A note on partially degenerate Hermite-Bernoulli polynomials of the first kind

Volume 30, Issue 3, pp 255--271 https://doi.org/10.22436/jmcs.030.03.05

Publication Date: January 12, 2023 Submission Date: June 11, 2022 Revision Date: August 02, 2022 Accteptance Date: August 31, 2022

Download PDF

Download XML

• Export citations
  • CITATIONS & REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • ARTICLE CITATION
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)

• 441 Downloads
• 1104 Views

Authors

W. A. Khan - Department of Mathematics and Natural Sciences, Prince Mohammad Bin Fahd University, P.O Box 1664, Al Khobar 31952, Saudi Arabia. M. S. Alatawi - Department of Mathematics, Faculty of Science, University of Tabuk, Tabuk 71491, Saudi Arabia.

Abstract

In this paper, we introduce a new class of partially degenerate Hermite-Bernoulli polynomials of the first kind and generalized Gould-Hopper-partially degenerate Bernoulli polynomials of the first kind and present some properties and identities of these polynomials. A new class of polynomials generalizing different classes of Hermite polynomials such as the real Gould-Hopper, as well as the 1-d and 2-d holomorphic, ternary and polyanalytic complex Hermite polynomials and their relationship to the partially degenerate Hermite-Bernoulli polynomials of the first kind are also discussed.

Share and Cite

Keywords

• Hermite polynomials
• partially degenerate Bernoulli polynomials of the first kind
• partially degenerate Hermite-Bernoulli polynomials of the first kind
• symmetry identities

MSC

•  11B68
•  33C45
•  11S40
•  11S80

References

• [1] N. Alam, W. A. Khan, S. Obeidat, G. Muhiuddin, N. S Khalifa, H. N. Zaidi, A. Altaleb, L. Bachioua, A note on Bell-based Bernoulli and Euler polynomials of complex variable, Computer Modelling in Engineering and Sciences, (2022), 24 pages
  • View Article
  • Google Scholar


• [2] N. Alam, W. A. Khan, C. S. Ryoo, A note on Bell-based Apostol-type Frobenius-Euler polynomials of a complex variable with its certain applications, Mathematics, 2022 (2022), 26 pages
  • View Article
  • Google Scholar


• [3] E. T. Bell, Exponential polynomials, Ann. of Math. (2), 35 (1934), 258–277
  • View Article
  • MathSciNet
  • Google Scholar


• [4] Y. A. Brychkov, On multiple sums of special functions, Integral Transforms Spec. Funct., 21 (2010), 877–884
  • View Article
  • MathSciNet
  • Google Scholar


• [5] G. Dattoli, C. Cesarano, D. Sacchetti, A note on truncated polynomials, Appl. Math. Comput., 134 (2003), 595–605
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [6] G. Dattoli, C. Chiccoli, S. Lorenzutta, G. Maino, A. Torre, Theory of generalized Hermite polynomials, Comput. Math. Appl., 28 (1994), 71–83
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [7] D. V. Dolgy, W. A. Khan, A note on type two degenerate poly-Changhee polynomials of the second kind, Symmetry, 2021 (2021), 12 pages
  • View Article
  • Google Scholar


• [8] U. Duran, M. Mehmet, S. Araci, Bell-based Bernoulli polynomials with applications, Axioms, 2021 (2021), 24 pages
  • View Article
  • Google Scholar
  • MATH


• [9] A. Ghanmi, K. Lamsaf, A unified generalization of real Gould-Hopper, 1-d and 2-d holomorphic and polyanalytic Hermite polynomials, arXiv, 2019 (2019), 11 pages
  • View Article
  • Google Scholar


• [10] F. Gori, Flattened Gaussian beams, Optics Commun., 107 (1994), 335–341
  • View Article
  • Google Scholar


• [11] H.W. Gould, A. T. Hopper, Operational formulas connected with two generalization of Hermite polynomials, Duke Math. J., 29 (1962), 51–63
  • View Article
  • MathSciNet
  • Google Scholar


• [12] H. Haroon, W. A. Khan, Degenerate Bernoulli numbers and polynomials associated with degenerate Hermite polynomials, Commun. Korean. Math. Soc., 33 (2018), 651–669
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [13] N. A. Jedda, A. Ghanmi, On a class of two-index real Hermite polynomials, Palest. J. Math., 3 (2014), 185–190
  • View Article
  • Google Scholar


• [14] J. Y. Kang, W. A. Khan, A new class of q-Hermite based Apostol-type Frobenius Genocchi polynomials, Commun. Korean Math. Soc., 35 (2020), 759–771
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [15] W. A. Khan, S. Araci, M. Acikgoz, H. Haroon, A new class of partially degenerate Hermite-Genocchi polynomials, J. Nonlinear Sci. Appl., 10 (2017), 5072–5081
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [16] W. A. Khan, M. Kamarujjama, Daud, Construction of partially degenerate Bell-Bernoulli polynomials of the first kind, Analysis (Berlin), 42 (2022), 171–184
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [17] W. A. Khan, H. Haroon, Some symmetric identities for the generalized Bernoulli, Euler and Genocchi polynomials associated with Hermite polynomials, SpringerPlus, 5 (2016), 1–21
  • View Article
  • Google Scholar


• [18] W. A. Khan, M. A. Pathan, On generalized Lagrange–Hermite–Bernoulli and related polynomials, Acta et Commentationes Universitatis Tartuensis de Mathematica, 23 (2019), 211–224
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [19] W. A. Khan, S. K. Sharma, A new class of Hermite-Based Higher-order Central Fubini polynomials, Int. J. Appl. Comput. Math., 87 (2020), 14 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [20] W. A. Khan, J. Younis, M. Nadeem, Construction of partially degenerate Laguerre-Bernoulli polynomials of the first kind, Appl. Math. Sci. Eng., 30 (2022), 362–375
  • View Article
  • MathSciNet
  • Google Scholar


• [21] T. Kim, On explicit formulas of p-adic q-L-functions, Kyushu J. Math., 48 (1994), 73–86
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [22] T. Kim, q–Volkenborn integration, Russ. J. Math. Phys., 9 (2002), 288–299
  • View Article


• [23] B. Kurt, Identities and relations on the Hermite-based tangent polynomials, TWMS J. Appl. Eng. Math., 10 (2020), 321–337
  • View Article
  • Google Scholar


• [24] B. Kurt, Unified degenerate Apostol-type Bernoulli, Euler, Genocchi, and Fubini polynomials, J. Math. Comput. Sci., 25 (2020), 259–268
  • View Article
  • Google Scholar


• [25] J. K. Kwon, S. H. Rim, J.-W. Park, A note on the Appell type Daehee polynomials, Global J. Pure Appl. Math., 11 (2015), 2745–2753
  • View Article
  • Google Scholar


• [26] G. Muhiuddin, W. A. Khan, U. Duran, D. Al-Kadi, A new class of higher-order hypergeometric Bernoulli polynomials associated with Lagrange-Hermite polynomials, Symmetry, 2021 (2021), 11 pages
  • View Article
  • Google Scholar


• [27] M. A. Pathan, W. A. Khan, Some implicit summation formulas and symmetric identities for the generalized Hermite- Bernoulli polynomials, Mediterr. J. Math., 12 (2015), 679–695
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [28] M. A. Pathan,W. A. Khan, A new class of generalized polynomials associated with Hermite and poly-Bernoulli polynomials, Miskolc Math. J., 22 (2021), 317–330
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [29] M. A. Pathan, W. A. Khan, On -Changhee-Hermite polynomials, Analysis (Berlin), 42 (2022), 57–69
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [30] M. A. Pathan, W. A. Khan, On a class of generalized Humbert-Hermite, Turkish J. Math., 46 (2022), 929–945
  • View Article
  • MathSciNet
  • Google Scholar


• [31] F. Qi, D. V. Dolgy, T. Kim, On the partially degenerate Bernoulli polynomials of the first kind, Global J. Pure Appl. Math., 11 (2015), 2407–2412
  • View Article
  • Google Scholar