial as a linear combination of q-Bernoulli polynomials with explicit coefficients. Also, we establish some connections between q-Bernoulli polynomials and higher-order q-Bernoulli polynomials.
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Açikgöz M, Erdal D, Araci S. A new approach to q-Bernoulli numbers and q-Bernoulli polynomials related to q-Bernstein polynomials. Adv Difference Equ, 2010, Article ID: 951764
Araci S, Acikgoz M, Kilicman A. Extended p-adic q-invariant integrals on Z p associated with applications of Umbral calculus. Adv Difference Equ, 2013, 2013: 96
Bayad A, Kim T. Identities involving values of Bernstein, q-Bernstein, q-Bernoulli, and q-Euler polynomials. Russ J Math Phys, 2011, 18: 133–143
Carlitz L. q-Bernoulli numbers and polynomials. Duke Math J, 1948, 15: 987–1000
Hegazi A S, Mansour M. A note on q-Bernoulli numbers and polynomials. J Nonlinear Math Phys, 2006, 13: 9–18
Ismail M E H, Rahman M. Inverse operators, q-fractional integrals, and q-Bernoulli polynomials. J Approx Theory, 2002, 114: 269–307
Kim D S, Kim T, Lee S H, et al. Some identities of Bernoulli, Euler and Abel polynomials arising from umbral calculus. Adv Difference Equ, 2013, 2013: 15
Kim T. q-Generalized Euler numbers and polynomials. Russ J Math Phys, 2006, 13: 293–298
Kim T. q-Bernoulli numbers and polynomials associated with Gaussian binomial coefficients. Russ J Math Phys, 2008, 15: 51–57
Kupershmidt B O. Reflection symmetries of q-Bernoulli polynomials. J Nonlinear Math Phys, 2005, 12: 412–422
Kurt V, Cenkci M. A new approach to q-Genocchi numbers and polynomials. Bull Korean Math Soc, 2010, 47: 575–583
Mahmudov N I. On a class of q-Bernoulli and q-Euler polynomials. Adv Difference Equ, 2013, 2013: 108
Mahmudov N I, Keleshteri M E. On a class of generalized q-Bernoulli and q-Euler polynomials. Adv Difference Equ, 2013, 2013: 115
Rim S H, Bayad A, Moon E J, et al. A new construction on the q-Bernoulli polynomials. Adv Difference Equ, 2011, 2011: 34
Roman S. More on the umbral calculus, with emphasis on the q-umbral calculus. J Math Anal Appl, 1985, 107: 222–254
Roman S. The Umbral Calculus. New York: Dover, 2005
Sharma A. q-Bernoulli and Euler numbers of higher order. Duke Math J, 1958, 25: 343–353
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Kim, D.S., Kim, T.K. q-Bernoulli polynomials and q-umbral calculus. Sci. China Math. 57, 1867–1874 (2014). https://doi.org/10.1007/s11425-014-4821-3
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DOI: https://doi.org/10.1007/s11425-014-4821-3