Certain combinatoric Bernoulli polynomials and convolution sums of divisor functions
© Kim and Yildiz Ikikardes; licensee Springer. 2013
Received: 26 July 2013
Accepted: 23 September 2013
Published: 8 November 2013
It is known that certain convolution sums can be expressed as a combination of divisor functions and Bernoulli formula. One of the main goals in this paper is to establish combinatoric convolution sums for the divisor sums . Finally, we find a formula of certain combinatoric convolution sums and Bernoulli polynomials.
KeywordsBernoulli numbers convolution sums
can be evaluated explicitly in terms of divisor functions and a combinatorial convolution sum. We prove the following.
Equations (1.3) and (1.4) are in (1.2) and [, Corollary 2.4]. Using these combinatoric convolution sums, we obtain the following.
where and .
Thus, we can pose a general question regarding Bernoulli polynomials.
The problem of convolution sums of the divisor function and the theory of Eisenstein series has recently attracted considerable interest with the emergence of quasimodular tools. In connection with the classical Jacobi theta and Euler functions, other aspects of the function are explored by Simsek in . Finally, we prove the following.
2 Properties of convolution sums derived from divisor functions
Proposition 5 ()
This proves the theorem. □
(ii) and (iii) are applied in a similar way. □
3 Bernoulli polynomials and convolution sums
Proposition 9 ()
It is well known that . Using Proposition 9, we get this lemma.
- (ii)If n is an odd integer, then(3.1)
- (iii)In (3.1), put , we get
Others cases follow in a similar way. This completes the proof. □
and (iii) are applied in a similar way. □
by (3.1) and (3.2).
- Hahn H: Convolution sums of some functions on divisors. Rocky Mt. J. Math. 2007, 37(5):1593-1622. 10.1216/rmjm/1194275937View ArticleGoogle Scholar
- Besge M: Extrait d’une lettre de M. Besge à M. Liouville. J. Math. Pures Appl. 1862, 7: 375-376.Google Scholar
- Cheng N, Williams KS: Evaluation of some convolution sums involving the sum of divisors functions. Yokohama Math. J. 2005, 52: 39-57.MathSciNetGoogle Scholar
- Lahiri DB:On Ramanujan’s function and the divisor function , I. Bull. Calcutta Math. Soc. 1946, 38: 193-206.MathSciNetGoogle Scholar
- Melfi G: On some modular identities. In Number Theory. de Gruyter, Berlin; 1998:371-382. (Eger, 1996)Google Scholar
- Huard JG, Ou ZM, Spearman BK, Williams KS: Elementary evaluation of certain convolution sums involving divisor functions. II. Number Theory for the Millennium 2002, 229-274.Google Scholar
- Kim D, Kim A, Sankaranarayanan A: Bernoulli numbers, convolution sums and congruences of coefficients for certain generating functions. J. Inequal. Appl. 2013., 2013: Article ID 225Google Scholar
- Williams KS London Mathematical Society Student Texts 76. In Number Theory in the Spirit of Liouville. Cambridge University Press, Cambridge; 2011.Google Scholar
- Kim D, Kim A, Kim M: A remark on algebraic curves derived from convolution sums. J. Inequal. Appl. 2013., 2013: Article ID 58Google Scholar
- Simsek Y: Elliptic analogue of the Hardy sums related to elliptic Bernoulli functions. Gen. Math. 2007, 15(2-3):3-23.MathSciNetGoogle Scholar
- Kim D, Bayad A: Convolution identities for twisted Eisenstein series and twisted divisor functions. Fixed Point Theory Appl. 2013., 2013: Article ID 81Google Scholar
- Cho, B, Kim, D, Park, H: Combinatorial convolution sums derived from divisor functions and Faulhaber sums (2013, submitted)Google Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.