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# Bernoulli Identities and Combinatoric Convolution Sums with Odd Divisor Functions

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Abstract:

We study the combinatoric convolution sums involving odd divisor functions, their relations to Bernoulli numbers, and some interesting applications. 1. Introduction The Bernoulli polynomials , which are usually defined by the exponential generating function play an important role in different areas of mathematics, including number theory and the theory of finite differences. It is well known that are rational numbers. It can be shown that for and is alternatively positive and negative for even . The are called Bernoulli numbers. Let denote the set of positive integers. Further, let , where . Throughout this paper, we define divisor functions as follows: We also make use of the following convention: Ramanujan [1] proved that using elementary arguments. Let be the complex upper half plane and let be for . Denote by the Dedekind function and by the th coefficient of . Alaca and Williams [2] proved that It turns out that we need not only divisor functions but also the coefficients of certain modular functions. For other divisor functions, Hahn [3] showed that and Glaisher [4–6] extended Besgue’s formula by replacing in the convolution sum in (4) by other sums ; for example, Recently, the combinatorial convolution sum is studied [7–10]. In [10] Williams proved the following. Proposition 1. Let and . Then Cho et al. found out the linear sum for combinatorial convolution sum of in [7]. Proposition 2. For and , one has where Denote by . The generating function ？？of is an even function and is zero for all odd positive integer . The aim of this paper is to study two combinatorial convolution sums of the analogous type of Proposition 2. When we write the convolution sums as linear sum of divisor function, in the result by Williams the coefficients are and ours are . More precisely, we prove the following theorems. Theorem 3. For and , Equation (7) is a special case when for the following theorem because and . Theorem 4. For and , Remark 5. The product of two modular forms is another modular form of bigger weight. The dimension of space of modular forms on is approximately linear for and the space generated by generating functions of divisor functions is clearly 2 as grows. More precisely speaking, for the Eisenstein series and which will be defined in Section 2 where is the space of cusp form of weight on and it is orthogonal complement of？？ in？？ . Since？？ = and = , for suitable constants？？ . On the other hand, Theorems 3 and 4 show that the combinatorial convolution sums are written as only divisor functions; that is, The disappearance of is observed in Examples

References

 [1] S. Ramanujan, “On certain arithmetical functions,” Transactions of the Cambridge Philosophical Society, vol. 22, pp. 159–184, 1916. [2] S. Alaca and K. S. Williams, “Evaluation of the convolution sums and ,” Journal of Number Theory, vol. 124, no. 2, pp. 491–510, 2007. [3] H. Hahn, “Convolution sums of some functions on divisors,” The Rocky Mountain Journal of Mathematics, vol. 37, no. 5, pp. 1593–1622, 2007. [4] 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,” Messenger of Mathematics, vol. 15, pp. 33–36, 1885. [5] J. W. L. Glaisher, “On certain sums of products of quantities depending upon the divisors of a number,” Messenger of Mathematics, vol. 15, pp. 1–20, 1885. [6] J. W. L. Glaisher, “On the square of the series in which the coefficients are the sums of the divisors of the exponents,” Messenger of Mathematics, vol. 14, pp. 156–163, 1884. [7] B. Cho, D. Kim, and H. Park, “Evaluation of a certain combinatorial convolution sum in higher level cases,” Journal of Mathematical Analysis and Applications, vol. 406, no. 1, pp. 203–210, 2013. [8] D. Kim and A. Bayad, “Convolution identities for twisted Eisenstein series and twisted divisor functions,” Fixed Point Theory and Applications, vol. 2013, article 81, 2013. [9] D. Kim, A. Kim, and A. Sankaranarayanan, “Bernoulli numbers, convolution sums and congruences of coefficients for certain generating functions,” Journal of Inequalities and Applications, vol. 2013, article 225, 2013. [10] K. S. Williams, Number Theory in the Spirit of Liouville, vol. 76 of London Mathematical Society Student Texts, Cambridge University Press, Cambridge, UK, 2011.

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