Some identities on the q-Euler polynomials of higher order and q-stirling numbers by the fermionic p-adic integral on ℤ p



A systemic study of some families of q-Euler numbers and families of polynomials of Nörlund type is presented by using the multivariate fermionic p-adic integral on ℤ p . The study of these higher-order q-Euler numbers and polynomials yields an interesting q-analog of identities for Stirling numbers.


Euler Number Stirling Number Euler Polynomial Generalize Euler Number 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    I. N. Cangul, H. Ozden, and Y. Simsek, “A New Approach to q-Genocchi Numbers and Their Interpolation Functions,” Nonlinear Anal. (in press; doi:10.1016/ Scholar
  2. 2.
    L. Comtet, Advanced Combinatorics (D. Reidel, Dordrecht, 1974).zbMATHGoogle Scholar
  3. 3.
    E. Y. Deeba and D. M. Rodriguez, “Stirling’s Series and Bernoulli Numbers,” Amer. Math. Monthly 98, 423–426 (1991).zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    M. Cenkci, M. Can, and V. Kurt, “p-Adic Interpolation Functions and Kummer-Type Congruences for q-Twisted Euler Numbers,” Adv. Stud. Contemp. Math. 9, 203–216 (2004).zbMATHMathSciNetGoogle Scholar
  5. 5.
    T. Kim, S.-D. Kim, and D.-W. Park, “On Uniform Differentiability and q-Mahler Expansions,” Adv. Stud. Contemp. Math. 4, 35–41 (2001).zbMATHGoogle Scholar
  6. 6.
    T. Kim, “The Modified q-Euler Numbers and Polynomials,” Adv. Stud. Contemp. Math. 16, 161–170 (2008).zbMATHGoogle Scholar
  7. 7.
    T. Kim, “Euler Numbers and Polynomials Associated with Zeta Functions,” Abstr. Appl. Anal. (2008), 11 pages (Article ID 581582).Google Scholar
  8. 8.
    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, 113–125 (1) (1985).zbMATHMathSciNetGoogle Scholar
  9. 9.
    T. Kim, “Note on the Euler q-Zeta Functions,” J. Number Theory (in press, doi:10.1016/j.jnt, 2009).Google Scholar
  10. 10.
    T. Kim, “q-Volkenborn Integration,” Russ. J. Math. Phys. 9(2), 288–299 (2002).zbMATHMathSciNetGoogle Scholar
  11. 11.
    T. Kim, “A Note on p-Adic q-Integral on Zp Associated with q-Euler Numbers,” Adv. Stud. Contemp. Math. 15(2), 133–138 (2007).zbMATHMathSciNetGoogle Scholar
  12. 12.
    T. Kim, “On p-Adic Interpolating Function for q-Euler Numbers and Its Derivatives,” J. Math. Anal. Appl. 339(1), 598–608 (2008).zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    T. Kim, “q-Extension of the Euler Formula and Trigonometric Functions,” Russ. J. Math. Phys. 14(3), 275–278 (2007).zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    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).zbMATHMathSciNetGoogle Scholar
  15. 15.
    T. Kim, “Non-Archimedean q-Integrals Associated with Multiple Changhee q-Bernoulli Polynomials,” Russ. J. Math. Phys. 10(1), 91–98 (2003).zbMATHMathSciNetADSGoogle Scholar
  16. 16.
    T. Kim, “q-Euler Numbers and Polynomials Associated with p-Adic q-Integrals,” J. Nonlinear Math. Phys. 14(1), 15–27 (2007).zbMATHCrossRefMathSciNetADSGoogle Scholar
  17. 17.
    H. Ozden, Y. Simsek, S.-H. Rim, and I. N. Cangul, “A Note on p-Adic q-Euler Measure,” Adv. Stud. Contemp. Math. 14(2), 233–239 (2007).MathSciNetGoogle Scholar
  18. 18.
    M. Schork, “Ward’s “Calculus of Sequences,” q-Calculus and the Limit q → −1,” Adv. Stud. Contemp. Math. 13(2), 131–141 (2006).zbMATHMathSciNetGoogle Scholar
  19. 19.
    M. Schork, “Combinatorial Aspects of Normal Ordering and Its Connection to q-Calculus,” Adv. Stud. Contemp. Math. 15(1), 49–57 (2007).zbMATHMathSciNetGoogle Scholar
  20. 20.
    Y. Simsek, “On p-Adic Twisted q-L-Functions Related to Generalized Twisted Bernoulli Numbers,” Russ. J. Math. Phys. 13(3), 340–348 (2006).zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Y. Simsek, “Theorems on Twisted L-Function and Twisted Bernoulli Numbers,” Adv. Stud. Contemp. Math. 11(2), 205–218 (2005).zbMATHMathSciNetGoogle Scholar
  22. 22.
    Y. Simsek, “q-Dedekind Type Sums Related to q-Zeta Function and Basic L-Series,” J. Math. Anal. Appl. 318(1), 333–351 (2006).zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    H. J. H. Tuenter, “A Symmetry of Power Sum Polynomials and Bernoulli,” Amer. Math. Monthly 108(3), 258–261 (2001).zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2009

Authors and Affiliations

  1. 1.Division of General Education-MathematicsKwangwoon UniversitySeoulS. Korea

Personalised recommendations