A new class of q-Hermite-based Apostol-type polynomials and its applications

Waseem A. Khan and Divesh Srivastava
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 26, 2020, Number 1, Pages 75—85
DOI: 10.7546/nntdm.2020.26.1.75-85
Download full paper: PDF, 196 Kb


Authors and affiliations

Waseem A. Khan
Department of Mathematics and Natural Sciences
Prince Mohammad Bin Fahd University
P.O Box 1664, Al Khobar 31952, Saudi Arabia

Divesh Srivastava
Department of Mathematics, Faculty of Science
Integral University
Lucknow-226026, India


The present article is to introduce a new class of q-Hermite based Apostol-type
polynomials and to investigate their properties and characteristics. In particular, the generating functions, series expression and explicit and recurrence relations for these polynomials are established. We derive some relationships for q-Hermite based Apostol-type polynomials associated with q-Apostol-type Bernoulli polynomials, q-Apostol-type Euler and q-Apostol-type Genocchi polynomials.


  • q-polynomials
  • q-Hermite-based Apostol-type polynomials
  • q-recurrence relations

2010 Mathematics Subject Classification

  • 05A10
  • 05A15
  • 11B68
  • 16B65
  • 33C45


  1. Andrews, G. E. & Askey, R. (1999). Special Functions Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge.
  2. Andrews, G. E. & Askey, R. (1985). Classical orthogonal polynomials. Macmillan
    Publishing Company, New York, USA.
  3. Carlitz, L. (1959). Eulerian numbers and polynomials. Math. Mag., 32, 247–260.
  4. Ernst, T. (2015). On certain generalized q-Apple polynomial expansions. Ann. Univ. Mariae Curie-Sklodowska Sect. A, 68 (2), 27–50.
  5. Jang, L. C., Kim, D. S., Jang, G. W. & Kwon, J. (2018). Some identities for q-Bernoulli numbers and polynomials arising from q-Bernstein polynomials, Adv. Stud. Contemp. Math. (Kyungshang), 28 (4), 659–667.
  6. Kurt, V. (2013). Some symmetry identities for the Apostol-type polynomials related to multiple alternating sums. Adv. Differ. Equ., 2013, 32.
  7. Khan, S. & Nahid, T. (2018). Determinant forms, difference equations and zeros of the q-Hermite-Apple polynomials. Mathematics. DOI:10.3390/math6110258.
  8. Khan, S. & Nahid, T. (2019). A unified family of generalized q-Hermite apostol type polynomials and its applications. Commun. Adv. Math. Sci., 2 (1), 1–8.
  9. Khan, W. A. (2015). Some Properties of the Generalized Apostol type Hermite-Based Polynomials. Kyung. Math. J., 55, 597–614.
  10. Luo, Q. M. (2006). Apostol–Euler polynomials of higher order and the Gaussian
    hypergeometric function. Taiwan. J. Math., 10, 917–925.
  11. Luo, Q. M. & Srivastava, H. M. (2005). Some generalizations of the Apostol–Bernoulli and Apostol–Euler polynomials, J. Math. Anal. Appl., 308, 290–302.
  12. Luo, Q. M. & Srivastava, H. M. (2006). Some relationships between the Apostol–Bernoulli and Apostol–Euler polynomials, Comput. Math. Appl., 51(3-4), 631–642.
  13. Luo, Q. M. & Srivastava, H. M. (2011). Some generalizations of the Apostol–Genochhi polynomials and the Stirling numbers of the second kind, Appl. Math. Comput., 217, 5702–5728.
  14. Lu, Q. D. & Srivastava, H. M. (2011). Some series identities involving the generalized Apostol type and related polynomials. Comput. Math. Appl., 62, 3591–3602.
  15. Mahmudov, N. I. (2013). On a class of q-Bernoulli and q-Euler polynomials. Adv. Differ. Equ., 2013, 108.
  16.  Mahmudov, N. I. (2014). Difference equations of q-Appell polynomials, Appl. Math. Comput., 245, 539–543.
  17. Özarslan, M. A. (2011). Unified Apostol–Bernoulli, Euler and Genocchi polynomials. Comput. Math. Appl., 62, 2452–2462.
  18. Nisar, K. S. & Khan, W. A. (2020). Note on q-Hermite-based unified Apostol-type
    polynomials. J. Interdiscpl. Math. 22 (7), 1185–1203.
  19. Simsek, Y. (2013). Generating functions for generalized Stirling type numbers, Array type polynomials, Eulerian type polynomials and their applications, Fixed Point Th. Appl., Springer, Article number: 87 (2013) DOI: 1186/1687-1812-2013-87.
  20. Srivastava, H. M. & Manocha, H. L. (1984). A Treatise on Generating Functions, Ellis Horwood Limited Co., New York.

Related papers

Cite this paper

Khan, W. A.  & Srivastava D. (2020). A new class of q-Hermite-based Apostol-typepolynomials and its applications. Notes on Number Theory and Discrete Mathematics, 26(1), 75-85, doi: 10.7546/nntdm.2020.26.1.75-85.

Comments are closed.