A New Family of q-analogue of Genocchi Numbers and Polynomials of Higher Order

• Journal title : Kyungpook mathematical journal
• Volume 54, Issue 1,  2014, pp.131-141
• Publisher : Department of Mathematics, Kyungpook National University
• DOI : 10.5666/KMJ.2014.54.1.131
Title & Authors
A New Family of q-analogue of Genocchi Numbers and Polynomials of Higher Order
Araci, Serkan; Acikgoz, Mehmet; Seo, Jong Jin;

Abstract
In the present paper, we introduce the new generalization of q-Genocchi polynomials and numbers of higher order. Also, we give some interesting identities. Finally, by applying q-Mellin transformation to the generating function for q-Genocchi polynomials of higher order put we define novel q-Hurwitz-Zeta type function which is an interpolation for this polynomials at negative integers.
Keywords
Genocchi numbers and polynomials;q-Genocchi numbers and polynomials of higher order;q-Mellin transformation;q-Hurwitz-zeta function;q-Gamma function;q-Exponential function;
Language
English
Cited by
1.
Bounds for q-integrals of ψ r + 1 r + 1 ${}_{r+1}\psi_{r+1}$ with applications, Journal of Inequalities and Applications, 2015, 2015, 1
References
1.
S. Araci, Novel identities involving Genocchi numbers and polynomials arising from applications of umbral calculus, Applied Mathematics and Computation, 233(2014), 599-607.

2.
M. Acikgoz, S. Araci and I. N. Cangul, A note on the modified q-Bernstein polynomials for functions of several variables and their reflections on q-Volkenborn integration, Applied Mathematics and Computation, 218(3)(2011), 707-712.

3.
S. Araci, M. Acikgoz, E. Sen, On the extended Kim's p-adic q-deformed fermionic integrals in the p-adic integer ring, Journal of Number Theory, 133(2013), 3348-3361.

4.
S. Araci, M. Acikgoz and K. H. Park, A note on the q-analogue of Kim's p-adic log gamma type functions associated with q-extension of Genocchi and Euler numbers with weight ${\alpha}$, Bull. Korean Math. Soc., 50(2)(2013), 583-588.

5.
S. Araci, D. Erdal and J. J. Seo, A study on the fermionic p-adic q-integral representation on $\mathbb{Z}_p$ associated with weighted q-Bernstein and q-Genocchi polynomials, Abstract and Applied Analysis, Volume 2011, Article ID 649248, 10 pages.

6.
S. Araci, M. Acikgoz and A. Gursul, Analytic continuation of weighted q-Genocchi numbers and polynomials, Commun. Korean Math. Soc., 28(3)(2013), 457-462.

7.
S. Araci and M. Acikgoz, A note on the Frobenius-Euler numbers and polynomials associated with Bernstein polynomials, Advanced Studies in Contemporary Mathematics, 22(3)(2012), 399-406.

8.
S. Araci, M. Acikgoz, H. Jolany, J. J. Seo, A unified generating function of the q-Genocchi polynomials with their interpolation functions, Proc. Jangjeon Math. Soc.,15(2)(2012), 227-233.

9.
S. Araci, J. J. Seo and D. Erdal, New construction weighted (h, q)-Genocchi numbers and polynomials related to Zeta-type functions, Discrete Dynamics in Nature and Society, 2011(2011), Article ID 487490, 7 pages.

10.
K. Brahim, and R. Guanes, Some applications of the q-Mellin Transform, Tamsui Oxford Journal of Mathematical Sciences, 26(3)(2010), 335-343.

11.
Y. He and C.Wang, Some formulae of products of the Apostol-Bernoulli and Apostol-Euler polynomials, Discrete Dynamics in Nature and Society, vol. 2012, Article ID 927953, 11 pages, 2012.

12.
F. H. Jackson, On q-definite integrals, The Quarterly Journal of Pure and Applied Mathematics, 41(1910), 193-2036.

13.
L. C. Jang, The q-analogue of twisted Lerch type Euler Zeta functions, Bull. Korean Math. Soc., 47(6)(2010), 1181-1188.

14.
H. Jolany and H. Sharifi, Some results for Apostol-Genocchi Polynomials of higher order, Bulletin of the Malaysian Mathematical Sciences Society, 36(2)(2013).

15.
V. G. Kac and P. Cheung, Quantum Calculus, Universitext, Springer-Verlag, New York, 2002.

16.
T. Kim, q-Generalized Euler numbers and polynomials, Russ. J. Math. Phys., 13(3)(2006), pp. 293-308

17.
T. Kim, Some identities on the q-Euler polynomials of higher order and q-stirling numbers by the fermionic p-adic integral on $\mathbb{Z}_p$, Russian J. Math. Phys., 16(2009), 484-491.

18.
T. Kim, On the q-extension of Euler and Genocchi numbers, J. Math. Anal. Appl., 326(2007), 1458-1465.

19.
T. Kim, On the analogs of Euler numbers and polynomials associated with p-adic q-integral on $\mathbb{Z}_p$ at q = -1, J. Math. Anal. Appl., 331 (2007), 779-792.

20.
T. Kim, Some Identities on the integral representation of the product of several q-Bernstein-type polynomials, Abstract and Applied Analysis, Volume 2011, Article ID 634675, 11 pages.

21.
T. Kim, S. H. Lee, H. H. Han and C. S. Ryoo, On the values of the weighted q-Zeta and L-functions, Discrete Dynamics in Nature and Society, Volume 2011, Article ID 476381, 7 pp.

22.
T. Kim, Euler numbers and polynomials associated with Zeta functions, Abstract and Applied Analysis, Volume 2008, Article ID 581582, 11 pages.

23.
T. Kim, Power series and asymptotic series associated with the q-analog of the twovariable p-adic L-function, Russian Journal of Mathematical Physics, 12(2)(2005), 186-196.

24.
T. Kim, On the q-extension of Euler and Genocchi numbers, J. Math. Anal. Appl., 326(2007), 1458-1465.

25.
D. Milicic, Notes on the Riemann's Zeta function, http://www.math.utah.edu/-milicic/zeta.pdf.

26.
S-H. Rim, J-H. Jin, E-J. Moon and S-J. Lee, On multiple interpolation functions of the q-Genocchi polynomials, Journal of Inequalities and Applications, Volume 2010, Article ID 351419, 13 pages.

27.
Richard L. Rubin, A $q^2$-analogue operator for $q^2$-analogue Fourier Analysis, J. Math. Analys. App., 212(1997), 571-582.