Acta Mathematica Hungarica

, Volume 120, Issue 3, pp 281–299 | Cite as

Generating functions of the (h, q) extension of twisted Euler polynomials and numbers

Article

Abstract

By using p-adic q-deformed fermionic integral on ℤp, we construct new generating functions of the twisted (h, q)-Euler numbers and polynomials attached to a Dirichlet character χ. By applying Mellin transformation and derivative operator to these functions, we define twisted (h, q)-extension of zeta functions and l-functions, which interpolate the twisted (h, q)-extension of Euler numbers at negative integers. Moreover, we construct the partially twisted (h, q)-zeta function. We give some relations between the partially twisted (h, q)-zeta function and twisted (h, q)-extension of Euler numbers.

Key words and phrases

p-adic Volkenborn integral twisted q-Euler numbers and polynomials zeta and l-functions 

2000 Mathematics Subject Classification

28B99 11B68 11S40 11S80 33D05 44A05 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [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. Contep. Math., 9 (2004), 203–216.MATHMathSciNetGoogle Scholar
  2. [2]
    T. Kim, An analogue of Bernoulli numbers and their applications, Rep. Fac. Sci. Engrg. Saga Univ. Math., 22 (1994), 21–26.MATHMathSciNetGoogle Scholar
  3. [3]
    T. Kim, q-Volkenborn integration, Russian J. Math Phys., 19 (2002), 288–299.Google Scholar
  4. [4]
    T. Kim, Non-archimedean q-integrals associated with multiple Changhee q-Bernoulli Polynomials, Russian J. Math Phys., 10 (2003), 91–98.MATHGoogle Scholar
  5. [5]
    T. Kim, q-Riemann zeta function, Internat J. Math. Sci., (2003), 185–192.Google Scholar
  6. [6]
    T. Kim, p-adic q-integrals associated with the Changhee-Barnes’ q-Bernoulli Polynomials, Integral Transform. Spec. Funct., 15 (2004), 415–420.MATHCrossRefGoogle Scholar
  7. [7]
    T. Kim, Analytic continuation of multiple q-zeta functions and their values at negative integers, Russian J. Math Phys., 11 (2004), 71–76.MATHGoogle Scholar
  8. [8]
    T. Kim, A new approach to q-zeta function, Adv. Stud. Contep. Math., 11 (2005), 157–162.MATHGoogle Scholar
  9. [9]
    T. Kim, On the q-extension of Euler and Genocchi numbers, J. Math. Anal. Appl., 326 (2007), 1458–1465.MATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    T. Kim, q-Euler numbers and polynomials associated with p-adic q-integrals, J. Nonlinear Math. Phys., 14 (2007), 15–27.CrossRefMathSciNetGoogle Scholar
  11. [11]
    T. Kim, On the analogs of Euler numbers and polynomials associated with p-adic q-integral on Z p at q = −1, J. Math. Anal. Appl., 331 (2007), 779–792.MATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    T. Kim, On p-adic q-l-functions and sums of powers, J. Math. Anal. Appl., 329 (2007), 1472–1481.CrossRefMathSciNetGoogle Scholar
  13. [13]
    T. Kim, An invariant p-adic q-integral on ℤp, Appl. Math. Letters, In Press, Corrected Proof, Available online 20 February 2007.Google Scholar
  14. [14]
    T. Kim, The modified q-Euler numbers and polynomials, ArXive:math.NT/0702523.Google Scholar
  15. [15]
    T. Kim and S-H. Rim, Generalized Carlitz’s q-Bernoulli numbers in the p-adic number field, Adv. Stud. Contep. Math., 2 (2000), 9–19.MATHMathSciNetGoogle Scholar
  16. [16]
    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), 13–21.MATHMathSciNetGoogle Scholar
  17. [17]
    T. Kim, M-S. Kim, L-C. Jang and S-H. Rim, New q-Euler numbers and polynomials associated with p-adic q-integrals, Adv. Stud. Contep. Math., 15 (2007), 140–153, arXiv:MathNT/0709.0089.MathSciNetGoogle Scholar
  18. [18]
    T. Kim, S-H. Rim and Y. Simsek, A note on the alternating sums of powers of consecutive q-integers, Adv. Stud. Contemp. Math., 13 (2006), 159–164.MATHMathSciNetGoogle Scholar
  19. [19]
    T. Kim and S-H. Rim, On the Twisted q-Euler numbers and polynomials associated with basic q-l-functions, J. Math. Anal. (to appear).Google Scholar
  20. [20]
    N. Koblitz, A new proof of certain formulas for p-adic L-functions, Duke Math. J., 46 (1979), 455–468.MATHCrossRefMathSciNetGoogle Scholar
  21. [21]
    N. Koblitz, On Carlitz’s q-Bernoulli numbers, J. Number Theory, 14 (1982), 332–339.MATHCrossRefMathSciNetGoogle Scholar
  22. [22]
    N. Koblitz, p-adic Analysis: A short course on recent work, London Math. Soc. Lecture Note Ser., 46 (1980).Google Scholar
  23. [23]
    H. Ozden, Y. Simsek, S-H. Rim and I. N. Cangul, A note on p-adic q-Euler measure, Advan. Stud. Contemp. Math., 14 (2007), 233–239.MATHMathSciNetGoogle Scholar
  24. [24]
    H. Ozden and Y. Simsek, A new extension of q-Euler numbers and polynomials related to their interpolation functions, to appear in Applied Mathematics Letters.Google Scholar
  25. [25]
    S-H. Rim and T. Kim, New Changhee q-Euler numbers and polynomials associated with p-adic q-integral, Computers & Math. Appl., 54 (2007), 484–489, arXiv:Math:NT/0611791.MATHCrossRefMathSciNetGoogle Scholar
  26. [26]
    Y. Simsek, Theorems on twisted L-functions and twisted Bernoulli numbers, Adv. Stud. Contep. Math., 11 (2005), 205–218.MATHMathSciNetGoogle Scholar
  27. [27]
    Y. Simsek, q-analogue of the twisted l-series and q-twisted Euler numbers, J. Number Theory, 110 (2005), 267–278.MATHCrossRefMathSciNetGoogle Scholar
  28. [28]
    Y. Simsek, Twisted (h, q)-Bernoulli numbers and polynomials related to twisted (h, q)-zeta function and L-function, J. Math. Anal. Appl., 324 (2006), 790–804.MATHCrossRefMathSciNetGoogle Scholar
  29. [29]
    Y. Simsek, On twisted q-Hurwitz zeta function and q-two-variable L-function, Appl. Math. Comput., 187 (2007), 466–473.MATHCrossRefMathSciNetGoogle Scholar
  30. [30]
    Y. Simsek, V. Kurt and D. Kim, New approach to the complete sum of products of the twisted (h, q)-Bernoulli numbers and polynomials, J. Nonlinear Math. Phys., 14 (2007), 44–56.CrossRefMathSciNetGoogle Scholar
  31. [31]
    H. M. Srivastava, T. Kim and Y. Simsek, q-Bernoulli numbers and polynomials associated with multiple q-zeta functions and basic L-series, Russian J. Math. Phys., 12 (2005), 241–268.MATHMathSciNetGoogle Scholar
  32. [32]
    H. M. Srivastava and A. Pinter, Remarks on some relationships between the Bernoulli and Euler polynomials, Appl. Math. Lett., 17 (2004), 375–380.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Faculty of Arts and Science, Department of MathematicsUniversity of UludagBursaTurkey
  2. 2.Faculty of Arts and Science, Department of MathematicsUniversity of AkdenizAntalyaTurkey

Personalised recommendations