TRIPLE AND FIFTH PRODUCT OF DIVISOR FUNCTIONS AND TREE MODEL†

Journal of Applied Mathematics & Informatics.
2016.
Jan,
34(1_2):
145-156

- Received : June 25, 2015
- Accepted : September 16, 2015
- Published : January 30, 2016

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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.
AMS Mathematics Subject Classfication : 11B68, 11A05
B_{n}
(
x
) is usually defined by means of the following generating function:
The corresponding Bernoulli number
B_{n}
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.
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
= 2
l
+ 1, then by Proposition 2.1, we have
where
b
= 2
j
+ 1,
c
= 2
s
+ 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
= 2
l
+ 1, then by Proposition 2.1
where
b
= 2
j
+ 1,
c
= 2
s
+ 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.
Values of T (k, n, q ).
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.
Values of F (l, n, q ).
where
D
represents various divisor functions,
i
is the current growth step, and
n
- 1 is the final iteration number of the
i
th growth step. Here,
D^{i}
(
B^{i}
(
x, y
)
a
) is a divisor function that determines the pattern of the number of branches,
D^{i}
(
T^{i}
(
x, y
)
a
) is a divisor function that determines the number of twigs, and
D^{i}
(
L^{i}
(
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
D^{i}
=
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.
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
).
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
.
Using this divisor model, we can find the total number of leaves (see
Table 4
).
Total leaves of tree
Daeyeoul Kim
National Institute for Mathematical Sciences Doryong-dong Yuseong-gu Daejeon 305-340, Republic of Korea .
e-mail: daeyeoul@nims.re.kr
Cheoljo Cheong
Chonbuk National university Department of Mathematics 567 Beakje-daero jeonju-si jeollabuk-do 561-756, Republic of Korea.
e-mail: jcj-f@hanmail.net
Hwasin Park
Chonbuk National university Department of Mathematics 567 Beakje-daero jeonju-si jeollabuk-do 561-756, Republic of Korea.
e-mail: Park@jbnu.ac.kr

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
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2. Proof of Theorems

In this section, we will discuss some relationships between the Bernoulli polynomials and the combinatoric convolution sums of divisor functions.
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Values ofT(k, n, q).

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Values ofY(k, n, q).

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Values ofF(l, n, q).

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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
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Total leaves of tree

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BIO

Berndt B. C.
1989
Ramanujan's Notebooks, Part II
Springer-Verlag
New York

Besge M.
(1862)
Extrait d'une lettre de M. Besge à M. Liouville
J. Math. Pures Appl.
7
256 -

Glaisher J.W.L.
(1885)
On certain sums of products of quantities depending upon the divisors of a number
Mess. Math.
15
1 -
20

Hahn H.
(2007)
Convolution sums of some functions on divisors
Rocky Mt. J. Math.
37
1593 -
1622
** DOI : 10.1216/rmjm/1194275937**

Huard J.G.
,
Ou Z.M.
,
Spearman B.K.
,
Williams K.S.
(2002)
Elementary evaluation of certain convolution sums involving divisor functions
Elementary evaluation of certain convolution sums involving divisor functions
II
229 -
274

Kim D.
,
Bayad A.
(2013)
Convolution identities for twisted Eisenstein series and twisted divisor functions
Fixed Point Theory and Appl.
81

Kim D.
,
Bayad A.
,
Ikikardes Nazli Yildiz
(2015)
Certain Combinatoric convolution sums and their relations to Bernoulli and Euler Polynomials
J. Korean Math. Soc.
52
537 -
565
** DOI : 10.4134/JKMS.2015.52.3.537**

Kim D.
,
Park Y.K.
(2014)
Bernoulli identities and combinatoric convolution sums with odd divisor functions
Abstract and Applied Analysis
Article ID 890973
2014

Kim D.
,
Park Y.K.
(2014)
Certain combinatoric convolution sums involving divisor functions product formula
Taiwan J. Math.
18
973 -
988

Kim J.
,
Kim D.
,
Cho H.
(2013)
Procedural Modeling of Trees based on Convolution Sums of Divisor Functions for Real-time Virtual Ecosystems
Comp. Anim. Virtual Worlds
24
237 -
246
** DOI : 10.1002/cav.1506**

Kim J.
,
Kim D.
,
Cho H.
(2013)
Tree Growth Model Design for Realistic Game Landscape Pro- duction(Korean)
Journal of Korea Game Society
13
49 -
58
** DOI : 10.7583/JKGS.2013.13.2.49**

Melfi G.
1998
On some modular identities, Number Theory (Eger., 1996)
De Gruyter
Berlin

Williams K. S.
2011
Number Theory in the Spirit of Liouville
Student Texts
London Mathematical Society
Cambridge
76

Citing 'TRIPLE AND FIFTH PRODUCT OF DIVISOR FUNCTIONS AND TREE MODEL†
'

@article{ E1MCA9_2016_v34n1_2_145}
,title={TRIPLE AND FIFTH PRODUCT OF DIVISOR FUNCTIONS AND TREE MODEL†}
,volume={1_2}
, url={http://dx.doi.org/10.14317/jami.2016.145}, DOI={10.14317/jami.2016.145}
, number= {1_2}
, journal={Journal of Applied Mathematics & Informatics}
, publisher={Korean Society of Computational and Applied Mathematics}
, author={KIM, DAEYEOUL
and
CHEONG, CHEOLJO
and
PARK, HWASIN}
, year={2016}
, month={Jan}