A novel approach for obtaining new identities for the lambda extension of q-Euler polynomials arising from the q-umbral calculus
-
2211
Downloads
-
3915
Views
Authors
Serkan Araci
- Department of Economics, Faculty of Economics, Administrative and Social Science, Hasan Kalyoncu University, TR-27410 Gaziantep, Turkey.
Mehmet Acikgoz
- Department of Mathematics, Faculty of Science and Arts, University of Gaziantep, TR-27310 Gaziantep, Turkey.
Toka Diagana
- Department of Mathematics, Howard University, 2441 6th Street, NW Washington 20059, D.C., U.S.A.
H. M. Srivastava
- Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3R4, Canada.
- China Medical University, Taichung 40402, Taiwan, Republic of China.
Abstract
In this article, a new q-generalization of the Apostol-Euler polynomials is introduced using the usual q-exponential function.
We make use of such a generalization to derive several properties arising from the q-umbral calculus.
Share and Cite
ISRP Style
Serkan Araci, Mehmet Acikgoz, Toka 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, Journal of Nonlinear Sciences and Applications, 10 (2017), no. 4, 1316--1325
AMA Style
Araci Serkan, Acikgoz Mehmet, Diagana Toka, Srivastava H. M., 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. (2017); 10(4):1316--1325
Chicago/Turabian Style
Araci, Serkan, Acikgoz, Mehmet, Diagana, Toka, Srivastava, H. M.. "A novel approach for obtaining new identities for the lambda extension of q-Euler polynomials arising from the q-umbral calculus." Journal of Nonlinear Sciences and Applications, 10, no. 4 (2017): 1316--1325
Keywords
- \(q\)-Apostol-Euler polynomials
- \(q\)-numbers
- \(q\)-exponential function
- \(q\)-umbral calculus
- (\(\lambda،q\))-Euler numbers
- (\(\lambda،q\))-Euler polynomials
- properties and identities.
MSC
References
-
[1]
S. Araci, Novel identities involving Genocchi numbers and polynomials arising from applications of umbral calculus, Appl. Math. Comput., 233 (2014), 599–607.
-
[2]
S. Araci, M. Acikgoz, A. Kilicman, Extended p-adic q-invariant integrals on \(\mathbb{Z}_p\) associated with applications of umbral calculus, Adv. Difference Equ., 2013 (2013 ), 14 pages.
-
[3]
S. Araci, M. Acikgoz, E. Sen, On the extended Kim’s p-adic q-deformed fermionic integrals in the p-adic integer ring, J. Number Theory, 133 (2013), 3348–3361.
-
[4]
S. Araci, M. Acikgoz, J. J. Seo, A new family of q-analogue of Genocchi numbers and polynomials of higher order, Kyungpook Math. J., 54 (2014), 131–141.
-
[5]
A. G. Bagdasaryan, An elementary and real approach to values of the Riemann zeta function, Phys. Atom. Nucl., 73 (2010), 251–254.
-
[6]
J.-S. 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.
-
[7]
J.-S. Choi, P. J. Anderson, 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.
-
[8]
R. Dere, Y. Simsek, H. M. Srivastava, A 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.
-
[9]
E. C. Ihrig, M. E. H. Ismail, A q-umbral calculus, J. Math. Anal. Appl., 84 (1981), 178–207.
-
[10]
M. E. H. Ismail, M. Rahman, Inverse operators, q-fractional integrals, and q-Bernoulli polynomials, J. Approx. Theory, 114 (2002), 269–307.
-
[11]
T. Kim, q-generalized Euler numbers and polynomials, Russ. J. Math. Phys., 13 (2006), 293–298.
-
[12]
D. S. Kim, T. Kim, q-Bernoulli polynomials and q-umbral calculus, Sci. China Math., 57 (2014), 1867–1874.
-
[13]
D. S. Kim, T. Kim, Some identities of q-Euler polynomials arising from q-umbral calculus, J. Inequal. Appl., 2014 (2014 ), 12 pages.
-
[14]
D. S. Kim, T. Kim, Umbral calculus associated with Bernoulli polynomials, J. Number Theory, 147 (2015), 871–882.
-
[15]
D. S. Kim, T. Kim, S.-H. Lee, J.-J. Seo, A note on q-Frobenius-Euler numbers and polynomials, Adv. Studies Theor. Phys., 7 (2013), 881–889.
-
[16]
T. Kim, T. Mansour, S.-H. Rim, S.-H. Lee, Apostol-Euler polynomials arising from umbral calculus, Adv. Difference Equ., 2013 (2013 ), 7 pages.
-
[17]
B. A. Kupershmidt, Reflection symmetries of q-Bernoulli polynomials, J. Nonlinear Math. Phys., 12 (2005), 412–422.
-
[18]
Q.-M. Luo, The multiplication formulas for the Apostol-Bernoulli and Apostol-Euler polynomials of higher order, Integral Transforms Spec. Funct., 20 (2008), 377–391.
-
[19]
N. I. Mahmudov, On a class of q-Bernoulli and q-Euler polynomials, Adv. Difference Equ., 2013 (2013), 11 pages.
-
[20]
N. I. Mahmudov, M. E. Keleshteri, On a class of generalized q-Bernoulli and q-Euler polynomials, Adv. Difference Equ., 2013 (2013 ), 10 pages.
-
[21]
Á . Pintér, H. M. Srivastava, Addition theorems for the Appell polynomials and the associated classes of polynomial expansions, Aequationes Math., 85 (2013), 483–495.
-
[22]
S.-H. Rim, J.-H. Jeong, On the modified q-Euler numbers of higher order with weight, Adv. Stud. Contemp. Math. (Kyungshang), 22 (2012), 93–98.
-
[23]
S. Roman, The theory of the umbral calculus, I, J. Math. Anal. Appl., 87 (1982), 58–115.
-
[24]
S. Roman, The theory of the umbral calculus, III, J. Math. Anal. Appl., 95 (1983), 528–563.
-
[25]
S. Roman, The umbral calculus, Pure and Applied Mathematics, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York (1984)
-
[26]
S. Roman, More on the umbral calculus, with emphasis on the q-umbral calculus, J. Math. Anal. Appl., 107 (1985), 222–254.
-
[27]
E. Sen, Theorems on Apostol-Euler polynomials of higher order arising from Euler basis, Adv. Stud. Contemp. Math. (Kyungshang), 23 (2013), 337–345.
-
[28]
H. M. Srivastava,/ , Some generalizations and basic (or q-) extensions of the Bernoulli, Euler and Genocchi polynomials, Appl. Math. Inf. Sci., 5 (2011), 390–444.
-
[29]
H. M. Srivastava, J.-S. Choi, Zeta and q-Zeta functions and associated series and integrals, Elsevier, Inc., Amsterdam (2012)