A Note on the p-Adic Interpolation Function for Multiple Generalized Genocchi Numbers

Serkan Araci^1, , Mehmet Acikgoz² and Erdoğan Şen³

¹Hatay, Turkey

²Department of Mathematics, Faculty of Science and Arts, University of Gaziantep, Gaziantep, Turkey

³Department of Mathematics, Faculty of Science and Letters, Namik Kemal University, Tekirdağ, Turkey

Cite this paper:
Serkan Araci, Mehmet Acikgoz and Erdoğan Şen. A Note on the p-Adic Interpolation Function for Multiple Generalized Genocchi Numbers. Turkish Journal of Analysis and Number Theory. 2013; 1(1):17-22. doi: 10.12691/tjant-1-1-5

Abstract

In the present paper, we deal with multiple generalized Genocchi numbers and polynomials. Also, we introduce analytic interpolating function for the multiple generalized Genocchi numbers attached to χ at negative integers in complex plane and we de.ne the multiple Genocchi p-adic L-function. Finally, we derive the value of the partial derivative of our multiple p-adic l-function at s = 0.

Keywords:
multiple generalized Genocchi numbers and poly-nomials Euler-Gamma function p-adic interpolation function multiple gen-eralized zeta function

This work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

References:

[1]	F. Qi, Integral representations and properties of Stirling numbers of the first kind, Journal of Number Theory, Volume 133, issue 7 (July, 2013), p. 2307-2319.
[2]	S. Araci, Novel identities for q-Genocchi numbers and polynomials, Journal of Function Spaces and Applications, Volume 2012, Article ID 214961, 13 pages.
[3]	S. Araci, D. Erdal and J. J. Seo, A study on the fermionic p-adic q-integral representation on p associated with weighted q-Bernstein and q-Genocchi polynomials, Abstract and Applied Analysis, Volume 2011, Article ID 649248, 10 pages.
[4]	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, Volume 2011, Article ID 487490, 7 pages.
[5]	S. Araci, M. Acikgoz and E. Sen, On the extended Kim.s p-adic q-deformed fermionic integrals in the p-adic integer ring, J. Number Theory (2013), http://dx.doi.org/10.1016/j.jnt.2013.04.007.
[6]	S. Araci, M. Acikgoz and H. Jolany, On p-adic interpolating function associated with modified Dirichlet’s type of twisted q-Euler numbers and polynomials with weight alpha, Journal of Classical Analysis, Volume 2, Number 1 (2013), 35-48.
[7]	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 (2013), No. 2, pp. 583-588.
[8]	E. Cetin, M. Acikgoz, I. N. Cangul and S. Araci, A note on the (h; q)-Zeta-type function with weight α, Journal of Inequalities and Applications 2013, 2013:100.
[9]	B. C. Kellner, On irregular prime power divisors of the Bernoulli numbers, Mathematics of Computation, Volume 76, Number 257, January 2007, Pages 405-441.
[10]	D.V. Kruchinin and V. V. Kruchinin, Application of a composition of generating functions for obtaining explicit formulas of polynomials, Journal of Mathematical Analysis and Applications, Volume 404, Issue 1, 1 August 2013, Pages 161-171.
[11]	T. Kim, On a p-adic interpolation function for the q-extension of the generalized Bernoulli polynomials and its derivative, Discrete Mathematics, 309 (2009) 1593-1602.
[12]	T. Kim, On the q-extension of Euler and Genocchi numbers, J. Math. Anal. Appl. 326 (2007) 1458-1465.
[13]	T. Kim, On the multiple q-Genocchi and Euler numbers, Russian J. Math. Phys. 15 (4) (2008) 481-486.
[14]	T. Kim, p-adic q-integrals associated with Changhee-Barnes.q-Bernoulli polynomials, Integral Transforms Spec. Funct. 15 (2004) 415-420.
[15]	T. Kim, Multiple p-adic L-function, Russ. J. Math. Phys. 13 (2006), no. 2, 151-157.
[16]	T. Kim, A Note on the q-Genocchi numbers and polynomials, Journal of Inequalities and Applications Article ID 71452, 8 pages.
[17]	T. Kim, q-Volkenborn integration, Russ. J. Math. phys. 9 (2002), 288-299.
[18]	T. Kim, An invariant p-adic q-integrals on p, Applied Mathematics Letters, vol. 21 (2008), pp. 105-108.
[19]	T. Kim, and J-S. Cho, A note on multiple Dirichlet.s q-L-function, Adv. Stud. Contemp. Math. (Kyungshang) 11 (2005), no. 1, 57-60.
[20]	T. Kim, A note on the q-multiple zeta function, Adv. Stud. Contemp. Math. (Kyungshang) 8 (2004), no. 2, 111-113.
[21]	T. Kim, Remark on the multiple Bernoulli numbers, Proc. Jangjeon Math. Soc. 6 (2003), no. 2, 185-192.
[22]	T. Kim, q-Euler numbers and polynomials associated with p-adic q-integrals, J. Nonlinear Math. Phys., 14 (2007), no. 1, 15-27.
[23]	T. Kim, New approach to q-Euler polynomials of higher order, Russ. J. Math. Phys., 17 (2010), no. 2, 218-225.
[24]	T. Kim, Some identities on the q-Euler polynomials of higher order and q-Stirling numbers by the fermionic p-adic integral on p, Russ. J. Math. Phys., 16 (2009), no.4,484-491.
[25]	T. Kim and S.H. Rim, On the twisted q-Euler numbers and polynomials associated with basic q-l-functions, Journal of Mathematical Analysis and Applications, vol. 336 (2007), no. 1, pp. 738-744.
[26]	T. Kim, On p-adic q-l-functions and sums of powers, J. Math. Anal. Appl. (2006).
[27]	T. Kim, On the analogs of Euler numbers and polynomials associated with p-adic q-integral on p at q = -1, J. Math. Anal. Appl. 331 (2) (2007) 779-792.
[28]	T. Kim, On p-adic interpolating function for q-Euler numbers and its derivatives, J. Math. Anal. Appl. 339 (1) (2008) 598-608.
[29]	T. Kim, q-Bernoulli numbers and polynomials associated with Gaussian binomial coe¢ cients, Russ. J. Math. Phys. 15 (1) (2008) 51-57.
[30]	T. Kim, Euler numbers and polynomials associated with zeta functions, Abstr. Appl. Anal. (2008) Art. ID 581582, 11 pp.
[31]	T. Kim, Analytic continuation of multiple q-zeta functions and their values at negative integers, Russ. J. Math. Phys. 11 (1) (2004) 71-76.
[32]	T. Kim, Barnes-type multiple q-zeta functions and q-Euler polynomials, J. Phys. A 43 (2010), no. 25, 255201, 11 pp.
[33]	T. Kim, On Euler-Barnes multiple zeta functions, Russ. J. Math. Phys. 10 (2003), no. 3, 261-267.
[34]	T. Kim, Non-Archimedean q-integrals associated with multiple Changhee q-Bernoulli polynomials, Russ. J. Math. Phys. 10 (2003), no. 1, 91-98.
[35]	T. Kim, On a p-adic interpolation function for the multiple generalized Euler numbers and its derivatives. J. Comput. Anal. Appl. 14 (2012), no. 3, 410-416.
[36]	G. D. Liu, Generating functions and generalized Euler numbers, Proc. Japan Acad. Ser. A Math. Sci., 84 (2008), 29-34.
[37]	C. Frappier, Representation formulas for entire functions of exponential type and Generalized Bernoulli polynomials, J. Austral. Math. Soc. Ser., 64 (1998), No. 3, 307-316.
[38]	H-S. Vandiver, On Generalizations of the Numbers of Bernoulli and Euler, Vol. 23, No. 10 (1937), 555-559.
[39]	H. Tsumura, On a p-adic interpolation of the generalized Euler numbers and it applications, Tokyo J. Math. 10 (2) (1987) 281-293.
[40]	K. Shiratani and S. Yamamoto, On a p-adic interpolation function for the Euler numbers and its derivatives, Mem. Fac. Sci. Kyushu Univ. Ser. A 39 (1) (1985) 113-125.
[41]	L-C. Washington, Introduction to cyclotomic fields, Second edition, Springer-Verlag, 1997.
[42]	G. J. Fox, A method of Washington applied to the derivation of a two-variable p-adic L-function, Paci.c J. Math. 209 (2003), 31-40.
[43]	J. Diamond, On the values of p-adic L-functions at positive integers, Acta Arith. 35 (1979), 223-237.