Degenerate Hermite poly-Bernoulli numbers and polynomials with q-parameter
DOI:
https://doi.org/10.24193/subbmath.2020.1.01Keywords:
Hermite polynomials, degenerate q-poly-Bernoulli polynomials, degenerate Hermite q-poly-Bernoulli polynomials, summation formulae, symmetric identities.Abstract
In this paper, we introduce a new class of degenerate Hermite polyBernoulli polynomials with q-parameter and give some identities of these polynomials related to the Stirling numbers of the second kind. Some implicit summation formulae and general symmetry identities are derived by using different analytical means and applying generating functions. These results extend some known summations and identities of degenerate Hermite poly-Bernoulli numbers and polynomials.
Mathematics Subject Classification (2010): 11B68, 11B73, 11B75, 33C45.
References
Andrews, L.C., Special functions for engineers and mathematicians, Macmillan Co. New York, 1985.
Bayad, A., Hamahata, Y., Multiple-polylogarithms and multi-poly-Bernoulli polynomials, Funct. Approx. Comment. Math., 46(2012), 45-61.
Bell, E.T., Exponential polynomials, Ann. of Math., 35(1934), 258-277.
Carlitz, L., Degenerate Stirling Bernoulli and Eulerian numbers, Util. Math., 15(1979), 51-88.
Cenkcia, M., Komatsu, T., Poly-Bernoulli numbers and polynomials with a q-parameter, Journal of Number Theory, 152(2015), 38-54.
Cesarano, C., Cennamo, G.M., Placidi, L., Humbert polynomials and functions in terms of Hermite polynomials towards applications to wave propagation, WSEAS Transactions on Mathematics, 14(2014), 595-602.
Dattoli, G., Lorenzutta, S., Ricci, P.E., Cesarano, C., On a family of hybrid polynomials, Integral Trasforms and Special Functions, 15(2004), 485-490.
Dattoli, G., Lorenzutta, S., Cesarano, C., Finite sums and generalized forms of Bernoulli polynomials, Rendiconti di Mathematica, 19(1999), 385-391.
Dattoli, G., Srivastava, H.M., Cesarano, C., The Laguerre and Legendre polynomials from an operational point of view, Applied Mathematics and Computation, 124(2001), 117-127.
Jolany, H., Mohebbi, H., Some results on generalized multi-poly Bernoulli and Euler polynomials, Int. J. Math. Comb., 2(2011), 117-129.
Jung, N.S., Ryoo, C.S., Symmetric identities for degenerate q-poly-Bernoulli numbers and polynomials, J. Appl. Math. & Informatics, 36(2018), 29-38.
Kaneko, M., Poly-Bernoulli numbers, J. de Theorie de Nombres, 9(1997), 221-228. [13] Khan, W.A., A note on degenerate Hermite-poly numbers and polynomials, J. Class. Anal., 1(2016), 65-76.
Khan, W.A., Some properties of the generalized Apostol type Hermite-Based polynomials, Kyungpook Math. J., 55(2015), 597-614.
Khan, W.A., A note on Hermite-based poly-Euler and multi poly-Euler polynomials, Palestine J. Math., 5(2016), 17-26.
Khan, W.A., A new class of Hermite poly-Genocchi polynomials, J. Anal. and Number Theory, 4(2016), 1-8.
Kim, D.S., Dolgy, T., Komatsu, D.V., Barnes type degenerate Bernoulli polynomials, Adv. Stud. Contemp. Math., 25(2015), 121-146.
Kim, D.S., Kim, T., A note on degenerate poly-Bernoulli numbers and polynomials, Advanc. Diff. Equat., (2015), DOI 10.1186/s13662-015-0595-3.
Pathan, M.A., Khan, W.A., Some implicit summation formulas and symmetric identities for the generalized Hermite-Bernoulli polynomials, Mediterr. J. Math., 12(2015), 679- 695.
Pathan, M.A., Khan, W.A., A new class of generalized polynomials associated with Hermite and Euler polynomials, Mediterr. J. Math., 13(2016), 913-928.
Pathan, M.A., Khan, W.A., Some implicit summation formulas and symmetric identities for the generalized Hermite-Euler polynomials, East-West J. Maths., 16(2014), 92-109.
Pathan, M.A., Khan, W.A., Some new classes of generalized Hermite-based Apostol- Euler and Apostol-Genocchi polynomials, Fasciculli Math., 55(2015), 153-170.
Ryoo, C.S., On degenerate q-tangent polynomials of higher order, J. Appl. Math. & Informatics, 35(2017), 113-120.
Srivastava, H.M., Manocha, H.L., A Treatise on Generating Functions, Ellis Horwood Limited, New York, 1984.
Young, P.T., Degenerate Bernoulli polynomials generalized factorials sums and their application, J. Number Theory, 128(2008), 738-758.
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2020 Studia Universitatis Babeș-Bolyai Mathematica
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
