×

\(q\)-Bernstein polynomials related to \(q\)-Frobenius-Euler polynomials, \(l\)-functions, and \(q\)-Stirling numbers. (English) Zbl 1262.11039

Summary: The aim of this paper is to derive new identities and relations associated with the \(q\)-Bernstein polynomials, \(q\)-Frobenius–Euler polynomials, \(l\)-functions, and \(q\)-Stirling numbers of the second kind. We also give some applications related to theses polynomials and numbers.

MSC:

11B68 Bernoulli and Euler numbers and polynomials
11B73 Bell and Stirling numbers
05A30 \(q\)-calculus and related topics
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Kac, Quantum Calculus (2002)
[2] Kim, A note on p-adic q-integral on Zp associated with q-Euler numbers, Advanced Studies in Contemporary Mathematics 15 pp 133– (2007)
[3] Kim, q-Volkenborn integration, Russian Journal of Mathematical Physics 19 pp 288– (2002)
[4] Kim, On the analogs of Bernoulli and Euler numbers, related identities and zeta and L-functions, Journal of the Korean Mathematical Society 45 pp 435– (2008) · Zbl 1223.11145
[5] Luo, q-Extensions of some relationships between the Bernoulli and Euler polynomials, Taiwanese Journal of Mathematics 15 pp 241– (2011)
[6] Phillips, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC 14 (2003)
[7] Satoh, q analogue of Riemann’s {\(\zeta\)}-function and q-Euler numbers, Journal Number Theory 31 pp 346– (1989) · Zbl 0675.12010
[8] Satoh, A construction of q analogue of Dedekind sums, Nagoya Mathematical Journal 127 pp 129– (1992) · Zbl 0761.11023
[9] Simsek, On twisted generalized Euler numbers, Bulletin of the Korean Mathematical Society 41 pp 299– (2004) · Zbl 1143.11305
[10] Simsek, q analogue of the twisted l-series and q-twisted Euler numbers, Journal Number Theory 110 pp 267– (2005) · Zbl 1114.11019
[11] Simsek, On q-deformed Stirling numbers, International Journal of Mathematics and Computation pp 70– (2012)
[12] Simsek, On interpolation functions of the twisted generalized Frobenius-Euler numbers, Advanced Studies in Contemporary Mathematics 15 pp 187– (2007)
[13] Simsek, A new generating function of q-Bernestein type polynomials and their interpolation function, Abstract and Applied Analysis 2010 (2010) · Zbl 1185.33013
[14] Simsek Y Acikgoz M Bayad A Lokesha V q -Frobenius-Euler polynomials related to the ( q -)Bernstein type polynomials. Numerical Analysis and Applied Mathematics Vols I-III Book Series; AIP Conference Proceedings 2010 1281 1156 1159
[15] Srivastava, q-Bernoulli numbers and polynomials associated with multiple q-zeta functions and basic L-series, Russian Journal of Mathematical Physics 12 pp 241– (2005) · Zbl 1200.11018
[16] Tsumura, A note on q-analogues of the Dirichlet series and q-Bernoulli numbers, Journal Number Theory 39 pp 251– (1991) · Zbl 0735.11009
[17] Bernstein, Démonstration du théorème de Weierstrass fondée sur la calcul des probabilités, Communications of the Mathematical Society Charkow Sér. 2 t 13 pp 1– (1912)
[18] Goldman, Pyramid Algorithms: A Dynamic Programming Approach to Curves and Surfaces for Geometric Modeling (2002)
[19] Goldman, Graphics Gems V pp 149– (1995)
[20] Tsumura, On a p-adic interpolation of generalized Euler numbers and its applications, Tokyo Journal of Mathematics 10 pp 281– (1987) · Zbl 0641.12007
[21] Carlitz, q-Bernoulli numbers and polynomials, Duke Mathematical Journal 15 pp 987– (1948) · Zbl 0032.00304
[22] Cangul, A note on interpolation functions of the Frobenius-Euler numbers, Applications of Mathematics in Technical and Natural Sciences, AIP Conference Proceedings 1301 pp 59– (2010)
[23] Choi, The multiple Hurwitz zeta function and the multiple Hurwitz-Euler eta function, Taiwanese Journal of Mathematics 15 pp 501– (2011) · Zbl 1273.11133
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.