ISRN Mathematical Analysis

VolumeΒ 2011Β (2011), Article IDΒ 908508, 14 pages

http://dx.doi.org/10.5402/2011/908508

## Rational Divide-and-Conquer Relations

^{1}Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University, Pathumthani 12120, Thailand^{2}Department of Mathematics, Faculty of Science, Kasetsart University, and Centre of Excellence in Mathematics, CHE, Mahidol University, Si Ayutthaya Road, Bangkok 10400, Thailand

Received 28 October 2010; Accepted 12 December 2010

Academic Editor: G. L.Β Karakostas

Copyright Β© 2011 Charinthip Hengkrawit et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A rational divide-and-conquer relation, which is a natural generalization of the classical divide-and-conquer relation, is a recursive equation of the form , where is a positive integer ; a rational function in variables and a given function. Closed-form solutions of certain rational divide-and-conquer relations which can be used to characterize the trigonometric cotangent-tangent and the hyperbolic cotangent-tangent function solutions are derived and their global behaviors are investigated.

#### 1. Introduction

The classical divide-and-conquer relation is a recursive relation of the form ([1β3]) where are positive integers and is a given function. This class of recurrence relations arises frequently in the analysis of recursive computer algorithms. Such algorithms split a problem of size into subproblems each of size , with extra operations being required when this split of a problem of size into smaller problems is made. Although, there are certain cases, see for example the table on page 273 of [3], where the relation (1.1) can be solved explicitly, it is generally impossible to solve (1.1) for all values of . However, when a starting value is given, a solution for can be found by making a change of variablesββ which turns (1.1) into a first order difference equation of the form ([1, page 137]) and this last recursive equation can be easily solved. Another aspect of importance in the study of divide-and-conquer relations deals with the size of which is used in analyzing the complexity of corresponding divide-and-conquer algorithms ([2, Section 5.3]).

Generalizing the above notion, by a *rational divide-and-conquer (RDAC) relation*, we refer to a recursive relation of the form
where a rational function in , and a given function. Here we aim to find explicit closed form solutions of certain nonlinear divide-and-conquer relations which is closely related to identities of the trigonometric and hyperbolic cotangent identities. Our investigation arises from an observation that the trigonometric cotangent function satisfies, among a number of other identities, the following identity:
which leads to an RDAC relation of the form
This relation can be rewritten as
which is a simpler looking RDAC relation of the form
that can be immediately solved. Let us mention in passing that similar substitution techniques have been employed earlier in [4, 5].

Our first objective here is to find, in the next section, a closed form solution of an RDAC relation generalizing (1.7). Experiences from (1.7) with the cotangent function lead us to apply the results from our first objective to use such RDAC relations to characterize the trigonometric and hyperbolic tangent and cotangent functions, and this will be carried out in the following section as applications.

#### 2. Closed Form Solutions

Before stating our main result, it is convenient to introduce a new notation. For , let us write where denote the customary multinomial coefficients. Our main result is:

Theorem 2.1. *Let , and . If the sequence satisfies the RDAC relation
**
then for , one has
*

*Proof. *Taking principal logarithms of (2.2), the relation becomes
where . For , evaluating (2.4) at , we get
Using the notation introduced above, we see at once that
To finish the proof, we need only show that for all
We proceed by induction. For any , assume that (2.7) holds up to . Thus, by (2.4) and the induction hypothesis one has

The cases and 3 are of particular interest and we record them here for future reference.

Corollary 2.2. * (I) Let . If the sequence satisfies the RDAC relation
**
then for , one has
**(II) Let . If the sequence satisfies the RDAC relation
**
then for , one has
*

#### 3. Applications

We now apply the result of Theorem 2.1 and Corollary 2.2 to several RDAC relations including those that can be used to characterize the trigonometric and hyperbolic tangent and cotangent functions.

Proposition 3.1. *(I) Suppose that the sequence satisfies an RDAC relation of the form
**
For and , if the condition is fulfilled, then
**
where
**(II) Assume that the sequence satisfies an RDAC relation of the form
**
For and , if is not an odd multiple of , then
**
where
**(III) If the sequence satisfies
**
then, for , , one has
**
where
**(IV) If the sequence satisfies
**
then, for , , one has
**
where
*

*Proof. *(I) As seen in Section 1, the RDAC relation (3.1) is equivalent to
whose solution is, by virtue of Corollary 2.2, . Thus,
Setting , one has
provided .

(II) Substituting by turns (3.4) into (3.1) and so the result follows at once from part (I).

(III) Substituting by in (3.7) turns it into a rational recursive equation of the form (3.1) and so part (I) yields the desired result.

(IV) Replacing by in (3.10), we get a rational recursive equation of the form (3.4) and part (II) yields the result.

*Remark 3.2. * Although the substitution by employed in part (II) of Proposition 3.1 allows us to obtain a closed form solution of the RDAC relation (3.4), there remains a difficulty should there exist integer such that . To overcome this shortcoming, we may either interpret the infinite value of the two expressions on both sides of the solution as equal or repeat the technique used in the proof of Proposition 3.1 to solve (3.4).

Proposition 3.3. *(I) Suppose that the sequence satisfies an RDAC relation of the form
**
For and , if the condition
**
is fulfilled, then
**
where
**(II) Assume that the sequence satisfies an RDAC relation of the form
**
For and , if is not an odd multiple of , then
**
where
**(III) If the sequence satisfies
**
then, for ,, one has
**
where
**(IV) If the sequence satisfies
**
then, for , , one has
**
where
*

*Proof. *(I) As seen in Section 1, the RDAC relation (3.16) is equivalent to
whose solution is, by virtue of Corollary 2.2, . Thus,
Setting , one has
provided .

(II) Substituting by turns (3.20) into (3.16) and so the result follows at once from part (I).

(III) Substituting by in (3.23) turns it into a rational recursive equation of the form (3.16), and so part (I) yields the desired result.

(IV) Replacing by in (3.26), we get a rational recursive equation of the form (3.20), and part (II) yields the result.

*Remark 3.4. *As in the remark following Proposition 3.1, the substitution by in part (II) causes no harm should there exists integer such that by either interpreting the infinite value of the expressions on both sides of the solution as equal. Alternately, we may repeat the technique used in the proof of Proposition 3.3 to solve (3.20).

As for the case of general , an entirely analogous proof as that in Proposition 3.3, which we omit here, leads to Proposition 3.5.

Proposition 3.5. *Let . **(I) Suppose that the sequence satisfies an RDAC relation of the form
**
For and , if the condition
**
is fulfilled, then
**
where
**(II) Assume that the sequence satisfies an RDAC relation of the form
**
For and , if is not an odd multiple of , then
**
where
**(III) If the sequence satisfies
**
then, for , , one has
**
where
**(IV) If the sequence satisfies
**
then, for , , one has
**
where
*

When all the exponents in (2.2) are equal to 1, RDAC relations, even more general than those in Proposition 3.5 which can be explicitly solved by our device, are given in the next proposition.

Proposition 3.6. *Let ,ββ,ββ. If the sequence satisfies
**
then for and one has
**
provided the values exist, where
*

*Proof. *Rewriting (3.45), we get
or
Theorem 2.1 thus yields for and
that is,
and the result follows.

#### 4. Global Behaviors

It is often desirable to know about global behaviors of the solutions of recursive equations, such as those in [6]. Using the explicit forms found above, this question is easily solved for RDAC relations in Proposition 3.5 with .

Proposition 4.1. *Let the notation be as in Proposition 3.1. **(I) Suppose that the sequence satisfies an RDAC relation of the form
**
For each fixed and , *(a)*if is a rational multiple of , then either diverges in finitely many steps or is periodic;*(b)*if is not a rational multiple of , then exists for all and the sequence is never periodic.**(II) Suppose that the sequence satisfies an RDAC relation of the form
**
For each fixed , *(a)*if is a rational multiple of , then either diverges in finitely many steps or is periodic; *(b)*if is not a rational multiple of , then exists for all and the sequence is never periodic.**(III) Suppose that the sequence satisfies
**
For each fixed , *(a)*if , then the sequence does not exist; *(b)*if , the sequence is strictly decreasing in the interval ; *(c)*if , then is strictly increasing in the interval .**(IV) Suppose that the sequence satisfies
**
For each fixed , *(z)*if , then is the zero sequence; *(b)*if , then the sequence is strictly increasing in ; *(c)*if , the sequence is strictly decreasing in .*

*Proof. *(I) From part (I) of Proposition 3.1, we know that
provided . Consider the case where is a rational multiple of , say,
If is a multiple of 2, then it is easily checked that diverges in finitely many steps. If is not a multiple of 2, let , where . Observe that for all large , when evaluating the values of cotangent, we need only look at
which is equivalent to looking at
Since each takes at most values and the sequence is infinite, there are positive integers such that , which in turn implies that is periodic.

Finally, if is not a rational multiple of , then is not a multiple of showing that the sequence is well defined and never periodic.

The proof of part (II) is similar to that of part (I).

(III) If , then the values become infinite for all and part (a) follows. Since is a strictly decreasing (resp. increasing) function of according as (resp. ), the results in (b) and (c) are immediate.

(IV) If , then . Arguments for the other two cases and are similar to those in part (III).

Note from Proposition 4.1 that global behaviors of solutions in the case depend solely on the single value . The situation when is more complex since their global behaviors depend heavily on the variable as we see in the following illustration. Keeping the notation of Proposition 3.5, suppose that the sequence satisfies an RDAC relation of the form From part (I) of Proposition 3.5, we know that This explicit form shows that, for each fixed , the behavior of considered as a function of depends on all , and we can merely infer that the values are well defined (i.e., finite) if and only if

#### Acknowledgment

This paper is supported by the Commission on Higher Education, the Thailand Research Fund RTA5180005, and the Centre of Excellence in Mathematics, Thailand.

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