TRIPLE AND FIFTH PRODUCT OF DIVISOR FUNCTIONS AND TREE MODEL

Abstract

It is known that certain convolution sums can be expressed as a combination of divisor functions and Bernoulli formula. In this article, we consider relationship between fifth-order combinatoric convolution sums of divisor functions and Bernoulli polynomials. As applications of these identities, we give a concrete interpretation in terms of the procedural modeling method.

1. Introduction

Throughout this paper, the symbols ℕ and ℤ denote the set of natural numbers and the ring of integers respectively. The classical Bernoulli polynomials Bn(x) is usually defined by means of the following generating function:

The corresponding Bernoulli number Bn is given by

The Bernoulli polynomial is expressed through the respective numbers and polynomials

For n ∈ ℕ, k ∈ ℤ and l ∈ {0, 1} we define some divisor functions

The identity

for the basic convolution sum first appeared in a letter from Besge to Liouville in 1862 (See [2]). Recently, the study of convolution formulas for divisor functions can be found in B.C. Berndt [1], J.W.L. Glaisher [3], H. Hahn [4], J.G. Huard et al. [5], D. Kim et al. [8], G. Melfi [12] and K.S. Williams [13]. We are motivated by Ramanujan’s recursion formula for sums of the product of two Eisenstein series (See [1]) and its proof, and also the following identities (See [13]):

In this paper we focus on the combinatorial convolution sums. For positive integers l and N, the combinatorial convolution sum

can be evaluated explicitly in terms of divisor functions. The aim of this article is to study fourth and fifth-order combinatorial convolution sums of the analogous types of (1.5) and (1.6). More precisely, we prove the following results.

Theorem 1.1. For k, q, n ∈ ℕ and k, n ≥ 2, we have

(a)

(b)

Theorem 1.2. Let n ≥ 4 be an even integer with l, q ∈ ℕ - {1}. Then

Finally, we introduce a divisor tree model using Theorem 1.1 in Section 3.

 

2. Proof of Theorems

In this section, we will discuss some relationships between the Bernoulli polynomials and the combinatoric convolution sums of divisor functions.

Propositin 2.1 ([6,7,9]). Let n ≥ 2, N ≥ 4, k, l ∈ ℕ and N be an even integer. Then we have

and

Remark 2.2. The different expression of (2.1) is in [8, Theorem 3].

To prove of Theorem 1.1, we need the following lemma.

Lemma 2.3. Let k be a positive integer. Then

Proof. By (1.2) we directly get this lemma. □

Proof of Theorem1.1 For k, n, q ∈ ℕ and k, n ≥ 2.

(a) If k be an odd positive integer and let k = 2l + 1, then by Proposition 2.1, we have

where b = 2j + 1, c = 2s + 1.

From the binomial theorem it follows that

By the property of Bernoulli polynomial

we get

Similarly, we obtain

This completes the proof of (a).

(b) If k be an odd positive integer and let k = 2l + 1, then by Proposition 2.1

where b = 2j + 1, c = 2s + 1.

From the binomial theorem, Lemma 2.3 and (2.4) we obtain

and

Example 2.4. We can find some values of T(k, n, q) and Y(k, n, q) in Theorem 1.1.

Table 1.Values of T(k, n, q).

Table 2.Values of Y(k, n, q).

Proof of Theorem 1.2 Let n ≥ 4 be an even integer and l, q ∈ ℕ - {1}.

By Proposition 2.1, we have

By the same method in Theorem 1.1, we derive the following 4 terms below;

Thus we have

Example 2.5. We can find some values of F(l, n, q) in Theorem 1.2.

Table 3.Values of F(l, n, q).

 

3. The tree-modeling method

Th procedural modeling method using convolution sums of divisor functions (MCD) was suggested for a variety of natural trees in a virtual ecosystem [10], [11]. Th basic structure of MCD is that it defines the growth grammar including the branch propagation, a growth pattern of branches and leaves, and a process of growth deformation for various generations of tree. Theorems 1.1 gives us a basic background for efficient and diverse generations and expressions of trees composing virtual ecosystem or real-time animation processing.

In order to apply MCD to the growth structure of a tree model, (1.3) is modified and expressed in

where D represents various divisor functions, iis the current growth step, and n - 1 is the final iteration number of the ith growth step. Here, Di(Bi(x, y)a) is a divisor function that determines the pattern of the number of branches, Di(Ti(x, y)a) is a divisor function that determines the number of twigs, and Di(Li(x, y)a) is a divisor function that determines the number of leaves with l different types of trees and grasses in the virtual system. We put Di = σ∗. Using Theorem 1.1, we obtain approximate total numbers for MCD. We suggest an example for

The basic models of tree consist of main column, bough, twig, and leaf. means the bough grown in the main column, means the twig grown in the bough, means leaf grown in the twig. If N = 2, q = 2, then

One bough are grown in the main column, and one twig are grown in the bough, and two leaves are grown in the twig.

Fig. 1.N = 2, q = 2:

If N = 3, q = 2, then

Consider the first sum of right hand side of (3.1) This number represents the following : New twigs are grown in the first bough, and two leaves are grown in the new twigs(See Figure 2).

Fig. 2.N = 3, q = 2:

Similarly, we consider the second sum of right hand side of (3.1) This number represents the following : Two new boughs are grown in the main column, and one twig are grown each in the bough, and two leaves are grown each in the twig (See Figure 2).

Similarly, we obtain Figure 3.

Fig. 3.

Using this divisor model, we can find the total number of leaves (see Table 4).

Table 4.Total leaves of tree