• Kim, Min-Soo (Department of Mathematics Kyungnam University) ;
  • Son, Jin-Woo (Department of Mathematics Kyungnam University)
  • Published : 2007.02.28


In this paper, we consider the q-Volkenborn integral of uniformly differentiable functions on the p-adic integer ring. By using this integral, we obtain the generating functions of twisted q-generalized Bernoulli numbers and polynomials. We find some properties of these numbers and polynomials.


q-Volkenborn integral;$I_q$-Fourier transforms


  1. M. Cenkci, M. Can, and V. Kurt, p-adic interpolation functions and Kummer-type congruences for q-twisted and q-generalized twisted Euler numbers, Adv. Stud. Contemp. Math. 9 (2004), no. 2, 203-216
  2. K. Iwasawa, Lectures on p-Adic L-Functions, Ann. of Math. Stud. 74, Princeton Univ. Press, Princeton, 1972
  3. M.-S. Kim and J.-W. Son, On Bernoulli numbers, J. Korean Math. Soc. 37 (2000), no. 3, 391-410
  4. M.-S. Kim and J.-W. Son, Some remarks on a q-analogue of Bernoulli numbers, J. Korean Math. Soc. 39 (2002), no. 2, 221-236
  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 explicit formulas of p-adic q-L-functions, Kyushu J. Math. 48 (1994), no. 1, 73-86
  7. 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
  8. T. Kim, q-Volkenborn integration, Russ. J. Math. Phys. 9 (2002), no. 3, 288-299
  9. T. Kim, A new approach to q-zeta function, Adv. Stud. Contemp. Math. 11 (2005), no. 2, 157-162
  10. T. Kim, On a p-adic interpolation function for the q-extension of the generalized Bernoulli polynomials and its derivative, available at math.NT/0502460, preprint 2005
  11. 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
  12. N. Koblitz, p-Adic Analysis: a Short Course on Recent Work, Cambridge University Press, Mathematical Society Lecture Notes, Series 46, 1980
  13. N. Koblitz, p-adic Numbers, p-adic Analysis, and Zeta-Functions, 2nd ed., Springer-Verlag, New York, 1984
  14. Q.-M. Luo and F. Qi, Relationships between generalized Bernoulli numbers and poly-nomials and generalized Euler numbers and polynomials, Adv. Stud. Contemp. Math. (Kyungshang) 7 (2003), no. 1, 11-18
  15. C. S. Ryoo, H. Song, and R. P. Agarwal, On the roots of the q-analogue of Euler-Barnes' polynomials, Adv. Stud. Contemp. Math. (Kyungshang) 9 (2004), no. 2, 153-163
  16. W. H. Schikhof, Ultrametric Calculus, An introduction to p-adic analysis, Cambridge Studies in Adv. Math. 4, Cambridge Univ. Press, Cambridge, 1984
  17. Y. Simsek, Theorems on twisted L-function and twisted Bernoulli numbers, Adv. Stud. Contemp. Math. (Kyungshang) 11 (2005), no. 2, 205-218
  18. H. M. Srivastava, T. Kim, and Y. Simsek, q-Bernoulli numbers and polynomials associ-ated with multiple q-Zeta functions and basic L-series, Russ. J. Math. Phys. 12 (2005), no. 2, 241-268
  19. L. C. Washington, Introduction to Cyclotomic Fields, 2nd ed., Springer-Verlag, New York, 1997
  20. C. F. Woodcock, Fourier analysis for p-adic Lipschitz functions, J. London Math. Soc. (2) 7 (1974), 681-693
  21. C. F. Woodcock, Convolutions on the ring of p-adic integers, J. London Math. Soc. (2) 20 (1979), no. 1, 101-108
  22. T. Kim, A note on the Fourier transform of p-adic q-integrals, available at ArXiv math.NT/0511573, preprint 2005
  23. T. Kubota and H. W. Leopoldt, Eine p-adische Theorie der Zetawerte. I. Einfuhrung der p-adischen Dirichletschen L-Funktionen, J. Reine Angew. Math. 214/215(1964), 328-339
  24. J. Satoh, q-analogue of Riemann's $\zeta$ -function and q-Euler numbers, J. Number Theory 31 (1989), no. 3, 346-362

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