Russian Journal of Mathematical Physics

, Volume 26, Issue 1, pp 40–49 | Cite as

A Note on Polyexponential and Unipoly Functions

  • D. S. KimEmail author
  • T. KimEmail author


In this paper, we introduce polyexponential functions as an inverse to the polylogarithm functions, construct type 2 poly-Bernoulli polynomials by using this and derive various properties of type 2 poly-Bernoulli numbers. Then we introduce unipoly functions attached to each suitable arithmetic function as a universal concept which includes the polylogarithm and polyexponential functions as special cases. By making use of unipoly functions, we define unipoly-Bernoulli polynomials, type 2 unipoly-Bernoulli numbers, and unipoly-Bernoulli numbers of the second kind, and show some basic properties for them.


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  1. 1.
    A. Bayad and J. Chikhi, “Non Linear Recurrences for Apostol–Bernoulli–Euler Numbers of Higher Order,” Adv. Stud. Contemp. Math.(Kyungshang) 22 (1), 1–6 (2012).MathSciNetzbMATHGoogle Scholar
  2. 2.
    L. Carlitz, “Some Polynomials Related to the Bernoulli and Euler Polynomials,” Util. Math. 19, 81–127 (1981).MathSciNetzbMATHGoogle Scholar
  3. 3.
    D. Ding and J. Yang, “Some Identities Related to the Apostol–Euler and Apostol–Bernoulli Polynomials,” Adv. Stud. Contemp. Math. (Kyungshang) 20 (1), 7–21 (2010).MathSciNetzbMATHGoogle Scholar
  4. 4.
    M. S. P. Eastham, “On Polylogarithms,” Proc. Glasgow Math. Assoc. 6, 169–171 (1964).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    S. Gaboury, R. Tremblay, and B.–J. Fugre, “Some Explicit Formulas for Certain New Classes of Bernoulli, Euler and Genocchi Polynomials,” Proc. Jangjeon Math. Soc. 17 (1), 115–123 (2014).MathSciNetzbMATHGoogle Scholar
  6. 6.
    M. Kaneko, “Poly–Bernoulli Numbers,” J. Théor. Nombres Bordeaux 9 (1), 221–228 (1997).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    D. S. Kim and T. Kim, “Some p–Adic Integrals on ℤp Associated with Trigonometric Functions,” Russ. J. Math. Phys. 25 (3), 300–308 (2018).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    D. S. Kim, T. Kim, C. S. Ryoo, and Y. Yao, “On p–Adic Integral Representation of q–Bernoulli Numbers Arising from Two Variable q–Bernstein Polynomials,” Symmetry 10 (10), Art. 451, pp. 11 (2018).Google Scholar
  9. 9.
    T. Kim and D. S. Kim, “Degenerate Laplace Transform and Degenerate Gamma Function,” Russ. J. Math. Phys. 24 (2), 241–248 (2017).MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    T. Kim and D. S. Kim, Identities for Degenerate Bernoulli Polynomials and Korobov Polynomials of the First Kind (Science China Mathematics, 10.1007/s11425–018–9338–5, available at:
  11. 11.
    T. Kim, D. S. Kim, G.–W. Jang, and J. Kwon, “Fourier Series of Sums of Products of Higher–Order Euler Functions,” J. Comput. Anal. Appl. 27 (2), 345–360 (2019).Google Scholar
  12. 12.
    T. Kim, D. S. Kim, G.–W. Jang, and J. Kwon, “Fourier Series of Sums of Product of Poly–Bernoulli and Euler Functions and Their Applications,” J. Comput. Anal. Appl. 26 (6), 1127–1145 (2019).Google Scholar
  13. 13.
    Y. H. Kim and K.–W. Hwang, “Symmetry of Power Sum and Twisted Bernoulli Polynomials,” Adv. Stud. Contemp. Math. (Kyungshang) 18 (2), 127–133 (2009).MathSciNetzbMATHGoogle Scholar
  14. 14.
    T. Komatsu, “Poly–Cauchy Numbers,” Kyushu J. Math. 67, 143–153 (2013).MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    V. Kurt, “On the Alternating Sums and Application to Apostol–Bernoulli Polynomials,” Proc. Jangjeon Math. Soc 16 (2), 251–258 (2013).MathSciNetzbMATHGoogle Scholar
  16. 16.
    H. Ozden, I.N. Cangul, and Y. Simsek, “Remarks on q–Bernoulli Numbers Associated with Daehee Numbers,” Adv. Stud. Contemp. Math. (Kyungshang) 18 (1), 41–48 (2009).MathSciNetzbMATHGoogle Scholar
  17. 17.
    S. Roman, The Umbral Calculus (Pure and Applied Mathematics, 111. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1984).zbMATHGoogle Scholar
  18. 18.
    Y. Simsek, “Identities on the Changhee Numbers and Apostol–Type Daehee Polynomials,” Adv. Stud. Contemp. Math. (Kyungshang) 27 (2), 199–212 (2017).MathSciNetzbMATHGoogle Scholar
  19. 19.
    Z. Zhang and H. Yang, “Some Closed Formulas for Generalized Bernoulli–Euler Numbers and Polynomials,” Proc. Jangjeon Math. Soc. 11 (2), 191–198 (2008).MathSciNetzbMATHGoogle Scholar

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© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Department of MathematicsSogang UniversitySeoulRepublic of Korea
  2. 2.Department of MathematicsKwangwoon UniversitySeoulRepublic of Korea

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