×

On the families of \(q\)-Euler polynomials and their applications. (English) Zbl 1311.05010

Summary: In this paper, we focus on applications of \(q\)-Euler polynomials and obtain some new combinatorial relations by using \(q\)-adic \(q\)-integral on \(\mathbb{Z}_p\). Moreover, we derive distribution formula (multiplication theorem) for Dirichlet type of \(q\)-Euler numbers and polynomials with weight \(\alpha\). Also we apply the method of analytic continuation of \(q\)-Euler polynomials which is the main result of this paper.

MSC:

05A10 Factorials, binomial coefficients, combinatorial functions
05A30 \(q\)-calculus and related topics
11B65 Binomial coefficients; factorials; \(q\)-identities
11B68 Bernoulli and Euler numbers and polynomials
11B73 Bell and Stirling numbers
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Araci, S.; Acikgoz, M.; Bagdasaryan, A.; Şen, E., The Legendre polynomials associated with Bernoulli, Euler, Hermite and Bernstein polynomials, Turkish J. Anal. Number Theory, 1, 1-3 (2013), doi:10.12691/tjant-1-1-1
[2] Choi, J.; Kim, H-M.; Kim, Y. H., Some identities on the high order \(q\)-Euler numbers and polynomials with weight 0, Abstract Appl. Anal., 2013, 10 (2013), Art. ID 459763 · Zbl 1276.34005
[3] Kim, T., Analytic continuation of \(q\)-Euler numbers and polynomials, Appl. Math. Lett., 21, 1320-1323 (2008) · Zbl 1183.33038
[4] Kim, T., On \(p\)-adic interpolating function for \(q\)-Euler numbers and its derivatives, J. Math. Anal. Appl., 339, 598-608 (2008) · Zbl 1160.11013
[5] Kim, T., On the weighted \(q\)-Bernoulli numbers and polynomials, Adv. Stud. Contemp. Math., 21, 2, 207-215 (2011) · Zbl 1256.11017
[6] Kim, T., \(q\)-Volkenborn integration, Russ. J. Math. Phys., 9, 288-299 (2002) · Zbl 1092.11045
[7] Kim, T., Some identities on the \(q\)-Euler polynomials of higher order and \(q\)-Stirling numbers by the fermionic \(p\)-adic integral on \(Z_p\), Russ. J. Math. Phys., 16, 4, 484-491 (2009) · Zbl 1192.05011
[8] Kim, T., Analytic continuation of multiple \(q\)-zeta functions and their values at negative integers, Russ. J. Math. Phys., 11, 1, 71-76 (2004) · Zbl 1115.11068
[9] Kim, T., Identities on the weighted \(q\)-Euler numbers and \(q\)-Bernstein polynomials, Adv. Stud. Contemp. Math. (Kyungshang), 22, 1, 7-12 (2012) · Zbl 1320.11014
[10] Kim, T.; Choi, J.; Kim, H.-M., A note on the weighted Lebesgue-Radon-Nikodym theorem with respect to \(p\)-adic invariant integral on \(Z_p\), J. Appl. Math. Inform., 30, 1-2, 211-217 (2012) · Zbl 1262.11100
[11] Kim, T.; Choi, J.; Kim, Y. H.; Ryoo, C. S., A note on the weighted \(p\)-adic \(q\)-Euler measure on \(Z_p\), Adv. Stud. Contemp. Math. (Kyungshang), 21, 1, 35-40 (2011) · Zbl 1276.11032
[12] Kim, T.; Lee, B.; Choi, J.; Kim, Y. H.; Rim, S.-H., On the \(q\)-Euler numbers and weighted \(q\)-Bernstein polynomials, Adv. Stud. Contemp. Math. (Kyungshang), 21, 1, 13-18 (2011) · Zbl 1276.11033
[13] Dolgy, D. V.; Kim, T., A note on the weighted \(q\)-Bernoulli numbers and the weighted \(q\)-Bernstein polynomials, Honam Math. J., 33, 4, 519-527 (2011) · Zbl 1243.11011
[14] Kim, T.; Lee, S. H.; Han, H.-H.; Ryoo, C. S., On the values of the weighted \(q\)-zeta and L-functions, Discrete Dyn. Nat. Soc., 7 (2011), Art. ID 476381 · Zbl 1248.11095
[15] Kim, T.; Dolgy, D. V.; Lee, B.; Rim, S.-H., Identities on the weighted \(q\)-Bernoulli numbers of higher order, Discrete Dyn. Nat. Soc., 6 (2011), Art. ID 918364 · Zbl 1288.11018
[16] Kim, T.; Kim, Y.-H; Ryoo, C. S., Some identities on the weighted \(q\)-Euler numbers and \(q\)-Bernstein polynomials, J. Inequal. Appl., 64 (2011), 7 pp · Zbl 1266.11041
[17] Kim, T.; Bayad, A.; Kim, Y. H., A study on the \(p\)-adic \(q\)-integral representation on \(Z_p\) associated with the weighted \(q\)-Bernstein and \(q\)-Bernoulli polynomials, J. Inequal. Appl., 8 (2011), Art. ID 513821 · Zbl 1221.11230
[18] Kim, T., New approach to \(q\)-Euler polynomials of higher order, Russ. J. Math. Phys., 17, 2, 218-225 (2010) · Zbl 1259.11030
[19] Cetin, E.; Acikgoz, M.; Cangul, I. N.; Araci, S., A note on the \((h, q)\)-Zeta-type function with weight \(\alpha \), J. Inequal. Appl., 100 (2013) · Zbl 1294.11015
[20] Ozden, H., \(q\)-Dirichlet type L-functions with weight \(\alpha \), Adv. Dif. Equ., 40 (2013) · Zbl 1370.11038
[21] Jolany, H.; Araci, S.; Acikgoz, M.; Seo, J. J., A note on the generalized \(q\)-Genocchi measure with weight \(\alpha \), Bol. Soc. Paran. Math., 31, 1, 17-27 (2013) · Zbl 1413.11125
[22] Simsek, Y., On \(p\)-adic twisted \(q\)-L-functions related to generalized twisted Bernoulli numbers, Russ. J. Math. Phys., 13, 3, 340-348 (2006) · Zbl 1163.11312
[23] Araci, S.; Acikgoz, M.; Gürsul, A., Analytic continuation of weighted \(q\)-Genocchi numbers and polynomials, Commun. Korean Math. Soc., 28, 3, 457-462 (2013) · Zbl 1273.05003
[24] Ryoo, C. S., A note on the weighted \(q\)-Euler numbers and polynomials, Adv. Stud. Contemp. Math., 21, 47-54 (2011) · Zbl 1276.11037
[25] Srivastava, H. M.; Choi, J., Zeta and \(q\)-Zeta Functions and Associated Series and Integrals (2012), Elsevier Science Publishers: Elsevier Science Publishers Amsterdam, London and New York · Zbl 1239.33002
[26] Srivastava, H. M.; Kim, T.; Simsek, Y., \(q\)-Bernoulli numbers and polynomials associated with multiple \(q\)-zeta functions and basic L-series, Russ. J. Math. Phys., 12, 241-268 (2005) · Zbl 1200.11018
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.