ON THE ANALOGS OF BERNOULLI AND EULER NUMBERS, RELATED IDENTITIES AND ZETA AND L-FUNCTIONS

- Journal title : Journal of the Korean Mathematical Society
- Volume 45, Issue 2, 2008, pp.435-453
- Publisher : The Korean Mathematical Society
- DOI : 10.4134/JKMS.2008.45.2.435

Title & Authors

ON THE ANALOGS OF BERNOULLI AND EULER NUMBERS, RELATED IDENTITIES AND ZETA AND L-FUNCTIONS

Kim, Tae-Kyun; Rim, Seog-Hoon; Simsek, Yilmaz; Kim, Dae-Yeoul;

Kim, Tae-Kyun; Rim, Seog-Hoon; Simsek, Yilmaz; Kim, Dae-Yeoul;

Abstract

In this paper, by using q-deformed bosonic p-adic integral, we give -Bernoulli numbers and polynomials, we prove Witt's type formula of -Bernoulli polynomials and Gauss multiplicative formula for -Bernoulli polynomials. By using derivative operator to the generating functions of -Bernoulli polynomials and generalized -Bernoulli numbers, we give Hurwitz type -zeta functions and Dirichlet's type -L-functions; which are interpolated -Bernoulli polynomials and generalized -Bernoulli numbers, respectively. We give generating function of -Bernoulli numbers with order r. By using Mellin transforms to their function, we prove relations between multiply zeta function and -Bernoulli polynomials and ordinary Bernoulli numbers of order r and -Bernoulli numbers, respectively. We also study on -Bernoulli numbers and polynomials in the space of locally constant. Moreover, we define -partial zeta function and interpolation function.

Keywords

Bernoulli numbers and polynomials;zeta functions;

Language

English

Cited by

1.

q-Analogues of some results for the Apostol-Euler polynomials,;

Advanced Studies in Contemporary Mathematics, 2010. vol.20. 1, pp.103-113

2.

Unification of Multiple Lerch-Zeta Type Functions,;;

Advanced Studies in Contemporary Mathematics, 2011. vol.21. 4, pp.367-374

3.

INTERPOLATION FUNCTIONS OF THE EULERIAN TYPE POLYNOMIALS AND NUMBERS,;

Advanced Studies in Contemporary Mathematics, 2013. vol.23. 2, pp.301-307

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

18.

19.

References

1.

E. W. Barnes, On the theory of the multiple gamma functions, Trans. Camb. Philos. Soc. 19 (1904), 374-425

2.

K. C. Garret and K. Hummel, A combinatorial proof of the sum of q-cubes, Electron. J. Combin. 11 (2004), no. 1, Research Paper 9

3.

K. Iwasawa, Lectures on p-adic L-function, Annals of Mathematics Studies, No. 74. Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1972

4.

L. C. Jang and H. K. Pak, Non-Archimedean integration associated with q-Bernoulli numbers, Proc. Jangjeon Math. Soc. 5 (2002), no. 2, 125-129

5.

T. Kim, An analogue of Bernoulli numbers and their congruences, Rep. Fac. Sci. Engrg. Saga Univ. Math. 22 (1994), no. 2, 21-26

6.

T. Kim, On a q-analogue of the p-adic log gamma functions and related integrals, J. Number Theory 76 (1999), no. 2, 320-329

7.

T. Kim, q-Volkenborn integration, Russ. J. Math. Phys. 9 (2002), no. 3, 288-299

8.

T. Kim, An invariant p-adic integral associated with Daehee numbers, Integral Transforms Spec. Funct. 13 (2002), no. 1, 65-69

9.

T. Kim, On Euler-Barnes multiple zeta functions, Russ. J. Math. Phys. 10 (2003), no. 3, 261-267

10.

T. Kim, A note on multiple zeta functions, JP J. Algebra Number Theory Appl. 3 (2003), no. 3, 471-476

11.

T. Kim, Non-archimedean q-integrals associated with multiple Changhee q-Bernoulli Polynomials, Russ. J. Math. Phys. 10 (2003), no. 1, 91-98

12.

T. Kim, Remark on the multiple Bernoulli numbers, Proc. Jangjeon Math. Soc. 6 (2003), no. 2, 185-192

13.

T. Kim, Sums of powers of consecutive q-integers, Adv. Stud. Contemp. Math. (Kyungshang) 9 (2004), no. 1, 15-18

14.

T. Kim, Analytic continuation of multiple q-zeta functions and their values at negative integers, Russ. J. Math. Phys. 11 (2004), no. 1, 71-76

15.

T. Kim, A note on multiple Dirichlet's q-L-function, Adv. Stud. Contemp. Math. (Kyungshang) 11 (2005), no. 1, 57-60

16.

T. Kim, Power series and asymptotic series associated with the q-analog of the two-variable p-adic L-function, Russ. J. Math. Phys. 12 (2005), no. 2, 186-196

18.

T. Kim, A new approach to p-adic q-L-functions, Adv. Stud. Contemp. Math. (Kyungshang) 12 (2006), no. 1, 61-72

19.

T. Kim, On the analogs of Euler numbers and polynomials associated with p-adic q-integral on Zp at q = -1, J. Math. Anal. Appl. (2006), doi:10.1016/j.jmaa.2006.09.027

20.

21.

T. Kim, L. C. Jang, S.-H. Rim, and H.-K. Pak, On the twisted q-zeta functions and q-Bernoulli polynomials, Far East J. Appl. Math. 13 (2003), no. 1, 13-21

22.

J. Satho, q-analogue of Riemann's ${\zeta}$ -function and q-Euler numbers, J. Number Theory 31 (1989), no. 3, 346-362

23.

M. Schlosser, q-analogues of the sums of consecutive integers, squares, cubes, quarts and quints, Electron. J. Combin. 11 (2004), no. 1, Research Paper 71

24.

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 (1985), no. 1, 113-125

25.

Y. Simsek, Theorems on twisted L-functions and twisted Bernoulli numbers, Adv. Stud. Contemp. Math. 11 (2005), no. 2, 205-218

26.

Y. Simsek, Twisted (h, q)-Bernoulli numbers and polynomials related to (h, q)-zeta function and L-function, J. Math. Anal. Appl. 324 (2006), 790-804

27.

Y. Simsek, D. Kim, T. Kim, and S.-H. Rim, A note on the sums of powes of consecutive q-integers, J. Appl. Funct. Different Equat. 1 (2006), 63-70

28.

Y. Simsek and S. Yang, Transformation of four Titchmarsh-type infinite integrals and generalized Dedekind sums associated with Lambert series, Adv. Stud. Contemp. Math. (Kyungshang) 9 (2004), no. 2, 195-202

29.

H. M. Srivastava, T. Kim, and Y. Simsek, q-Bernoulli multiple q-zeta functions and basic L-series, Russ. J. Math. Phys. 12 (2005), no. 2, 241-268