Q-Bernoulli numbers and polynomials associated with multiple q-zeta functions and basic L-series

Russian Journal of Mathematical Physics

Volumn 12, Issue 2, 2005, Pages 241-268

  Srivastava, H M ^a   Kim, T ^b   Simsek, Y ^c  

^a UNIVERSITY OF VICTORIA   (Canada)

^b Kongju National University   (South Korea)

^c MERSIN UNIVERSITY   (Turkey)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 20744457221     PISSN: 10619208     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (60)

1. G. E. Andrews, "q-Analogues of the Binomial Coefficient Congruences of Babbage, Wolstenhlome and Glaisher," Discrete Math. 204, 15-25 (1999).
2. T. M. Apostol, Introduction to Analytic Number Theory (Springer, New York, 1976).
3. R. Askey, "The q-Gamma and q-Beta Functions," Applicable Anal. 8, 125-141 (1978).
4. E. W. Barnes, "On the Theory of the Multiple Gamma Functions," Trans. Cambridge Philos. Soc. 19, 374-425 (1904).
5. L. Carlitz, "q-Bernoulli Numbers and Polynomials," Duke Math. J. 15, 987-1000 (1948).
6. I. Cherednik, "On q-Analogues of the Riemann's Zeta Function," Selecta Math. 7, 447-491 (2001).
7. E. Deeba and D. Rodriguez, "Stirling's Series and Bernoulli Functions," Amer. Math. Monthly 98, 423-426 (1991).
8. K. Dilcher, "Some q-Series Identities Related to Divisor Functions," Discrete Math. 145, 83-93 (1995).
9. E. Friedman and S. Ruijsenaars, "Shintani-Barnes Zeta and Gamma Functions," Adv. in Math. 187, 362-395 (2004).
10. F. T. Howard, "Applications of a Recurrence for the Bernoulli Numbers," J. Number Theory 52, 157-172 (1995).
11. K. Iwasawa, Lecture on p-Adic L-Functions, Annals of Mathematics Studies 74 (Princeton University Press, Princeton, 1972).
12. M. Jimbo and T. Miwa, "Quantum KZ Equation with |q| = 1 and Correlation Functions of the XXZ Model in the Gapless Regime," J. Phys. A: Math. Gen. 29, 2923-2958 (1996).
13. J. Katriel, "Stirling Numbers Identities Interconsistency of g-Analogues," J. Phys. A: Math. Gen. 31, 3559-3572 (1988).
14. T. Kim, "On Explicit Formulas of p-Adic q-L-Functions," Kyushu J. Math. 48, 73-86 (1994).
15. T. Kim, "On a q-Analogue of the p-Adic Log-Gamma Functions and Related Integrals," J. Number Theory 76, 320-329 (1999).
16. T. Kim, "A Note on p-Adic q-Dedekind Sums," C. R. Acad. Bulgare Sci. 54 (10), 37-42 (2001).
17. T. Kim, "An Invariant p-Adic Integral Associated with Daehee Numbers," Integral Transform. Spec. Funct. 13, 65-69 (2002).
18. T. Kim, "q-Volkenborn Integration," Russian J. Math. Phys. 19, 288-299 (2002).
19. T. Kim, "On p-Adic q-L-Functions and Sums of Powers," Discrete Math. 252, 179-187 (2002).
20. T. Kim, "Non-Archimedean q-Integrals Associated with Multiple Changhee q-Bernoulli Polynomials," Russian J. Math. Phys. 10, 91-98 (2003).
21. T. Kim, "q-Riemann Zeta Functions," Internat. J. Math. Math. Sci. 2004, 599-605 (2004).
22. T. Kim, "On Euler-Barnes Multiple Zeta Functions," Russian J. Math. Phys. 10, 261-267 (2003).
23. T. Kim, "A Note on Dirichlet Series," Proc. Jangjeon Math. Soc. 6, 161-166 (2003).
24. T. Kim, "Sums of Powers of Consecutive q-Integers," Adv. Stud. Contemp. Math. 9, 15-18 (2004).
25. T. Kim, "A Note on q-Zeta Functions," in Proceedings of the 15th International Conference of the Jangjeon Mathematical Society (Hapcheon, South Korea, August 5-7, 2004), pp. 110-114.
26. T. Kim, "A Note on the q-Multiple Zeta Function," Adv. Stud. Contemp. Math. 8, 111-113 (2004).
27. T. Kim, "p-Adic q-Integrals Associated with the Changhee-Barnes' q-Bernoulli Polynomials," Integral Transform. Spec. Funct. 15, 415-420 (2004).
28. T. Kim, "Analytic Continuation of Multiple q-Zeta Functions and Their Values at Negative Integers," Russian J. Math. Phys. 11, 71-76 (2004).
29. T. Kim and S.-H. Rim, "A Note on the q-Integrals and q-Series, " Adv. Stud. Contemp. Math. 2, 37-45 (2000).
30. T. Kim, L. C. Jang, S.-H. Rim, and J. W. Son, "On the Values of Zeta and L-Functions," Proc. Jangjeon Math. Soc. 1, 11-18 (2000).
31. T. Kim, S.D. Kim, and D.W. Park, "On Uniform Differentiability and q-Mahler Expansions," Adv. Stud. Contemp. Math. 4, 35-41 (2001).
32. T. Kim and S.-H. Rim, "On Changhee-Barnes' q-Euler Numbers and Polynomials," Adv. Stud. Contemp. Math. 9, 81-86 (2004).
33. A. Khrennikov, p-Adic Valued Distributions in Mathematical Physics (Kluwer Academic, Dordrecht, 1994.
34. A. Khrennikov, "p-Adic Discrete Dynamical Systems and Their Applications in Physics and Cognitive Science," Russian J. Math. Phys. 11, 45-70 (2004).
35. N. Koblitz, "q-Extension of the p-Adic Gamma Function," Trans. Amer. Math. Soc. 260, 449-457 (1980).
36. N. Koblitz, "On Carlitz's q-Bernoulli Numbers," J. Number Theory 14, 332-339 (1982).
37. T. H. Koornwinder, "Special Functions and q-Commuting Variables," Fields Inst. Comm. 14, 131-166 (1997).
38. N. Kurokawa, "Multiple Sine Functions and Selberg Zeta Functions," Proc. Japan Acad. Ser. A Math. Sci. 67, 61-64 (1991).
39. K. Matsumoto, "The Analytic Continuation and the Asymptotic Behaviour of Certain Multiple Zeta-Function. I," J. Number Theory 101, 223-243 (2003).
40. C. A. Nelson and M. G. Gartley, "On the Zeros of the q-Analogue of Exponential Function," J. Phys. A: Math. Gen. 27, 3857-3881 (1994).
41. q(z))," J. Phys. A: Math. Gen. 24, 8099-8115 (1996).
42. M. Nishizawa, "On a q-Analogue of the Multiple Gamma Functions," Lett. Math. Phys. 37, 201-209 (1996).
43. K. Ota, "On Kummer-Type Congruences for Derivatives of Barnes' Multiple Bernoulli Polynomials," J. Number Theory 92, 1-36 (2002).
44. K. Ota, "Derivatives of Dedekind Sums and Their Reciprocity Law," J. Number Theory 98, 280-309 (2003).
45. T. M. Rassias and H. M. Srivastava, "Some Classes of Infinite Series Associated with the Riemann Zeta and Polygamma Functions and Generalized Harmonic Numbers," Appl. Math. Comput. 131, 593-605 (2002).
46. A.M. Robert, A Course in p-Adic Analysis (Springer, New York, 2000).
47. S. N. M. Ruijsenaars, "On Barnes' Multiple Zeta and Gamma Functions," Adv. in Math. 156, 107-132 (2000).
48. B.E. Sagan, "Congruence Properties of q-Analogues," Adv. in Math. 95, 127-143 (1992).
49. W. H. Schikhof, Ultrametric Calculus: An Introduction to p-Adic Analysis, Cambridge Studies in Advanced Mathematics 4 (Cambridge University Press, Cambridge, 1984).
50. K. Shiratani, "On Euler Numbers," Mem. Fac. Sci. Kyushu Univ. 27, 1-5 (1973).
51. K. Shiratani and S. Yamamoto, "On a p-Adic Interpolation Function for the Euler Numbers and Its Derivative," Mem. Fac. Sci. Kyushu Univ. 39, 113-125 (1985).
52. Y. Simsek, "Theorems on Twisted L-Functions and Twisted Bernoulli Numbers," Proc. Jangjeon Math. Soc. 8 (2005).
53. Y. Simsek, "Generalixed Dedekind Sums Associated with the Abel Sum and the Eisenstein and Lambert Series," Adv. Stud. Contemp. Math. 9, 125-137 (2004).
54. H. M. Srivastava and J. Choi, Series Associated with the Zeta and Related Functions (Kluwer Academic, Dordrecht, 2001).
55. H. M. Srivastava and P. W. Karlsson, Multiple Gaussian Hypergeometric Series (John Wiley and Sons, New York, 1985).
56. H. Tsumura, "On a p-Adic Interpolation of Generalized Euler Numbers and Its Applications," Tokyo J. Math. 10, 281-293 (1987).
57. V. S. Vladimirov, I. V. Volovich, and E. I. Zelenov, p-Adic Analysis and Mathematical Physics, Series on Soviet and East European Mathematics 1 (World Scientific Publishing, Singapore, 1994).
58. P. T. Young, "On the Behavior of Some Two-Variable p-Adic L-Function," J. Number Theory 98, 67-86 (2003).
59. L.C. Washington, Introduction to Cyclotomic Fields, Graduate Texts in Mathematics 83, 2nd ed. (Springer, New York, 1997).
60. E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, 4th ed. (Cambridge University Press, Cambridge, 1927).