JOURNAL BROWSE
Search
Advanced SearchSearch Tips
ON THE ANALOGS OF BERNOULLI AND EULER NUMBERS, RELATED IDENTITIES AND ZETA AND L-FUNCTIONS
facebook(new window)  Pirnt(new window) E-mail(new window) Excel Download
 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;
  PDF(new window)
 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
4.
A formula for generating modular forms with weight 4,;

Advanced Studies in Contemporary Mathematics, 2015. vol.25. 2, pp.201-209 crossref(new window)
1.
A set of finite order differential equations for the Appell polynomials, Journal of Computational and Applied Mathematics, 2014, 259, 108  crossref(new windwow)
2.
Notes on generalization of the Bernoulli type polynomials, Applied Mathematics and Computation, 2011, 218, 3, 906  crossref(new windwow)
3.
A unified presentation of certain meromorphic functions related to the families of the partial zeta type functions and the L-functions, Applied Mathematics and Computation, 2012, 219, 8, 3903  crossref(new windwow)
4.
On the von Staudt–Clausen's theorem related to q-Frobenius–Euler numbers, Journal of Number Theory, 2016, 159, 329  crossref(new windwow)
5.
On a class of q-Bernoulli and q-Euler polynomials, Advances in Difference Equations, 2013, 2013, 1, 108  crossref(new windwow)
6.
q-Bernstein polynomials related to q-Frobenius-Euler polynomials, l-functions, and q-Stirling numbers, Mathematical Methods in the Applied Sciences, 2012, 35, 8, 877  crossref(new windwow)
7.
Values of twisted Barnes zeta functions at negative integers, Russian Journal of Mathematical Physics, 2013, 20, 2, 129  crossref(new windwow)
8.
q-Dirichlet type L-functions with weight α, Advances in Difference Equations, 2013, 2013, 1, 40  crossref(new windwow)
9.
A family of p-adic twisted interpolation functions associated with the modified Bernoulli numbers, Applied Mathematics and Computation, 2010, 216, 10, 2976  crossref(new windwow)
10.
A unified presentation of the generating functions of the generalized Bernoulli, Euler and Genocchi polynomials, Computers & Mathematics with Applications, 2010, 60, 10, 2779  crossref(new windwow)
11.
p-Adic distribution of the unification of the Bernoulli, Euler and Genocchi polynomials, Applied Mathematics and Computation, 2011, 218, 3, 970  crossref(new windwow)
12.
Generating functions for generalized Stirling type numbers, Array type polynomials, Eulerian type polynomials and their applications, Fixed Point Theory and Applications, 2013, 2013, 1, 87  crossref(new windwow)
13.
On a class of generalized q-Bernoulli and q-Euler polynomials, Advances in Difference Equations, 2013, 2013, 1, 115  crossref(new windwow)
14.
A unified presentation of three families of generalized Apostol type polynomials based upon the theory of the umbral calculus and the umbral algebra, Journal of Number Theory, 2013, 133, 10, 3245  crossref(new windwow)
15.
Higher-order Euler-type polynomials and their applications, Fixed Point Theory and Applications, 2013, 2013, 1, 40  crossref(new windwow)
 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 crossref(new window)

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 crossref(new window)

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

17.
T. Kim, Multiple p-adic L-function, Russ. J. Math. Phys. 13 (2006), 151-157 crossref(new window)

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 crossref(new window)

20.
T. Kim, A note on q-Bernoulli numbers and polynomials, J. Nonlinear Math. Phys. 13 (2006), 315-320 crossref(new window)

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 crossref(new window)

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 crossref(new window)

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