Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
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A REPRESENTATION OF DEDEKIND SUMS WITH QUASI-PERIODICITY EULER FUNCTIONS

  • KIM, MIN-SOO (Division of Mathematics, Science, and Computers, Kyungnam University)
  • Received : 2017.04.15
  • Accepted : 2017.05.15
  • Published : 2017.09.30
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Abstract

In this paper, we shall provide several properties of Dedekind sums with quasi-periodicity Euler functions. In particular, we present a representation of these Dedekind sums in terms of the Eulerian functions and the tangent functions.

Keywords

References

  1. T.M. Apostol, Modular functions and Dirichlet series in number theory, Graduate Texts in Mathematics 41, Springer-Verlag, New York, 1990.
  2. T.M. Apostol, Generalized Dedekind sums and transformation formulae of certain Lambert series, Duke Math. J. 17 (1950), 147-157. https://doi.org/10.1215/S0012-7094-50-01716-9
  3. T.M. Apostol, Theorems on generalized Dedekind sums, Pacific J. Math. 2 (1952), 1-9. https://doi.org/10.2140/pjm.1952.2.1
  4. A. Bayad, Arithmetical properties of elliptic Bernoulli and Euler numbers, Int. J. Algebra 4 (2010), no. 5-8, 353-372.
  5. L. Carlitz, Generalized Dedekind sums, Math. Z. 85 (1964), 83-90. https://doi.org/10.1007/BF01114880
  6. L. Carlitz, The reciprocity theorem for Dedekind sums, Pacific J. Math. 3 (1953), 523-527. https://doi.org/10.2140/pjm.1953.3.523
  7. L. Carlitz, Some theorems on generalized Dedekind sums, Pacific J. Math. 3 (1953), 513-522. https://doi.org/10.2140/pjm.1953.3.513
  8. L. Carlitz, Multiplication formulas for products of Bernoulli and Euler polynomials, Pacific J. Math. 9 (1959), 661-666. https://doi.org/10.2140/pjm.1959.9.661
  9. M. Cenkci, M. Can and V. Kurt, Degenerate and character Dedekind sums, J. Number Theory 124 (2007), no. 2, 346-363. https://doi.org/10.1016/j.jnt.2006.09.006
  10. R. Dedekind, Erlluterungen zu zwei Fragmenten von Riemamr, in "B. Riemann's Gesammelte Mathematische Werke" (H. Weber, Ed.), 2nd ed., pp. 466-472, Berlin, 1892. [Reprinted by Dover, New York, 1953.]
  11. S. Hu, D. Kim and M.-S. Kim, On reciprocity formula of Apostol-Dedekind sum with quasi-periodic Euler functions, J. Number Theory 162 (2016), 54-67. https://doi.org/10.1016/j.jnt.2015.10.022
  12. S. Hu and M.-S. Kim, The p-adic analytic Dedekind sums, J. Number Theory 171 (2017), 112-127. https://doi.org/10.1016/j.jnt.2016.07.022
  13. N.S. Jung and C.S. Ryoo, A research on a new approach to Euler polynomials and Bernstein polynomials with variable $[x]_q$, J. Appl. Math. Inform. 35 (2017), no. 1-2, 205-215. https://doi.org/10.14317/jami.2017.205
  14. T. Kim, Note on Dedekind type DC sums, Adv. Stud. Contemp. Math. 18 (2009), 249-260.
  15. M.-S. Kim, On the special values of Tornheim's multiple series, J. Appl. Math. Inform. 33 (2015), no. 3-4, 305-315. https://doi.org/10.14317/jami.2015.305
  16. M. Mikolas, On certain sums generating the Dedekind sums and their reciprocity laws, Pacific J. Math. 7 (1957), 1167-1178. https://doi.org/10.2140/pjm.1957.7.1167
  17. H. Rademacher, Zur Theorie der Modulfunktionen, Reine Angew. Math. 167 (1931), 312-366.
  18. H. Rademacher, Some remarks on certain generalized Dedekind sums, Acta Arith. 9 (1964), 97-105. https://doi.org/10.4064/aa-9-1-97-105
  19. H. Rademacher and E. Grosswald, Dedekind sums, The Carus Mathematical Monographs, No. 16, The Mathematical Association of America, Washington, D.C., 1972.
  20. Y. Simsek, Special functions related to Dedekind-type DC-sums and their applications, Russ. J. Math. Phys. 17 (2010), no. 4, 495-508. https://doi.org/10.1134/S1061920810040114
  21. C. Snyder, p-adic interpolation of Dedekind sums, Bull. Austral. Math. Soc. 37 (1988), no. 2, 293-301. https://doi.org/10.1017/S0004972700026848
  22. L. Takacs, On generalized Dedekind sums, J. Number Theory 11 (1979), no. 2, 264-272. https://doi.org/10.1016/0022-314X(79)90044-1

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