Multifarious results for q-Hermite-based Frobenius-type Eulerian polynomials

Waseem Ahmad Khan, Idrees Ahmad Khan, Mehmet Acikgoz and Ugur Duran
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 26, 2020, Number 2, Pages 127–141
DOI: 10.7546/nntdm.2020.26.2.127-141
Full paper (PDF, 220 Kb)


Authors and affiliations

Waseem Ahmad Khan
Department of Mathematics and Natural Sciences, Prince Mohammad Bin Fahd University
Al Khobar, 31952, Saudi Arabia

Idrees Ahmad Khan
Department of Mathematics, Integral University
Lucknow-226026, India

Mehmet Acikgoz
Department of Mathematics, University of Gaziantep
TR-27310 Gaziantep, Turkey

Ugur Duran
Department of the Basic Concepts of Engineering, Iskenderun Technical University
TR-31200 Hatay, Turkey


In this paper, a new class of q-Hermite-based Frobenius-type Eulerian polynomials is introduced by means of generating function and series representation. Several fundamental formulas and recurrence relations for these polynomials are derived via different generating methods. Furthermore, diverse correlations including the q-Apostol-Bernoulli polynomials, the q-Apostol-Euler poynoomials, the q-Apostol-Genocchi polynomials and the q-Stirling numbers of the second kind are also established by means of the their generating functions.


  • Hermite polynomials
  • Frobenius-type Eulerian polynomials
  • Hermite-based Frobenius-type Eulerian polynomials
  • q-numbers
  • q-polynomials

2010 Mathematics Subject Classification

  • 11B73
  • 11B83
  • 11B68


  1. Al-Salam, W. A. (1967). q-Appell polynomials, Ann. Mat. Pura Appl., 77, 31–45.
  2. Andrews, G. E., Akey, R. & Roy, R. (1999) Special Functions, Cambridge University Press, Cambridge.
  3. Carlitz, L. (1959). Eulerian numbers and polynomials, Math. Mag., 32, 247–260.
  4. Carlitz, L. (1960). Eulerian numbers and polynomials of higher order, Duke Math. J., 27, 401–423.
  5. Cheon, G. S., & Jung, J. H. (2016). The q-Sheffer sequence of a new type and associated orthogonal polynomials, Linear Algebra Appl., 491, 247–260.
  6. Chung, W. S., Kim, T., & Kwon, H. I. (2014). On the q-analog of the Laplace transforms, Russ. J. Math. Phys. 21, 156–168.
  7. Keleshteri, M. E., & Mahmudov, N. I. (2015). A study on q-Appell polynomials from determinantal point of view, Appl. Math. Comput., 260, 351–369.
  8. Keleshteri, M. E., & Mahmudov, N. I. (2015). On the class of 2D q-Appell polynomials, Preprint, arXiv:1512.03255v1.
  9. Kim, T. (2006). q-generalized Euler numbers and polynomials, Russ. J. Math. Phys., 13, 293–298.
  10. Kim, T. (2007). q-Extension of the Euler formula and trigonometric functions, Russ. J. Math. Phys., 14, 275–278.
  11. Kim, T., Choi, J., Kim, Y. H., & Ryoo, C. S. (2011). On q-Bernstein and q-Hermite
    polynomials, Proc. Jangjeon Math. Soc., 14, 215–221.
  12. Kim, D. S., Kim, T., Kim, W. J., & Dolgy, D. V. (2012). A note on Eulerian polynomials, Abstr. Appl. Anal., Article No. 269640, 10 pages.
  13. Kim, D. S., Kim, T., Kim, Y. H., & Dolgy, D. V. (2012). A note on Eulerian polynomials associated with Bernoulli and Euler numbers and polynomials, Adv. Stud. Contemp. Math., 22, 379–389.
  14. Mahmudov, N. I. (2013). On a class of q-Bernoulli and q-Euler polynomials, Adv. Differ. Equ., 108, 1–11.
  15. Mahmudov, N. I. (2014). Difference equations of q-Appell polynomials, Appl. Math. Comput., 245, 539–543.
  16. Mahmudov, N. I., & Keleshteri, M. E. (2014). q-extensions for the Apostol-type
    polynomials, J. Appl. Math., Article No. 868167, 8 pages.
  17. Mahmudov, N. I. & Momenzadeh, M. (2014). On a class of q-Bernoulli, q-Euler and q-Genocchi polynomials, Abstr. Appl. Anal., Article No. 696454, 10 pages.
  18. Pathan, M. A., & Khan, W. A. (2015). Some implicit summation formulas and symmetric identities for the generalized Hermite–Bernoulli polynomials, Mediterr. J. Math., 12, 679–695.
  19. Pathan, M. A., & Khan, W. A. (2016). A new class of generalized polynomials associated with Hermite and Euler polynomials, Mediterr. J. Math., 13, 913–928.
  20. Riyasat, M. & Khan, S. (2018). Some results on q-Hermite-based hybrid polynomials, Glasnik Matematicki, 53 (73), 9–31.
  21. Srivastava H. M. (2007). Eulerian and other integral representations for some families of hypergeometric polynomials, Inter. J. Appl. Math. Stat., 11, 149–171.
  22. Srivastava, H. M., Boutiche, M. A., & Rahmani, M. (2018). A class of Frobenius-type Eulerian polynomials, Rocky Mountain J. Math., 48, 1003–1013.
  23. Srivastava, H. M., & Manocha, H. L. (1984). A Treatise on Generating Functions, Ellis Horwood Limited, New York.

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Cite this paper

Khan, W. A., Khan, I. A., Acikgoz, M., & Duran, U. (2020). Multifarious results for q-Hermite-based Frobenius-type Eulerian polynomials. Notes on Number Theory and Discrete Mathematics, 26 (2), 127-141, DOI: 10.7546/nntdm.2020.26.2.127-141.

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