Glasnik Matematicki, Vol. 53, No. 1 (2018), 9-31.

SOME RESULTS ON q-HERMITE BASED HYBRID POLYNOMIALS

Mumtaz Riyasat and Subuhi Khan

Abstract.   In this article, a hybrid class of the q-Hermite based Apostol type Frobenius-Euler polynomials is introduced by means of generating function and series representation. Several important formulas and recurrence relations for these polynomials are derived via different generating function methods. Further, the 2D q-Hermite based Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials are introduced and important relations for these polynomials are also established. Finally, a new class of the 2D q-Hermite based Appell polynomials is investigated as the generalization of the above polynomials. The determinant definitions for the 2D q-Hermite based Appell and related polynomials are also explored.

2010 Mathematics Subject Classification.   11B73, 11B83, 11B68.

Key words and phrases.   q-Hermite type polynomials, Apostol type q-Frobenius-Euler polynomials, q-Hermite based Apostol type Frobenius-Euler polynomials.

DOI: 10.3336/gm.53.1.02

References:

1. W. A. Al-Salam, q-Appell polynomials, Ann. Mat. Pura Appl. (4) 17 (1967), 31-45.
   MathSciNet     CrossRef

2. G. E. Andrews, R. Askey and R. Roy, Special functions, Cambridge University Press, Cambridge, 1999.
   MathSciNet     CrossRef

3. L. Carlitz, Eulerian numbers and polynomials, Math. Mag. 32 (1958/1959), 247-260.
   MathSciNet     CrossRef

4. Gi-Sang Cheon and Ji-Hwan Jung, The q-Sheffer sequences of a new type and associated orthogonal polynomials, Linear Algebra Appl. 491 (2016), 171-186.
   MathSciNet     CrossRef

5. J. Choi, P. J. Anderson and H. M. Srivastava, Some q-extensions of the Apostol-Bernoulli and the Apostol-Euler polynomials of order n, and the multiple Hurwitz zeta function, Appl. Math. Comput. 199 (2008), 723-727.
   MathSciNet     CrossRef

6. J. Choi, P. J. Anderson and H. M. Srivastava, Carlitz's q-Bernoulli and q-Euler numbers and polynomials and a class of generalized q-Hurwitz zeta functions, Appl. Math. Comput. 215 (2009), 1185-1208.
   MathSciNet     CrossRef

7. R. Dere, Y. Simsek and H. M. Srivastava, Unified presentation of three families of generalized Apostol type polynomials based upon the theory of the umbral calculus and the umbral algebra, J. Number Theory 133 (2013), 3245-3263.
   MathSciNet     CrossRef

8. B. S. El-Desouky and R. S. Gomaa, A new unified family of generalized Apostol-Euler, Bernoulli and Genocchi polynomials, Appl. Math. Comput. 247 (2014), 695-702.
   MathSciNet     CrossRef

9. B. K. Karande and N. K. Thakare, On the unification of the Bernoulli and Euler polynomials, Indian J. Pure Appl. Math. 6 (1975), 98-107.
   MathSciNet    

10. M. Eini Keleshteri and N. I. Mahmudov, A study on q-Appell polynomials from determinantal point of view, Appl. Math. Comput. 260 (2015), 351-369.
    MathSciNet     CrossRef

11. M. Eini Keleshteri and N. I. Mahmudov, On the class of 2 D q-Appell polynomials, arXiv:1512.03255v1.

12. Subuhi Khan, G. Yasmin and M. Riyasat, Certain results for the 2-variable Apostol type and related polynomials, Comput. Math. Appl. 69 (2015), 1367-1382.
    MathSciNet     CrossRef

13. B. Kurt, A note on the Apostol type q-Frobenius-Euler polynomials and generalizations of the Srivastava-Pinter addition theorems, Filomat 30 (2016), 65-72.
    MathSciNet     CrossRef

14. B. Kurt and Y. Simsek, Frobenius-Euler type polynomials related to Hermite-Bernoulli polyomials, Numerical Analysis and Appl. Math. ICNAAM 2011 Conf. Proc. 1389 (2011), 385-388.

15. Q.-M. Luo and H. M. Srivastava, Some generalizations of the Apostol-Genocchi polynomials and the Stirling numbers of the second kind, Appl. Math. Comput. 217 (2011), 5702-5728.
    MathSciNet     CrossRef

16. N. I. Mahmudov, On a class of q-Bernoulli and q-Euler polynomials, Adv. Difference Equ. 108 (2013), 11 pp.
    MathSciNet     CrossRef

17. N. I. Mahmudov, Difference equations of q-Appell polynomials, Appl. Math. Comput. 245 (2014), 539-543.
    MathSciNet     CrossRef

18. N. I. Mahmudov and M. E. Keleshteri, On a class of generalized q-Bernoulli and q-Euler polynomials, Adv. Difference Equ. 115 (2013), 10 pp.
    MathSciNet     CrossRef

19. N. I. Mahmudov and M. E. Keleshteri, q-extensions for the Apostol type polynomials, J. Appl. Math. (2014) Art. ID 868167, 8 pp.
    MathSciNet     CrossRef

20. N. I. Mahmudov and M. Momenzadeh, On a class of q-Bernoulli, q-Euler and q-Genocchi polynomials, Abstr. Appl. Anal. (2014), Art. ID 696454, 10 pp.

21. M. A. Özarslan, Unified Apostol-Bernoulli, Euler and Genocchi polynomials, Comput. Math. Appl. 62 (2011), 2452-2462.
    MathSciNet     CrossRef

22. H. Ozden and Y. Simsek, Modification and unification of the Apostol-type numbers and polynomials, Appl. Math. Comput. 235 (2014), 338-351.
    MathSciNet     CrossRef

23. H. Ozden, Y. Simsek and H. M. Srivastava, A unified presentation of the generating function of the generalized Bernoulli, Euler and Genocchi polynomials, Comput. Math. Appl. 60 (2010), 2779-2787.
    MathSciNet     CrossRef

24. M. Riyasat, S. Khan and T. Nahid, q-difference equations for the composite 2D q-Appell polynomials and their applications, Cogent Math. 4 (2017), 1-23.

25. M. Riyasat, S. Khan and N. I. Mahmudov, A numerical computation of zeros and finding determinant forms for some new families of q-special polynomials, Azerbaijan Journal of Mathematics, to appear.

26. Y. Simsek, Generating functions for q-Apostol type Frobenius-Euler numbers and polynomials, Axioms 1 (2012), 395-403.

27. Y. Simsek, Generating functions for generalized Stirling type numbers, Array type polynomials, Eulerian type polynomials and their applications, Fixed Point Theory Appl. 87 (2013), 28 pp.
    MathSciNet     CrossRef