Bernoulli Identities and Combinatoric Convolution Sums with Odd Divisor Functions

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 17 and 18.
All calculations in Lemmas 6 and 7 and Theorems 9 and 10 are obtained by usins  .

2. Modular Forms

In this section, we observe the convolution sums as a view point of generating functions of divisor functions.

The normalized Eisenstein series is defined by

For the generating function of we denote

Let be a finite index subgroup of  . The modular form of weight on is a holomorphic function on such that for a positive integer . The vector space over of holomorphic modular forms of weight on is finite dimensional and is denoted by .

Note that (if ) and (if ) for . Moreover, the product of two modular forms of weights is also modular form of weight .

The is the discriminant function with Ramanujan -function as its coefficient. It is modular of weight on .

Define the following two weight modular forms and by using the Dedekind -function defined in (5):

We get the lemma.

Lemma 6. Consider the following:(1) (2) (3)

Proof. The functions , , and are modular functions of weight on . Note that is a -dimensional vector space over generated by , and because . Their -expansions are as follows: By comparing the above expansions with ones of for , we get our result.

Lemma 7. Consider the following:(1) (2) (3)

Proof. One can prove these by using a similar way to Lemma 6 and -dimensional vector space generated by , and over because their -expansions are as follows:

Remark 8. (1) is the normalized Hecke eigenform on the full modular group of weight :
(2) and are normalized newforms on of weight 14. The coefficients () satisfy that Moreover, for the Atkin-Lehner involution , if we define the action on the complex valued function as
By the help of Lemmas 6 and 7 we get the formulae for each convolution sum.

Theorem 9. Consider the following:(1) (2) (3)

Theorem 10. Consider the following:(1) (2) (3)

3. Proof of Theorems

In his series of eighteen papers published between 1858 and 1865, Joseph Liouville (1809–1882) stated without proof several elementary arithmetic formulae. One of these is the following formula.

Proposition 11 (see [10, page 112]). Let be an even function. Let . Then one obtains

Now we are ready to prove our theorems in Section 1.

Proof of Theorem 3. We apply () in Proposition 11. Then the left-hand side is On the other hand by using Bernoulli’s identity [10, page 42], the right-hand side is After dividing both sides by we get

Remark 12. When is a constant function in Proposition 11, it is a trivial formula such that

Corollary 13. For  ,

Proof. Applying in Theorem 3, our result is proved.

Corollary 14. Let  . Then one has

Proof. Let   in Theorem 3. Then the left-hand side is  The right-hand side of Theorem 3 for is
Since , we are done.

Remark 15. The above result is also in [8, Theorem 3.4], but we do not use the combinatoric convolution sums of -functions but odd divisor function.

Corollary 16. For the odd prime case, one has

Proof. In Theorem 3, by Corollary 13.

Proof of Theorem 4. Note that We reconsider Proposition 1 as the last one in the previous line:
Thus, by Proposition 1, Theorem 3, and (53).

The following examples show us that the coefficients of cusp forms disappear in the combinatorial convolution sum for the weights and , explicitly.

Example 17. Consider the case of Theorem 3 for . The left-hand side is
When we put the formula in Theorem 9 in the above, the coefficients of and are zero.

Example 18. The left-hand side of Theorem 3 when is By applying Theorem 10 to the above, one can check that our theorem is true and the coefficients and of cusp forms disappear.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.