Certain results of 2-variable \(q\)-generalized tangent-Apostol-type polynomials
Volume 22, Issue 3, pp 238--251
http://dx.doi.org/10.22436/jmcs.022.03.04
Publication Date: August 07, 2020
Submission Date: May 12, 2019
Revision Date: November 11, 2019
Accteptance Date: December 15, 2019
-
1345
Downloads
-
3158
Views
Authors
Ghazala Yasmin
- Department of Applied Mathematics, Aligarh Muslim University, Aligarh 202002, India.
Abdulghani Muhyi
- Department of Applied Mathematics, Aligarh Muslim University, Aligarh 202002, India.
- Department of Mathematics, Hajjah University, Hajjah, Yemen.
Abstract
The present article aims to introduce and investigate a new class of \(q\)-hybrid special polynomials, namely 2-variable \(q\)-generalized tangent-Apostol-type polynomials. The generating function, series definition and many other useful relations and identities of this class are established. In addition, certain members of 2-variable \(q\)-generalized tangent-Apostol-type family are investigated and some properties of these members are obtained. The graphical representations of these members are shown for several values of indices with the help of Matlab. Further, the distributions of zeros of these members are displayed.
Share and Cite
ISRP Style
Ghazala Yasmin, Abdulghani Muhyi, Certain results of 2-variable \(q\)-generalized tangent-Apostol-type polynomials, Journal of Mathematics and Computer Science, 22 (2021), no. 3, 238--251
AMA Style
Yasmin Ghazala, Muhyi Abdulghani, Certain results of 2-variable \(q\)-generalized tangent-Apostol-type polynomials. J Math Comput SCI-JM. (2021); 22(3):238--251
Chicago/Turabian Style
Yasmin, Ghazala, Muhyi, Abdulghani. "Certain results of 2-variable \(q\)-generalized tangent-Apostol-type polynomials." Journal of Mathematics and Computer Science, 22, no. 3 (2021): 238--251
Keywords
- \(q\)-calculus
- \(q\)-tangent polynomials and numbers
- \(q\)-Apostol-type polynomials and numbers
- generating functions
MSC
References
-
[1]
M. Acikgoz, S. Araci, U. Duran, New extensions of some known special polynomials under the theory of multiple $q$-calculus, Turkish J. Anal. Number Theory, 3 (2015), 128--139
-
[2]
W. A. Al-Salam, $q$-Appell polynomials, Ann. Mat. Pura Appl., 77 (1967), 31--45
-
[3]
G. E. Andrews, R. Askey, R. Roy, Special Functions, Cambridge University Press, Cambridge (1999)
-
[4]
T. M. Apostol, On the Lerch Zeta function, Pacific. J. Math., 1 (1951), 161--167
-
[5]
S. Araci, M. Acikgoz, T. Diagana, H. M. Srivastava, A novel approach for obtaining new identities for the $\lambda$ extension of $q$-Euler polynomials arising from the $q$-umbral calculus, J. Nonlinear Sci. Appl., 10 (2007), 1316--1325
-
[6]
A. Aral, V. Gupta, R. P. Agarwal, Applications of $q$-Calculus in Operator Theory, Springer, New York (2013)
-
[7]
C. Bildirici, M. Acikgoz, S. Araci, A note on analogues of tangent polynomials, J. Algebra Number Theory Acad., 4 (2014), 21 pages
-
[8]
J. Choi, P. J. Anderson, 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--737
-
[9]
T. Ernst, $q$-Bernoulli and $q$-Euler polynomials, an umbral approach, Int. J. Diff. Equ., 1 (2006), 31--80
-
[10]
T. Ernst, A comprehensive treatment of $q$-calculus, Springer Science & Business Media, Basel (2012)
-
[11]
T. Kim, A note on the $q$-Genocchi numbers and polynomials, J. Inequal. Appl., 2007 (2007), 8 pages
-
[12]
T. Kim, $q$-Euler numbers and polynomials associated with $p$-adic $q$-integral, J. Nonlinear Math. Phys., 14 (2007), 15--27
-
[13]
T. Kim, $q$-Bernoulli numbers and polynomials associated with Gaussian binomial coefficients, Russ. J. Math. Phys., 15 (2008), 51--57
-
[14]
D. S. Kim, T. Kim, T. Komatsu, J. J. Seo, An umbral calculus approach to poly-Cauchy polynomials with a $q$ parameter, J. Comput. Anal. Appl., 18 (2015), 762--792
-
[15]
T. Kim, D. S. Kim, H. I. Kwon, J. J. Seo, D. V. Dolgy, Some identities of $q$-Euler polynomials under the symmetric group of degree $n$, J. Nonlinear Sci. Appl., 9 (2016), 1077--1082
-
[16]
D. S. Kim, T. Kim, H. Y. Lee, $p$-adic $q$-integral on $Z_{p}$ associated with Frobenius-type Eulerian polynomials and umbral calculus, Adv. Stud Contemp. Math. (Kyungshang), 23 (2013), 243--251
-
[17]
B. Kurt, Notes on unified $q$-Apostol-type polynomials, Filomat, 30 (2016), 921--927
-
[18]
Q.-M. Luo, Apostol-Euler polynomials of higher order and the Gaussian hypergeometric function, Taiwanese J. Math., 10 (2006), 917--925
-
[19]
Q.-M. Luo, Extension for the Genocchi polynomials and its Fourier expansions and integral representations, Osaka J. Math., 48 (2011), 291--309
-
[20]
Q.-M. Luo, H. M. Srivastava, Some generalizations of the Apostol-Bernoulli and Apostol-Euler polynomials, J. Math. Anal. Appl., 308 (2005), 290--302
-
[21]
N. I. Mahmudov, $q$-Analogues of the Bernoulli and Genocchi polynomials and the Srivastava-Pintér addition theorems, Discrete Dyn. Nat. Soc., 2012 (2012), 8 pages
-
[22]
N. I. Mahmudov, On a class of $q$-Bernoulli and $q$-Euler polynomials, Adv. Difference Equ., 2013 (2013), 11 pages
-
[23]
N. I. Mahmudov, M. E. Keleshteri, $q$-Extensions for the Apostol type polynomials, J. Appl. Math., 2014 (2014), 7 pages
-
[24]
N. I. Mahmudov, M. Momenzadeh, On a class of $q$-Bernoulli, $q$-Euler, and $q$-Genocchi polynomials, Abstr. Appl. Anal., 2014 (2014), 10 pages
-
[25]
C. S. Ryoo, A note on the tangent numbers and polynomials, Adv. Studies Theor. Phys., 7 (2013), 447--454
-
[26]
C. S. Ryoo, Generalized tangent numbers and polynomials associated with $p$-adic integral on $z_{p}$, Appl. Math. Sci., 7 (2013), 4929--4934
-
[27]
C. S. Ryoo, Some properties of two dimensional $q$-tangent numbers and polynomials, Global J. Pur. Appl. Math., 12 (2016), 2999--3007
-
[28]
G. Yasmin, A. Muhyi, S. Araci, Certain Results of $q$-Sheffer--Appell Polynomials, Symmetry, 11 (2019), 19 pages