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Multiple two-variable p-adic q-L-function and its behavior at s = 0

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Abstract

The objective of this paper is to construct a multiple p-adic q-L-function of two variables which interpolates multiple generalized q-Bernoulli polynomials. By using this function, we solve a question of Kim and Cho. We also define a multiple partial q-zeta function which is related to the multiple q-L-function of two variables. Finally, we give a finite-sum representation of the multiple p-adic q-L-function of two variables and prove a multiple q-extension of the generalized formula of Diamond and Ferrero-Greenberg.

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References

  1. T. M. Apostol, Introduction to Analytic Number Theory, Undergraduate Texts in Mathematics (Springer, New York, 1985).

    Google Scholar 

  2. M. Cenkci, M. Can, and V. Kurt, “p-Adic Interpolation Functions and Kummer-Type Congruences for q-Twisted and Generalized q-Twisted Euler Numbers,” Adv. Stud. Contemp. Math. 9(2), 203–216 (2004).

    MATH  MathSciNet  Google Scholar 

  3. M. Cenkci and M. Can, “Some Results on q-Analogue of the Lerch Zeta Function,” Adv. Stud. Contemp. Math. 12(2), 213–223 (2006).

    MATH  MathSciNet  Google Scholar 

  4. J. Diamond, “The p-Adic Log Gamma Function and p-Adic Euler Constants,” Trans. Amer. Math. Soc. 233, 321–337 (1977).

    Article  MATH  MathSciNet  Google Scholar 

  5. B. Ferrero and R. Greenberg, “On the Behaviour of p-Adic L-Functions at s = 0,” Invent. Math. 50, 91–102 (1978).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  6. G. J. Fox, “A Method of Washington Applied to the Derivation of a Two-Varible p-Adic L-Function,“ Pacific J. Math. 209(1), 31–40 (2003).

    MATH  MathSciNet  Google Scholar 

  7. K. Iwasawa, Lectures on p-Adic L-Functions, Ann. Math. Studies 74 (Princeton University Press, Princeton, 1972).

    Google Scholar 

  8. L. Jang, T. Kim, and D.-W. Park, “Kummer Congruence for the Bernoulli Numbers of Higher Order,“ Appl. Math. Comput. 151, 589–593 (2004).

    Article  MATH  MathSciNet  Google Scholar 

  9. T. Kim, “On p-Adic q-L-Functions and Sums of Powers,” Discrete Math. 252, 179–187 (2002).

    Article  MATH  MathSciNet  Google Scholar 

  10. T. Kim, “q-Volkenborn Integration,” Russ. J. Math. Phys. 9(3), 288–299 (2002).

    MATH  MathSciNet  Google Scholar 

  11. T. Kim, “Non-Archimedean q-Integrals Associated with Multiple Changhee q-Bernoulli Polynomials,“ Russ. J. Math. Phys. 10(1), 91–98 (2003).

    MATH  ADS  MathSciNet  Google Scholar 

  12. T. Kim, “On Euler-Barnes Multiple Zeta Functions,” Russ. J. Math. Phys. 10(3), 261–267 (2003).

    MATH  MathSciNet  Google Scholar 

  13. T. Kim, “Sums of Powers of Consequtive q-Integers,” Adv. Stud. Contemp. Math. 9, 15–18 (2004).

    MATH  Google Scholar 

  14. T. Kim, “Analytic Continuation of Multiple q-Zeta Functions and Their Values at Negative Integers,“ Russ. J. Math. Phys. 11(1), 71–76 (2004).

    MATH  ADS  MathSciNet  Google Scholar 

  15. 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(2), 186–196 (2005).

    MATH  MathSciNet  Google Scholar 

  16. T. Kim, “A New Approach to p-Adic q-L-Function,” Adv. Stud. Contemp. Math. 12(1), 61–72 (2006).

    MATH  MathSciNet  Google Scholar 

  17. T. Kim, “Multiple p-Adic L-Function,” Russ. J. Math. Phys. 13(2), 151–157 (2006).

    Article  MATH  MathSciNet  Google Scholar 

  18. T. Kim and J.-S. Cho, “A Note on Multiple Dirichlet’s q-L-Function,” Adv. Stud. Contemp. Math. 11(1), 57–60 (2005).

    MATH  MathSciNet  Google Scholar 

  19. N. Koblitz, “A New Proof of Certain Formulas for p-Adic L-Functions,” Duke Math. J. 46(2), 455–468 (1979).

    Article  MATH  MathSciNet  Google Scholar 

  20. N. Koblitz, p-Adic Analysis: A Short Course on Recent Work, London Math. Soc. Lecture Notes Ser. 46 (Cambridge University Press, Cambridge-New York, 1980).

    MATH  Google Scholar 

  21. T. Kubota, H.-W. Leopoldt, “Eine p-adische Theorie der Zetawerte I, Einführung der p-adischen Dirichletschen L-Funktionen,” J. Reine Angew. Math., no. 214/215, 328–339 (1964).

    Google Scholar 

  22. C. A. Nelson and M.G. Gartley, “On the Zeros of the q-Analogue of Exponential Function,” J. Phys. A: Math. Gen. 24, 3857–3881 (1994).

    Article  ADS  MathSciNet  Google Scholar 

  23. C. A. Nelson and M.G. Gartley, “On the Two q-Analogues of Logarithmic Functions: lnq(w), ln(lnq(w)),“ J. Phys. A: Math. Gen. 27, 8099–8115 (1996).

    Article  ADS  MathSciNet  Google Scholar 

  24. N. Nörlund, Vorlesungen über Differenzenrechnung (Chelsea, New York, 1954).

    Google Scholar 

  25. K. Shiratani and S. Yamamato, “On a p-Adic Interpolation Function for the Euler Numbers and Its Derivative,” Mem. Fac. Sci. Kyushu Univ. 39, 113–125 (1985).

    Article  MATH  Google Scholar 

  26. Y. Simsek, “On p-Adic Twisted q-L-Functions Related to Generalized Twisted Bernoulli Numbers,“ Russ. J. Math. Phys. 13(3), 340–348 (2006).

    Article  MATH  MathSciNet  Google Scholar 

  27. Y. Simsek, “On Twisted q-Hurwitz Zeta Function and q-Two-Variable L-function,” Appl. Math. Comput. 187(1), 466–473 (2007).

    Article  MATH  MathSciNet  Google Scholar 

  28. Y. Simsek, “Twisted p-Adic (h, q)-L-Functions“ (submitted).

  29. Y. Simsek, “The Behavior of the Twisted p-Adic (h, q)-L-Functions at s = 0,” J. Korean Math. Soc. 44(4), 915–929 (2007).

    Article  MATH  MathSciNet  Google Scholar 

  30. Y. Simsek, D. Kim, and S.-H. Rim, “On the Two Variable q-L-Series,” Adv. Stud. Contemp. Math. 10(2), 131–142 (2005).

    MATH  MathSciNet  Google Scholar 

  31. H.M. Srivastava, T. Kim, and Y. Simsek, “q-Bernoulli Numbers and Polynomials Associated with Multiple q-Zeta Functions and Basic L-Series,” Russ. J. Math. Phys. 12(2), 241–268 (2005).

    MATH  MathSciNet  Google Scholar 

  32. L. C. Washington, “A Note on p-Adic L-Functions,” J. Number Theory 8, 245–250 (1976).

    Article  MATH  MathSciNet  Google Scholar 

  33. L. C. Washington, Introduction to Cyclotomic Fields, 2nd ed. (Springer, New York, 1997).

    MATH  Google Scholar 

  34. P. T. Young, “On the Behavior of Some Two-Variable p-Adic L-Function,” J. Number Theory 98, 67–86 (2003).

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Akdeniz University, 07058, Antalya, Turkey

    M. Cenkci, Y. Simsek & V. Kurt

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  1. M. Cenkci
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  2. Y. Simsek
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  3. V. Kurt
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Correspondence to Y. Simsek.

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Cenkci, M., Simsek, Y. & Kurt, V. Multiple two-variable p-adic q-L-function and its behavior at s = 0. Russ. J. Math. Phys. 15, 447–459 (2008). https://doi.org/10.1134/S106192080804002X

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