A Unified Family of Generalized $q$-Hermite Apostol Type Polynomials and its Applications

Year 2019, Volume: 2 Issue: 1, 1 - 8, 22.03.2019

Subuhi Khan , Tabinda Nahid

https://doi.org/10.33434/cams.475529

Abstract

The intended objective of this paper is to introduce a new class of generalized $q$-Hermite based Apostol type polynomials by combining the $q$-Hermite polynomials and a unified family of $q$-Apostol-type polynomials. The generating function, series definition and several explicit representations for these polynomials are established. The $q$-Hermite-Apostol Bernoulli, $q$-Hermite-Apostol Euler and $q$-Hermite-Apostol Genocchi polynomials are studied as special members of this family and corresponding relations for these polynomials are obtained.

Keywords

$q$-Hermite polynomials, Generalized $q$-Apostol type polynomials, Generalized $q$-Hermite Apostol type polynomials, Explicit representation

References

• [1] G.E. Andrews, R. Askey, R. Roy, Special Functions Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 1999.
• [2] G.E. Andrews, R. Askey, Classical orthogonal polynomials, C. Brenziniski et al. (editors), in ”Polynomes Orthogonaux et Applications”, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1171 1984, pp 36-63.
• [3] N. I. Mahmudov, Difference equations of q-Appell polynomials, Appl. Math. Comput., 245 (2014), 539-543.
• [4] Q. M. Luo, H. M. Srivastava, Some generalizations of the Apostol-Bernoulli and Apostol-Euler polynomials, J. Math. Anal. Appl., 308 (2005), 290-302.
• [5] Q. M. Luo, H. M. Srivastava, Some relationships between the Apostol-Bernoulli and Apostol-Euler polynomials, Comput. Math. Appl., 51(3-4) (2006), 631-642.
• [6] Q. M. Luo, Apostol Euler polynomials of higher orders and gaussian hypergeometric functions, Taiwanese J. Math., 10 (2006), 917-925.
• [7] Q. M. Luo, H. M. Srivastava, Some generalizations of the Apostol-Genochhi polynomials and the stirling numbers of the second kind, Appl. Math. Comput., 217 (2011), 5702-5728.
• [8] T. Ernst, On certain generalized q-Appell polynomial expansions, Ann. Univ. Mariae Curie-Sklodowska Sect. A, 68(2) (2015), 27-50.
• [9] M. A. Ozarslan, Unified Apostol-Bernoulli, Euler and Genocchi polynomials, Comput. Math. Appl., 62(6) (2011), 2452- 2462.
• [10] B. Kurt, Notes on unified q-Apostol type polynomials, Filomat, 30 (2016), 921-927.