Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
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A STUDY OF SUM OF DIVISOR FUNCTIONS AND STIRLING NUMBER OF THE FIRST KIND DERIVED FROM LIOUVILLE FUNCTIONS

  • KIM, DAEYEOUL (Department of Mathematics and Institute of Pure and Applied Mathematics, Chonbuk National University) ;
  • KIM, SO EUN (Department of Mathematics and Institute of Pure and Applied Mathematics, Chonbuk National University) ;
  • SO, JI SUK (Department of Mathematics and Institute of Pure and Applied Mathematics, Chonbuk National University)
  • Received : 2018.06.13
  • Accepted : 2018.08.15
  • Published : 2018.09.30
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Abstract

Using the theory of combinatoric convolution sums, we establish some arithmetic identities involving Liouville functions and restricted divisor functions. We also prove some relations involving restricted divisor functions and Stirling numbers of the first kind for divisor functions.

Keywords

References

  1. J.M. Conway, R.K. Guy, The Book of Numbers, Springer-Verlag, New York, 1996: 258-259.
  2. L. Comtet, Advanced combinatorics, D Reidel Publishing Company, Boston. 1974.
  3. A. Alaca, S. Alaca and K.S. Williams, The convolution sum ${\Sigma}_{m ${\sigma}(m){\sigma}(n-16m)$, Canad. Math. Bull. 51 (2008), 3-14. https://doi.org/10.4153/CMB-2008-001-1
  4. A. Alaca, S. Alaca and K.S. Williams, The convolution sums ${\Sigma}_{l+24m=n}$ ${\sigma}(l){\sigma}(m)$ and ${\Sigma}_{3l+8m=n}$ ${\sigma}(l){\sigma}(m)$, M. J. Okayama Univ. 49 (2007), 93-111.
  5. B.C. Berndt, Ramanujan's Notebooks, Part II. Springer-Verlag, New York, 1989.
  6. B. Cho, D. Kim and J.-K. Koo, Divisor functions arising from q-series, Publ. Math. Debrecen 76 (2010), 495-508.
  7. B. Cho, D. Kim and J.-K. Koo, Modular forms arising from divisor functions, J. Math. Anal. Appl. 356 (2009), 537-547. https://doi.org/10.1016/j.jmaa.2009.03.003
  8. J.W.L. Glaisher, On the square of the series in which the coefficients are the sums of the divisors of the exponents, Mess. Math. 14 (1884), 156-163.
  9. J.W.L. Glaisher, On certain sums of products of quantities depending upon the divisors of a number, Mess. Math. 15 (1885), 1-20.
  10. J.W.L. Glaisher, Expressions for the five powers of the series in which the coefficients are the sums of the divisors of the exponents, Mess. Math. 15 (1885), 33-36.
  11. H. Hahn, Convolution sums of some functions on divisors, Rocky Mountain J. Math. 37 (2007), 1593-1622. https://doi.org/10.1216/rmjm/1194275937
  12. J.G. Huard, Z.M. Ou, B.K. Spearman, and K.S. Williams, Elementary Evaluation of Certain Convolution Sums Involving Divisor Functions, Number theory for the millennium, II, (2002), 229-274.
  13. G. Melfi, On some modular identities, de Gruyter, Berlin, 1998, 371-382.
  14. S. Ramanujan, On certain arithmetical functions, Trans. Cambridge Philos. Soc. 22 (1916), no9, 159-184.