# A PRODUCT FORMULA FOR COMBINATORIC CONVOLUTION SUMS OF ODD DIVISOR FUNCTIONS

• Lee, Kwangchul (Department of Mathematics, Chonbuk National University) ;
• Kim, Daeyeoul (National Institute for Mathematical Sciences) ;
• Seo, Gyeong-Sig (Department of Mathematics, Institute of Pure and Applied Mathematics, Chonbuk National University)
• Accepted : 2016.02.29
• Published : 2016.06.25
• 78 18

#### Abstract

If we let $L(2K;n):=\sum_{s=0}^{k-1}(^{2k}_{2s+1})\sum_{m=1}^{n-1}{\sigma}_{2k-2s-1,1}(m;2){\sigma}_{2s+1,1}(n-m;2)$ with $${\sigma}_{k,l}(n;2):=\sum\limits_{{d{\mid}n}\atop{d{\equiv}l(mod2)}}d^k$$, then we get the formula of L(2u; p)L(2v; p)L(2w; p).

#### Keywords

Divisor functions;Convolution sums;Bernoulli polynomials

#### References

1. A. Alaca, S. Alaca, and K. S. Williams, The convolution sum ${{\Sigma}_{l+24m=n}}^{{\sigma}(l){\sigma}(m)}$ and ${{\Sigma}_{3l+8m=n}}^{{\sigma}(l){\sigma}(m)}$, Math. J. Okayama Univ. 49 (2007), 93-111.
2. A. Alaca, S. Alaca, and K. S. Williams, The convolution sum ${\Sigma}_{m{<}{\frac{n}{16}}^{{\sigma}(m){\sigma}(n-16m)}$, Canad. Math. Bull. 51 (2008), no. 1, 3-14. https://doi.org/10.4153/CMB-2008-001-1
3. B. C. Berndt, Ramanujan's Notebooks. Part II, Springer-Verlag, New York, 1989.
4. Dario Castellanos, A note on bernoulli polynomials, Univ. de Carabobo, Valencia, Venezuela (1989), 98-102.
5. 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.
6. J. W. L. Glaisher, On certain sums of products of quantities depending upon the divisors of a number, Mess. Math. 15 (1885), 1-20.
7. 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.
8. H. Hahn, Convolution sums of some functions on divisors, Rocky Mountain J. Math. 37 (2007), no. 5, 1593-1622. https://doi.org/10.1216/rmjm/1194275937
9. 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 (Urbana, IL, 2000), 229-274, A K Peters, Natick, MA, 2002.
10. D. Kim, A. Bayad, and N. Y. Ikikardes, Certain combinatoric convolution sums and their relations to Bernoulli and Euler Polynomials, J. Korean Math. Soc. 52 (2015), No. 3, pp. 537-565. https://doi.org/10.4134/JKMS.2015.52.3.537
11. K. S. Williams, Number Theory in the Spirit of Liouville, London Mathematical Society, Student Texts 76, Cambridge, 2011.