In this paper, by considering higher-order degenerate Bernoulli and Euler polynomials which were introduced by Carlitz, we investigate some properties of mixed-type of those polynomials. In particular, we give some identities of mixed-type degenerate special polynomials which are derived from the fermionic integrals on Zp and the bosonic integrals on Zp.
@article{bwmeta1.element.doi-10_1515_math-2015-0037,
author = {Dae San Kim and Taekyun Kim},
title = {Some identities of degenerate special polynomials},
journal = {Open Mathematics},
volume = {13},
year = {2015},
zbl = {1337.11011},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_1515_math-2015-0037}
}
Dae San Kim; Taekyun Kim. Some identities of degenerate special polynomials. Open Mathematics, Tome 13 (2015) . http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_1515_math-2015-0037/
[1] Araci, S., Acikgoz, M., A note on the Frobenius-Euler numbers and polynomials associated with Bernstein polynomials., Adv. Stud. Contemp. Math. (Kyungshang), 2012, 22(3), 399–406. | Zbl 1261.05003
[2] Bayad A., Chikhi J., Apostol-Euler polynomials and asymptotics for negative binomial reciprocals., Adv. Stud. Contemp. Math. (Kyungshang), 2014, 24(1), 33–37. | Zbl 1317.11026
[3] Carlitz L., Degenerate Stirling, Bernoulli and Eulerian numbers., Utilitas Math., 1979, 15, 51–88.
[4] Ding D., Yang J., Some identities related to the Apostol-Euler and Apostol-Bernoulli polynomials., Adv. Stud. Contemp. Math. (Kyungshang), 2010, 20(1), 7–21. | Zbl 1192.05001
[5] Gaboury S., Tremblay R., Fugère B.-J., Some explicit formulas for certain new classes of Bernoulli, Euler and Genocchi polynomials., Proc. Jangjeon Math. Soc., 2014, 17(1), 115–123. | Zbl 06315773
[6] He Y., Zhang W., A convolution formula for Bernoulli polynomials., Ars Combin., 2013, 108, 97–104. | Zbl 1289.11017
[7] Jeong J.-H., Jin J.-H., Park J.-W., Rim S.-H., On the twisted weak q-Euler numbers and polynomials with weight 0., Proc. Jangjeon Math. Soc., 2013, 16(2), 157–163. | Zbl 1297.11010
[8] Jolany H., and Darafsheh M. R., Some other remarks on the generalization of Bernoulli and Euler numbers., Sci. Magna, 2009, 5(3), 118–129. | Zbl 1296.11013
[9] Kim D. S., and Kim T., Higher-order Degenerate Euler Polynomials., Applied Mathematical Sciences, 2015, 9(2), 57–73.
[10] Kim D. S., Kim T., Komatsu T., and Lee S.-H., Barnes-type Daehee of the first kind and poly-Cauchy of the first kind mixed-type polynomials., Adv. Difference Equ., 2014, 2014:140, pp 22.
[11] Kim T., q-Bernoulli numbers and polynomials associated with Gaussian binomial coefficients., Russ. J. Math. Phys., 2008, 15(1), 51–57. [WoS] | Zbl 1196.11040
[12] Kim T., Symmetry p-adic invariant integral on Zp for Bernoulli and Euler polynomials., J. Difference Equ. Appl., 2008, 14(12), 1267–1277. | Zbl 1229.11152
[13] Kim T., Symmetry of power sum polynomials and multivariate fermionic p-adic invariant integral on Zp., Russ. J. Math. Phys., 2009, 16(1), 93–96. [WoS] | Zbl 1200.11089
[14] Luo Q.-M., Qi F., Relationships between generalized Bernoulli numbers and polynomials and generalized Euler numbers and polynomials., Adv. Stud. Contemp. Math. (Kyungshang), 2013, 7(1), 11–18. | Zbl 1042.11012
[15] Ozden H., q-Dirichlet type L-functions with weight α., Adv. Difference Equ., 2013, 2013:40, pp 5.
[16] Ozden H., Cangul, I. N., Simsek Y., Remarks on q-Bernoulli numbers associated with Daehee numbers., Adv. Stud. Contemp. Math. (Kyungshang), 2009, 18(1), 41–48. | Zbl 1188.05005
[17] Park J.-W., New approach to q-Bernoulli polynomials with weight or weak weight., Adv. Stud. Contemp. Math. (Kyungshang), 2014, 24(1), 39–44. | Zbl 1317.11028
[18] Roman, S., The umbral calculus, vol. 111 of Pure and Applied Mathematics., Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1984.
[19] Ryoo C. S., Song H., and Agarwal R. P., On the roots of the q-analogue of Euler-Barnes’ polynomials., Adv. Stud. Contemp. Math. (Kyungshang), 2004, 9(2), 153–163. | Zbl 1068.11013
[20] S¸ en E., Theorems on Apostol-Euler polynomials of higher order arising from Euler basis., Adv. Stud. Contemp. Math. (Kyungshang), 2013, 23(2), 337–345. | Zbl 1297.11008
[21] Simsek Y., Interpolation functions of the Eulerian type polynomials and numbers., Adv. Stud. Contemp. Math. (Kyungshang), 2013, 23(2), 301–307. | Zbl 06280544
[22] Volkenborn A., Ein P-adisches Integral und seine Anwendungen. I., Manuscripta Math., 1972, 7, 341–373. [Crossref] | Zbl 0245.10045
[23] Volkenborn A., Ein p-adisches Integral und seine Anwendungen. II., Manuscripta Math. 1974, 12, 17–46. [Crossref] | Zbl 0276.12018
[24] Zhang Z., and Yang H., Some closed formulas for generalized Bernoulli-Euler numbers and polynomials., Proc. Jangjeon Math. Soc., 2008, 11(2), 191–198. | Zbl 1178.05003