Table of Contents
ISRN Mathematical Analysis
VolumeΒ 2011, Article IDΒ 908508, 14 pages
http://dx.doi.org/10.5402/2011/908508
Research Article

Rational Divide-and-Conquer Relations

1Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University, Pathumthani 12120, Thailand
2Department 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 𝑓(𝑏𝑛)=𝑅(𝑓(𝑛),𝑓(𝑛),…,𝑓(π‘βˆ’1)𝑛)+𝑔(𝑛), where 𝑏 is a positive integer β‰₯2; 𝑅 a rational function in π‘βˆ’1 variables and 𝑔 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 ([1–3]) 𝑛𝐹(𝑛)=π‘ŽπΉπ‘ξ‚+𝐺(𝑛),(1.1) where π‘Ž,𝑏(β‰₯2) are positive integers and 𝐺(𝑛) 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 𝑛 into π‘Ž subproblems each of size [𝑛/𝑏], with 𝐺(𝑛) extra operations being required when this split of a problem of size 𝑛 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 𝑛. However, when a starting value F(π‘πœ†) is given, a solution for 𝑛=π‘π‘˜(π‘˜>πœ†) can be found by making a change of variables  𝐹(π‘π‘˜)=πœ™(π‘˜) which turns (1.1) into a first order difference equation of the form ([1, page 137]) πœ™ξ€·π‘(π‘˜)=π‘Žπœ™(π‘˜βˆ’1)+πΊπ‘˜ξ€Έ,(1.2) 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 𝐹(𝑛) 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 𝑓(𝑏𝑛)=𝑅(𝑓(𝑛),𝑓(2𝑛),…,𝑓(π‘βˆ’1)𝑛)+𝑔(𝑛),(1.3) where π‘βˆˆβ„•,𝑏β‰₯2,𝑅(π‘₯1,…,π‘₯π‘βˆ’1) a rational function in π‘₯1,…,π‘₯π‘βˆ’1, and 𝑔(𝑛) 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: cot(3𝐴)=cot2𝐴cotπ΄βˆ’1,cot2𝐴+cot𝐴(1.4) which leads to an RDAC relation of the form π‘₯3𝑛=π‘₯2𝑛π‘₯π‘›βˆ’1π‘₯2𝑛+π‘₯𝑛.(1.5) This relation can be rewritten as π‘₯3π‘›βˆ’π‘–π‘₯3𝑛=ξ‚΅π‘₯+π‘–π‘›βˆ’π‘–π‘₯𝑛π‘₯+𝑖2π‘›βˆ’π‘–π‘₯2π‘›ξ‚Άξ‚€βˆš+𝑖𝑖=ξ‚βˆ’1,(1.6) which is a simpler looking RDAC relation of the form π‘ˆ3𝑛=π‘ˆπ‘›π‘ˆ2π‘›ξ‚΅π‘ˆπ‘›π‘₯∢=π‘›βˆ’π‘–π‘₯𝑛+𝑖(1.7) 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 π‘ˆπ‘π‘›=π‘ˆπ›Ό1π‘›π‘ˆπ›Ό22π‘›β‹―π‘ˆπ›Όπ‘βˆ’1(π‘βˆ’1)𝑛,(1.8) 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 π‘˜βˆˆβ„•, let us write 𝛼1𝑉ℓ+𝛼2𝑉2β„“+β‹―+π›Όπ‘βˆ’1𝑉(π‘βˆ’1)β„“ξ€Έβˆ—π‘˜=𝑖1+𝑖2+β‹―+π‘–π‘βˆ’1=π‘˜ξ‚΅π‘˜π‘–1,𝑖2,…,π‘–π‘βˆ’1𝛼𝑖11𝛼𝑖22β‹―π›Όπ‘–π‘βˆ’1π‘βˆ’1𝑉1𝑖12𝑖2β‹―(π‘βˆ’1)π‘–π‘βˆ’1β„“,(2.1) where ξ€·π‘˜π‘–1,𝑖2,…,π‘–π‘βˆ’1ξ€ΈβˆΆ=π‘˜!/𝑖1!𝑖2!β‹―π‘–π‘βˆ’1! denote the customary multinomial coefficients. Our main result is:

Theorem 2.1. Let π‘βˆˆβ„•,𝑏β‰₯2, and 𝛼1,…,π›Όπ‘βˆ’1βˆˆβ„. If the sequence {π‘ˆπ‘›}𝑛β‰₯0 satisfies the RDAC relation π‘ˆπ‘π‘›=π‘ˆπ›Ό1π‘›π‘ˆπ›Ό22π‘›β‹―π‘ˆπ›Όπ‘βˆ’1(π‘βˆ’1)𝑛(𝑛β‰₯1),(2.2) then for β„“β‰’0(mod𝑏), one has π‘ˆπ‘π‘˜β„“ξ‘π‘–1+𝑖2+β‹―+π‘–π‘βˆ’1=β„“π‘ˆξ‚€π‘˜π‘–1,𝑖2,…,π‘–π‘βˆ’1𝛼𝑖11𝛼𝑖22β‹―π›Όπ‘–π‘βˆ’1π‘βˆ’11𝑖12𝑖2β‹―(π‘βˆ’1)π‘–π‘βˆ’1β„“(π‘˜βˆˆβ„•).(2.3)

Proof. Taking principal logarithms of (2.2), the relation becomes 𝑉𝑏𝑛=𝛼1𝑉𝑛+𝛼2𝑉2𝑛+β‹―+π›Όπ‘βˆ’1𝑉(π‘βˆ’1)𝑛(𝑛β‰₯1),(2.4) where 𝑉𝑖=logπ‘ˆπ‘–. For β„“β‰’0(mod𝑏), evaluating (2.4) at 𝑛=𝑏ℓ, we get 𝑉𝑏2β„“=𝛼1𝑉𝑏ℓ+𝛼2𝑉𝑏2β„“+β‹―+π›Όπ‘βˆ’1𝑉𝑏(π‘βˆ’1)β„“=𝛼1𝛼1𝑉ℓ+𝛼2𝑉2β„“+β‹―+π›Όπ‘βˆ’1𝑉(π‘βˆ’1)β„“ξ€Έ+𝛼2𝛼1𝑉2β„“+𝛼2𝑉22β„“+β‹―+π›Όπ‘βˆ’1𝑉2(π‘βˆ’1)β„“ξ€Έ+β‹―+π›Όπ‘βˆ’1𝛼1𝑉(π‘βˆ’1)β„“+𝛼2𝑉2(π‘βˆ’1)β„“+β‹―+π›Όπ‘βˆ’1𝑉(π‘βˆ’1)2β„“ξ€Έ=𝛼21𝑉ℓ+𝛼22𝑉22β„“+β‹―+𝛼2π‘βˆ’1𝑉(π‘βˆ’1)2β„“+2𝛼1𝛼2𝑉2β„“+β‹―+2𝛼𝑖𝛼𝑗𝑉𝑖𝑗ℓ+β‹―+2π›Όπ‘βˆ’2π›Όπ‘βˆ’1𝑉(π‘βˆ’2)(π‘βˆ’1)β„“.(2.5) Using the notation introduced above, we see at once that 𝑉𝑏ℓ=𝛼1𝑉ℓ+𝛼2𝑉2β„“+β‹―+π›Όπ‘βˆ’1𝑉(π‘βˆ’1)β„“ξ€Έβˆ—1𝑉𝑏2β„“=𝛼1𝑉ℓ+𝛼2𝑉2β„“+β‹―+π›Όπ‘βˆ’1𝑉(π‘βˆ’1)β„“ξ€Έβˆ—2.(2.6) To finish the proof, we need only show that for all π‘˜βˆˆβ„•π‘‰π‘π‘˜β„“=𝛼1𝑉ℓ+𝛼2𝑉2β„“+β‹―+π›Όπ‘βˆ’1𝑉(π‘βˆ’1)β„“ξ€Έβˆ—π‘˜.(2.7) We proceed by induction. For any β„“β‰’0(mod𝑏), assume that (2.7) holds up to π‘˜. Thus, by (2.4) and the induction hypothesis one has π‘‰π‘π‘˜+1β„“=𝛼1π‘‰π‘π‘˜β„“+𝛼2π‘‰π‘π‘˜2β„“+β‹―+π›Όπ‘βˆ’1π‘‰π‘π‘˜(π‘βˆ’1)β„“=𝛼1𝛼1𝑉ℓ+𝛼2𝑉2β„“+β‹―+π›Όπ‘βˆ’1𝑉(π‘βˆ’1)β„“ξ€Έβˆ—π‘˜+𝛼2𝛼1𝑉2β„“+𝛼2𝑉22β„“+β‹―+π›Όπ‘βˆ’1𝑉2(π‘βˆ’1)β„“ξ€Έβˆ—π‘˜+β‹―+π›Όπ‘βˆ’1𝛼1𝑉(π‘βˆ’1)β„“+𝛼2𝑉2(π‘βˆ’1)β„“+β‹―+π›Όπ‘βˆ’1𝑉(π‘βˆ’1)2β„“ξ€Έβˆ—π‘˜=𝛼1𝑖1+𝑖2+β‹―+π‘–π‘βˆ’1=π‘˜ξ‚΅π‘˜π‘–1,𝑖2,…,π‘–π‘βˆ’1𝛼𝑖11𝛼𝑖22β‹―π›Όπ‘–π‘βˆ’1π‘βˆ’1𝑉1𝑖12𝑖2β‹―(π‘βˆ’1)π‘–π‘βˆ’1β„“+𝛼2𝑖1+𝑖2+β‹―+π‘–π‘βˆ’1=π‘˜ξ‚΅π‘˜π‘–1,𝑖2,…,π‘–π‘βˆ’1𝛼𝑖11𝛼𝑖22β‹―π›Όπ‘–π‘βˆ’1π‘βˆ’1𝑉(1𝑖12)(2𝑖22)β‹―((π‘βˆ’1)π‘–π‘βˆ’12)β„“+β‹―+π›Όπ‘βˆ’1𝑖1+𝑖2+β‹―+π‘–π‘βˆ’1=π‘˜ξ‚΅π‘˜π‘–1,𝑖2,…,π‘–π‘βˆ’1𝛼𝑖11𝛼𝑖22β‹―π›Όπ‘–π‘βˆ’1π‘βˆ’1𝑉(1𝑖1(π‘βˆ’1))(2𝑖2(π‘βˆ’1))β‹―(π‘βˆ’1)π‘–π‘βˆ’1+1β„“=𝑖1+𝑖2+β‹―+π‘–π‘βˆ’1=π‘˜+1ξ‚΅π‘˜π‘–1,𝑖2,…,π‘–π‘βˆ’1𝛼𝑖11𝛼𝑖22β‹―π›Όπ‘–π‘βˆ’1π‘βˆ’1𝑉1𝑖12𝑖2β‹―(π‘βˆ’1)π‘–π‘βˆ’1β„“.(2.8)

The cases 𝑏=2 and 3 are of particular interest and we record them here for future reference.

Corollary 2.2. (I) Let π›Όβˆˆβ„. If the sequence {π‘ˆπ‘›}𝑛β‰₯0 satisfies the RDAC relation π‘ˆ2𝑛=π‘ˆπ›Όπ‘›(𝑛β‰₯1),(2.9) then for β„“β‰’0(mod2), one has π‘ˆ2π‘˜β„“=π‘ˆπ›Όπ‘˜β„“(π‘˜βˆˆβ„•).(2.10)
(II) Let 𝛼1,𝛼2βˆˆβ„. If the sequence {π‘ˆπ‘›}𝑛β‰₯0 satisfies the RDAC relation π‘ˆ3𝑛=π‘ˆπ›Ό1π‘›π‘ˆπ›Ό22𝑛(𝑛β‰₯1),(2.11) then for β„“β‰’0(mod3), one has π‘ˆ3π‘˜β„“=π‘ˆξ‚€π‘˜0ξ‚π›Όπ‘˜1𝛼02β„“π‘ˆξ‚€π‘˜1𝛼1π‘˜βˆ’1𝛼22β„“β‹―π‘ˆξ‚€π‘˜π‘˜ξ‚π›Ό01π›Όπ‘˜22π‘˜β„“(π‘˜βˆˆβ„•).(2.12)

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 {π‘₯𝑛}𝑛β‰₯0 satisfies an RDAC relation of the form π‘₯2𝑛=π‘₯2π‘›βˆ’12π‘₯𝑛(𝑛β‰₯1).(3.1) For β„“β‰’0(mod2) and π‘˜βˆˆβ„•, if the condition 2π‘˜πœƒβ„“β‰’0(mod2πœ‹) is fulfilled, then π‘₯2π‘˜β„“ξƒ©=cotβˆ’2π‘˜πœƒβ„“2ξƒͺξ€·2=cotπ‘˜arccotπ‘₯β„“ξ€Έ,(3.2) where πœƒβ„“=βˆ’2arccotπ‘₯β„“.(3.3)
(II) Assume that the sequence {π‘₯𝑛}𝑛β‰₯0 satisfies an RDAC relation of the form π‘₯2𝑛=2π‘₯𝑛1βˆ’π‘₯2𝑛(𝑛β‰₯0).(3.4) For β„“β‰’0(mod2) and π‘˜βˆˆβ„•, if 2π‘˜πœƒβ„“ is not an odd multiple of πœ‹, then π‘₯2π‘˜β„“ξƒ©=tanβˆ’2π‘˜πœƒβ„“2ξƒͺξ€·2=tanπ‘˜arctanπ‘₯β„“ξ€Έ,(3.5) where πœƒβ„“=βˆ’2arctanπ‘₯β„“.(3.6)
(III) If the sequence {π‘₯𝑛}𝑛β‰₯0 satisfies π‘₯2𝑛=π‘₯2𝑛+12π‘₯𝑛(𝑛β‰₯0),(3.7) then, for β„“β‰’0(mod2), π‘˜βˆˆβ„•, one has π‘₯2π‘˜β„“ξƒ©=cothβˆ’2π‘˜πœƒβ„“2ξƒͺξ€·2=cothπ‘˜arccothπ‘₯β„“ξ€Έ,(3.8) where πœƒβ„“=βˆ’2arccothπ‘₯β„“.(3.9)
(IV) If the sequence {π‘₯𝑛}𝑛β‰₯0 satisfies π‘₯2𝑛=2π‘₯𝑛1+π‘₯2𝑛(𝑛β‰₯0),(3.10) then, for β„“β‰’0(mod2), π‘˜βˆˆβ„•, one has π‘₯2π‘˜β„“ξƒ©=tanhβˆ’2π‘˜πœƒβ„“2ξƒͺξ€·2=tanhπ‘˜arctanhπ‘₯β„“ξ€Έ,(3.11) where πœƒβ„“=βˆ’2arctanhπ‘₯β„“.(3.12)

Proof. (I) As seen in Section 1, the RDAC relation (3.1) is equivalent to π‘ˆ2𝑛=π‘ˆ2π‘›ξ‚΅π‘ˆπ‘›=π‘₯π‘›βˆ’π‘–π‘₯𝑛+𝑖,(3.13) whose solution is, by virtue of Corollary 2.2, π‘ˆ2π‘˜β„“=π‘ˆ2π‘˜β„“. Thus, π‘₯2π‘˜β„“βˆ’π‘–π‘₯2π‘˜β„“=ξ‚΅π‘₯+π‘–β„“βˆ’π‘–π‘₯β„“ξ‚Ά+𝑖2π‘˜.(3.14) Setting π‘’π‘–πœƒβ„“=(π‘₯β„“βˆ’π‘–)/(π‘₯β„“+𝑖), one has π‘₯2π‘˜β„“ξ‚΅=𝑖1+𝑒𝑖2π‘˜πœƒβ„“1βˆ’π‘’π‘–2π‘˜πœƒβ„“ξ‚Άξƒ©=cotβˆ’2π‘˜πœƒβ„“2ξƒͺξ€·2=cotπ‘˜arccotπ‘₯β„“ξ€Έ,(3.15) provided 2π‘˜πœƒβ„“β‰’0(mod2πœ‹).
(II) Substituting π‘₯𝑛 by 1/π‘₯𝑛 turns (3.4) into (3.1) and so the result follows at once from part (I).
(III) Substituting π‘₯𝑛 by 𝑖π‘₯𝑛 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 π‘₯𝑛 by 𝑖π‘₯𝑛 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 π‘₯𝑛 by 1/π‘₯𝑛 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 𝑁 such that π‘₯𝑁=0. 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 {π‘₯𝑛}𝑛β‰₯0 satisfies an RDAC relation of the form π‘₯3𝑛=π‘₯2𝑛π‘₯π‘›βˆ’1π‘₯2𝑛+π‘₯𝑛(𝑛β‰₯1).(3.16) For β„“β‰’0(mod3) and π‘˜βˆˆβ„•, if the condition ξ‚΅π‘˜0ξ‚Άπœƒβ„“+ξ‚΅π‘˜1ξ‚Άπœƒ2β„“ξ‚΅π‘˜π‘˜ξ‚Άπœƒ+β‹―+2π‘˜β„“β‰’0(mod2πœ‹)(3.17) is fulfilled, then π‘₯3π‘˜β„“ξƒ©βˆ’ξ€·=cotπ‘˜0ξ€Έπœƒβ„“βˆ’ξ€·π‘˜1ξ€Έπœƒ2β„“ξ€·βˆ’β‹―βˆ’π‘˜π‘˜ξ€Έπœƒ2π‘˜β„“2ξƒͺπ‘˜0ξ‚Ά=cotξ‚΅ξ‚΅arccotπ‘₯β„“+ξ‚΅π‘˜1ξ‚Άarccotπ‘₯2β„“ξ‚΅π‘˜π‘˜ξ‚Ά+β‹―+arccotπ‘₯2π‘˜β„“ξ‚Ά,(3.18) where πœƒπ‘—=βˆ’2arccotπ‘₯π‘—ξ€·ξ€½π‘—βˆˆβ„“,2β„“,…,2π‘˜β„“ξ€Ύξ€Έ.(3.19)
(II) Assume that the sequence {π‘₯𝑛}𝑛β‰₯0 satisfies an RDAC relation of the form π‘₯3𝑛=π‘₯𝑛+π‘₯2𝑛1βˆ’π‘₯𝑛π‘₯2𝑛(𝑛β‰₯0).(3.20) For β„“β‰’0(mod3) and π‘˜βˆˆβ„•, if πœƒβ„“+ξ€·π‘˜1ξ€Έπœƒ2β„“ξ€·+β‹―+π‘˜π‘˜ξ€Έπœƒ2π‘˜β„“ is not an odd multiple of πœ‹, then π‘₯3π‘˜β„“ξƒ©βˆ’ξ€·=tanπ‘˜0ξ€Έπœƒβ„“βˆ’ξ€·π‘˜1ξ€Έπœƒ2β„“ξ€·βˆ’β‹―βˆ’π‘˜π‘˜ξ€Έπœƒ2π‘˜β„“2ξƒͺπ‘˜0ξ‚Ά=tanξ‚΅ξ‚΅arctanπ‘₯β„“+ξ‚΅π‘˜1ξ‚Άarctanπ‘₯2β„“ξ‚΅π‘˜π‘˜ξ‚Ά+β‹―+arctanπ‘₯2π‘˜β„“ξ‚Ά,(3.21) where πœƒπ‘—=βˆ’2arctanπ‘₯π‘—ξ€·ξ€½π‘—βˆˆβ„“,2β„“,…,2π‘˜β„“ξ€Ύξ€Έ.(3.22)
(III) If the sequence {π‘₯𝑛}𝑛β‰₯0 satisfies π‘₯3𝑛=π‘₯𝑛π‘₯2𝑛+1π‘₯𝑛+π‘₯2𝑛(𝑛β‰₯0),(3.23) then, for β„“β‰’0(mod3),π‘˜βˆˆβ„•, one has π‘₯3π‘˜β„“ξƒ©βˆ’ξ€·=cothπ‘˜0ξ€Έπœƒβ„“βˆ’ξ€·π‘˜1ξ€Έπœƒ2β„“ξ€·βˆ’β‹―βˆ’π‘˜π‘˜ξ€Έπœƒ2π‘˜β„“2ξƒͺπ‘˜0ξ‚Ά=cothξ‚΅ξ‚΅arccothπ‘₯β„“+ξ‚΅π‘˜1ξ‚Άarccothπ‘₯2β„“ξ‚΅π‘˜π‘˜ξ‚Ά+β‹―+arccothπ‘₯2π‘˜β„“ξ‚Ά,(3.24) where πœƒπ‘—=βˆ’2arccothπ‘₯π‘—ξ€·ξ€½π‘—βˆˆβ„“,2β„“,…,2π‘˜β„“ξ€Ύξ€Έ.(3.25)
(IV) If the sequence {π‘₯𝑛}𝑛β‰₯0 satisfies π‘₯3𝑛=π‘₯𝑛+π‘₯2𝑛1+π‘₯𝑛π‘₯2𝑛(𝑛β‰₯0),(3.26) then, for β„“β‰’0mod3, π‘˜βˆˆβ„•, one has π‘₯3π‘˜β„“ξƒ©βˆ’ξ€·=tanhπ‘˜0ξ€Έπœƒβ„“βˆ’ξ€·π‘˜1ξ€Έπœƒ2β„“ξ€·βˆ’β‹―βˆ’π‘˜π‘˜ξ€Έπœƒ2π‘˜β„“2ξƒͺπ‘˜0ξ‚Ά=tanhξ‚΅ξ‚΅arctanhπ‘₯β„“+ξ‚΅π‘˜1ξ‚Άarctanhπ‘₯2β„“ξ‚΅π‘˜π‘˜ξ‚Ά+β‹―+arctanhπ‘₯2π‘˜β„“ξ‚Ά,(3.27) where πœƒπ‘—=βˆ’2arctanhπ‘₯π‘—ξ€·ξ€½π‘—βˆˆβ„“,2β„“,…,2π‘˜β„“ξ€Ύξ€Έ.(3.28)

Proof. (I) As seen in Section 1, the RDAC relation (3.16) is equivalent to π‘ˆ3𝑛=π‘ˆπ‘›π‘ˆ2π‘›ξ‚΅π‘ˆπ‘›=π‘₯π‘›βˆ’π‘–π‘₯𝑛+𝑖,(3.29) whose solution is, by virtue of Corollary 2.2, π‘ˆ3π‘˜β„“=π‘ˆξ‚€π‘˜0ξ‚β„“π‘ˆξ‚€π‘˜12β„“β‹―π‘ˆξ‚€π‘˜π‘˜ξ‚2π‘˜β„“. Thus, π‘₯3π‘˜β„“βˆ’π‘–π‘₯3π‘˜β„“=ξ‚΅π‘₯+π‘–β„“βˆ’π‘–π‘₯β„“ξ‚Ά+π‘–ξ‚€π‘˜0π‘₯2β„“βˆ’π‘–π‘₯2β„“ξ‚Ά+π‘–ξ‚€π‘˜1⋯π‘₯2π‘˜β„“βˆ’π‘–π‘₯2π‘˜β„“ξ‚Ά+π‘–ξ‚€π‘˜π‘˜ξ‚.(3.30) Setting π‘’π‘–πœƒπ‘—=(π‘₯π‘—βˆ’π‘–)/(π‘₯𝑗+𝑖)(π‘—βˆˆ{β„“,2β„“,…,2π‘˜β„“}), one has π‘₯3π‘˜β„“βŽ›βŽœβŽœβŽ=𝑖1+π‘’π‘–π‘˜0ξ‚πœƒξ‚€ξ‚€β„“ξ‚€π‘˜π‘˜ξ‚πœƒ+β‹―+2π‘˜β„“ξ‚1βˆ’π‘’π‘–π‘˜0ξ‚πœƒξ‚€ξ‚€β„“ξ‚€π‘˜π‘˜ξ‚πœƒ+β‹―+2π‘˜β„“ξ‚βŽžβŽŸβŽŸβŽ ξƒ©βˆ’ξ€·=cotπ‘˜0ξ€Έπœƒβ„“ξ€·βˆ’β‹―βˆ’π‘˜π‘˜ξ€Έπœƒ2π‘˜β„“2ξƒͺπ‘˜0ξ‚Ά=cotξ‚΅ξ‚΅arccotπ‘₯β„“+ξ‚΅π‘˜1ξ‚Άarccotπ‘₯2β„“ξ‚΅π‘˜π‘˜ξ‚Ά+β‹―+arccotπ‘₯2π‘˜β„“ξ‚Ά,(3.31) provided ξ€·π‘˜0ξ€Έπœƒβ„“ξ€·+β‹―+π‘˜π‘˜ξ€Έπœƒ2π‘˜β„“β‰’0(mod2πœ‹).
(II) Substituting π‘₯𝑛 by 1/π‘₯𝑛 turns (3.20) into (3.16) and so the result follows at once from part (I).
(III) Substituting π‘₯𝑛 by 𝑖π‘₯𝑛 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 π‘₯𝑛 by 𝑖π‘₯𝑛 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 π‘₯𝑛 by 1/π‘₯𝑛 in part (II) causes no harm should there exists integer 𝑁 such that π‘₯𝑁=0 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 𝑏, an entirely analogous proof as that in Proposition 3.3, which we omit here, leads to Proposition 3.5.

Proposition 3.5. Let π‘βˆˆβ„•,𝑏β‰₯2.
(I) Suppose that the sequence {π‘₯𝑛}𝑛β‰₯0 satisfies an RDAC relation of the form π‘₯𝑏𝑛=π‘₯(π‘βˆ’1)𝑛π‘₯π‘›βˆ’1π‘₯(π‘βˆ’1)𝑛+π‘₯𝑛(𝑛β‰₯1).(3.32) For β„“β‰’0(mod𝑏) and π‘˜βˆˆβ„•, if the condition ξ‚΅π‘˜0ξ‚Άπœƒβ„“+ξ‚΅π‘˜1ξ‚Άπœƒ(π‘βˆ’1)β„“ξ‚΅π‘˜π‘˜ξ‚Άπœƒ+β‹―+(π‘βˆ’1)π‘˜β„“β‰’0mod2πœ‹(3.33) is fulfilled, then π‘₯π‘π‘˜β„“ξƒ©βˆ’ξ€·=cotπ‘˜0ξ€Έπœƒβ„“βˆ’ξ€·π‘˜1ξ€Έπœƒ(π‘βˆ’1)β„“ξ€·βˆ’β‹―βˆ’π‘˜π‘˜ξ€Έπœƒ(π‘βˆ’1)π‘˜β„“2ξƒͺπ‘˜0ξ‚Ά=cotξ‚΅ξ‚΅arccotπ‘₯β„“+ξ‚΅π‘˜1ξ‚Άarccotπ‘₯(π‘βˆ’1)β„“ξ‚΅π‘˜π‘˜ξ‚Ά+β‹―+arccotπ‘₯(π‘βˆ’1)π‘˜β„“ξ‚Ά,(3.34) where πœƒπ‘—=βˆ’2arccotπ‘₯π‘—ξ€·ξ€½π‘—βˆˆβ„“,(π‘βˆ’1)β„“,…,(π‘βˆ’1)π‘˜β„“ξ€Ύξ€Έ.(3.35)
(II) Assume that the sequence {π‘₯𝑛}𝑛β‰₯0 satisfies an RDAC relation of the form π‘₯𝑏𝑛=π‘₯𝑛+π‘₯(π‘βˆ’1)𝑛1βˆ’π‘₯𝑛π‘₯(π‘βˆ’1)𝑛(𝑛β‰₯0).(3.36) For β„“β‰’0(mod𝑏) and π‘˜βˆˆβ„•, if πœƒβ„“+ξ€·π‘˜1ξ€Έπœƒ(π‘βˆ’1)β„“ξ€·+β‹―+π‘˜π‘˜ξ€Έπœƒ(π‘βˆ’1)π‘˜β„“ is not an odd multiple of πœ‹, then π‘₯π‘π‘˜β„“ξƒ©βˆ’ξ€·=tanπ‘˜0ξ€Έπœƒβ„“βˆ’ξ€·π‘˜1ξ€Έπœƒ(π‘βˆ’1)β„“ξ€·βˆ’β‹―βˆ’π‘˜π‘˜ξ€Έπœƒ(π‘βˆ’1)π‘˜β„“2ξƒͺπ‘˜0ξ‚Ά=tanξ‚΅ξ‚΅arctanπ‘₯β„“+ξ‚΅π‘˜1ξ‚Άarctanπ‘₯(π‘βˆ’1)β„“ξ‚΅π‘˜π‘˜ξ‚Ά+β‹―+arctanπ‘₯(π‘βˆ’1)π‘˜β„“ξ‚Ά,(3.37) where πœƒπ‘—=βˆ’2arctanπ‘₯π‘—ξ€·ξ€½π‘—βˆˆβ„“,(π‘βˆ’1)β„“,…,(π‘βˆ’1)π‘˜β„“ξ€Ύξ€Έ.(3.38)
(III) If the sequence {π‘₯𝑛}𝑛β‰₯0 satisfies π‘₯𝑏𝑛=π‘₯𝑛π‘₯(π‘βˆ’1)𝑛+1π‘₯𝑛+π‘₯(π‘βˆ’1)𝑛(𝑛β‰₯0),(3.39) then, for β„“β‰’0(mod𝑏), π‘˜βˆˆβ„•, one has π‘₯π‘π‘˜β„“ξƒ©βˆ’ξ€·=cothπ‘˜0ξ€Έπœƒβ„“βˆ’ξ€·π‘˜1ξ€Έπœƒ(π‘βˆ’1)β„“ξ€·βˆ’β‹―βˆ’π‘˜π‘˜ξ€Έπœƒ(π‘βˆ’1)π‘˜β„“2ξƒͺπ‘˜0ξ‚Ά=cothξ‚΅ξ‚΅arccothπ‘₯β„“+ξ‚΅π‘˜1ξ‚Άarccothπ‘₯(π‘βˆ’1)β„“ξ‚΅π‘˜π‘˜ξ‚Ά+β‹―+arccothπ‘₯(π‘βˆ’1)π‘˜β„“ξ‚Ά,(3.40) where πœƒπ‘—=βˆ’2arccothπ‘₯π‘—ξ€·ξ€½π‘—βˆˆβ„“,(π‘βˆ’1)β„“,…,(π‘βˆ’1)π‘˜β„“ξ€Ύξ€Έ.(3.41)
(IV) If the sequence {π‘₯𝑛}𝑛β‰₯0 satisfies π‘₯𝑏𝑛=π‘₯𝑛+π‘₯(π‘βˆ’1)𝑛1+π‘₯𝑛π‘₯(π‘βˆ’1)𝑛(𝑛β‰₯0),(3.42) then, for β„“β‰’0(mod𝑏), π‘˜βˆˆβ„•, one has π‘₯π‘π‘˜β„“ξƒ©βˆ’ξ€·=tanhπ‘˜0ξ€Έπœƒβ„“βˆ’ξ€·π‘˜1ξ€Έπœƒ(π‘βˆ’1)β„“ξ€·βˆ’β‹―βˆ’π‘˜π‘˜ξ€Έπœƒ(π‘βˆ’1)π‘˜β„“2ξƒͺπ‘˜0ξ‚Ά=tanhξ‚΅ξ‚΅arctanhπ‘₯β„“+ξ‚΅π‘˜1ξ‚Άarctanhπ‘₯(π‘βˆ’1)β„“ξ‚΅π‘˜π‘˜ξ‚Ά+β‹―+arctanhπ‘₯(π‘βˆ’1)π‘˜β„“ξ‚Ά,(3.43) where πœƒπ‘—=βˆ’2arctanhπ‘₯π‘—ξ€·ξ€½π‘—βˆˆβ„“,(π‘βˆ’1)β„“,…,(π‘βˆ’1)π‘˜β„“ξ€Ύξ€Έ.(3.44)

When all the exponents 𝛼𝑗 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 π‘βˆˆβ„•,  𝑏β‰₯2,β€‰β€‰π‘€βˆˆβ„‚β§΅{0}. If the sequence {π‘₯𝑛}𝑛β‰₯0 satisfies π‘₯𝑏𝑛𝑀=ξ€·π‘₯𝑛π‘₯+𝑀2𝑛⋯π‘₯+𝑀(π‘βˆ’1)𝑛+ξ€·π‘₯+𝑀𝑛π‘₯βˆ’π‘€ξ€Έξ€·2𝑛⋯π‘₯βˆ’π‘€(π‘βˆ’1)π‘›ξ€Έβˆ’π‘€ξ€·π‘₯𝑛π‘₯+𝑀2𝑛⋯π‘₯+𝑀(π‘βˆ’1)π‘›ξ€Έβˆ’ξ€·π‘₯+𝑀𝑛π‘₯βˆ’π‘€ξ€Έξ€·2𝑛⋯π‘₯βˆ’π‘€(π‘βˆ’1)π‘›ξ€Έβˆ’π‘€,(3.45) then for β„“β‰’0(mod𝑏) and π‘˜βˆˆβ„• one has π‘₯π‘π‘˜β„“ξ‚΅π΄=𝑀++π΄βˆ’π΄+βˆ’π΄βˆ’ξ‚Ά,(3.46) provided the values exist, where 𝐴+ξ‘βˆΆ=𝑖1+𝑖2+β‹―+π‘–π‘βˆ’1=β„“ξ€·π‘₯1𝑖12𝑖2β‹―(π‘βˆ’1)π‘–π‘βˆ’1β„“ξ€Έ+π‘€ξ‚€π‘˜π‘–1,𝑖2,…,π‘–π‘βˆ’1ξ‚π΄βˆ’ξ‘βˆΆ=𝑖1+𝑖2+β‹―+π‘–π‘βˆ’1=β„“ξ€·π‘₯1𝑖12𝑖2β‹―(π‘βˆ’1)π‘–π‘βˆ’1β„“ξ€Έβˆ’π‘€ξ‚€π‘˜π‘–1,𝑖2,…,π‘–π‘βˆ’1.(3.47)

Proof. Rewriting (3.45), we get π‘₯π‘π‘›βˆ’π‘€π‘₯𝑏𝑛=ξ‚΅π‘₯+π‘€π‘›βˆ’π‘€π‘₯𝑛π‘₯+𝑀2π‘›βˆ’π‘€π‘₯2𝑛⋯π‘₯+𝑀(π‘βˆ’1)π‘›βˆ’π‘€π‘₯(π‘βˆ’1)𝑛+𝑀(3.48) or π‘ˆπ‘π‘›=π‘ˆπ‘›π‘ˆ2π‘›β‹―π‘ˆ(π‘βˆ’1)π‘›ξ‚΅π‘ˆπ‘—=π‘₯π‘—βˆ’π‘€π‘₯𝑗.+𝑀,π‘—βˆˆ{𝑛,2𝑛,…,(π‘βˆ’1)𝑛}(3.49) Theorem 2.1 thus yields for β„“β‰’0(mod𝑏) and π‘˜βˆˆβ„•π‘ˆπ‘π‘˜β„“=𝑖1+𝑖2+β‹―+π‘–π‘βˆ’1=β„“π‘ˆξ‚€π‘˜π‘–1,𝑖2,…,π‘–π‘βˆ’11𝑖12𝑖2β‹―(π‘βˆ’1)π‘–π‘βˆ’1β„“,(3.50) that is, π‘₯π‘π‘˜β„“βˆ’π‘€π‘₯π‘π‘˜β„“=+𝑀𝑖1+𝑖2+β‹―+π‘–π‘βˆ’1=β„“ξ‚΅π‘₯1𝑖12𝑖2β‹―(π‘βˆ’1)π‘–π‘βˆ’1β„“βˆ’π‘€π‘₯1𝑖12𝑖2β‹―(π‘βˆ’1)π‘–π‘βˆ’1β„“ξ‚Ά+π‘€ξ‚€π‘˜π‘–1,𝑖2,…,π‘–π‘βˆ’1,(3.51) 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 𝑏=2.

Proposition 4.1. Let the notation be as in Proposition 3.1.
(I) Suppose that the sequence {π‘₯𝑛}𝑛β‰₯0 satisfies an RDAC relation of the form π‘₯2𝑛=π‘₯2π‘›βˆ’12π‘₯n(𝑛β‰₯1).(4.1) For each fixed β„“β‰’0(mod2) and π‘˜βˆˆβ„•, (a)if πœƒβ„“ is a rational multiple of πœ‹, then either {π‘₯2π‘˜β„“}π‘˜βˆˆβ„• diverges in finitely many steps or {π‘₯2π‘˜β„“} is periodic;(b)if πœƒβ„“ is not a rational multiple of πœ‹, then π‘₯2π‘˜β„“ exists for all π‘˜βˆˆβ„• and the sequence {π‘₯2π‘˜β„“}π‘˜βˆˆβ„• is never periodic.
(II) Suppose that the sequence {π‘₯𝑛}𝑛β‰₯0 satisfies an RDAC relation of the form π‘₯2𝑛=2π‘₯𝑛1βˆ’π‘₯2𝑛(𝑛β‰₯0).(4.2) For each fixed β„“β‰’0(mod2), (a)if πœƒβ„“ is a rational multiple of πœ‹, then either {π‘₯2π‘˜β„“}π‘˜βˆˆβ„• diverges in finitely many steps or {π‘₯2π‘˜β„“} is periodic; (b)if πœƒβ„“ is not a rational multiple of πœ‹, then π‘₯2π‘˜β„“ exists for all π‘˜βˆˆβ„• and the sequence {π‘₯2π‘˜β„“}π‘˜βˆˆβ„• is never periodic.
(III) Suppose that the sequence {π‘₯𝑛}nβ‰₯0 satisfies π‘₯2𝑛=π‘₯2𝑛+12π‘₯𝑛(𝑛β‰₯0).(4.3) For each fixed β„“β‰’0(mod2), (a)if πœƒβ„“=0, then the sequence {π‘₯2π‘˜β„“}π‘˜β‰₯1 does not exist; (b)if πœƒβ„“>0, the sequence {π‘₯2π‘˜β„“} is strictly decreasing in the interval [coth(πœƒβ„“),1); (c)if πœƒβ„“<0, then {π‘₯2π‘˜β„“}π‘˜β‰₯1 is strictly increasing in the interval [coth(πœƒβ„“),βˆ’1).
(IV) Suppose that the sequence {π‘₯𝑛}𝑛β‰₯0 satisfies π‘₯2𝑛=2π‘₯𝑛1+π‘₯2𝑛(𝑛β‰₯0).(4.4) For each fixed β„“β‰’0(mod2), (z)if πœƒβ„“=0, then {π‘₯2π‘˜β„“}π‘˜βˆˆβ„• is the zero sequence; (b)if πœƒβ„“>0, then the sequence {π‘₯2π‘˜β„“} is strictly increasing in [tanh(πœƒβ„“),1); (c)if πœƒβ„“<0, the sequence is strictly decreasing in [tanh(πœƒβ„“),βˆ’1).

Proof. (I) From part (I) of Proposition 3.1, we know that π‘₯2π‘˜β„“ξ€·=cotβˆ’2π‘˜βˆ’1πœƒβ„“ξ€Έ(4.5) provided 2π‘˜πœƒβ„“β‰’0(mod2πœ‹). Consider the case where πœƒβ„“ is a rational multiple of πœ‹, say, πœƒβ„“=π‘šπœ‹π‘‘withπ‘š,𝑑(>0)βˆˆβ„€,gcd(π‘š,𝑑)=1.(4.6) If 𝑑 is a multiple of 2, then it is easily checked that {π‘₯2π‘˜β„“}π‘˜βˆˆβ„• diverges in finitely many steps. If 𝑑β‰₯2 is not a multiple of 2, let 𝑑=2𝑣𝑇, where 2π‘£βˆ£βˆ£π‘‘,𝑇β‰₯3. Observe that for all large π‘›βˆˆβ„•, when evaluating the values of cotangent, we need only look at 2π‘›πœƒβ„“=2π‘›π‘šπœ‹π‘‘=2π‘›βˆ’π‘£π‘šπœ‹π‘‡(mod2πœ‹),(4.7) which is equivalent to looking at πΊπ‘›βˆΆ=2π‘›βˆ’π‘£π‘š(mod2𝑇).(4.8) Since each 𝐺𝑛 takes at most 2𝑇 values and the sequence {π‘₯2π‘˜β„“}π‘˜βˆˆβ„• is infinite, there are positive integers 𝑁1<𝑁2 such that 𝐺𝑁1=𝐺𝑁2, which in turn implies that {π‘₯2π‘˜β„“}π‘˜βˆˆβ„• is periodic.
Finally, if πœƒβ„“ is not a rational multiple of πœ‹, then 2π‘˜βˆ’1πœƒβ„“ is not a multiple of πœ‹ showing that the sequence {π‘₯2π‘˜β„“}π‘˜βˆˆβ„• is well defined and never periodic.
The proof of part (II) is similar to that of part (I).
(III) If πœƒβ„“=0, then the values π‘₯2π‘˜β„“=coth(2π‘˜πœƒβ„“) become infinite for all π‘˜βˆˆβ„• and part (a) follows. Since π‘₯2π‘˜β„“=coth(2π‘˜πœƒβ„“) is a strictly decreasing (resp. increasing) function of π‘˜ according as πœƒβ„“>0 (resp. πœƒβ„“<0), the results in (b) and (c) are immediate.
(IV) If πœƒβ„“=0, then π‘₯2π‘˜β„“=tanh(πœƒβ„“)=0. Arguments for the other two cases πœƒβ„“>0 and πœƒβ„“<0 are similar to those in part (III).

Note from Proposition 4.1 that global behaviors of solutions in the case 𝑏=2 depend solely on the single value πœƒβ„“. The situation when 𝑏β‰₯3 is more complex since their global behaviors depend heavily on the variable π‘˜ as we see in the following illustration. Keeping the notation of Proposition 3.5, suppose that the sequence {π‘₯𝑛}𝑛β‰₯0 satisfies an RDAC relation of the form x𝑏𝑛=π‘₯(π‘βˆ’1)𝑛π‘₯π‘›βˆ’1π‘₯(π‘βˆ’1)𝑛+π‘₯𝑛(𝑛β‰₯1).(4.9) From part (I) of Proposition 3.5, we know that π‘₯π‘π‘˜β„“ξƒ©βˆ’ξ€·=cotπ‘˜0ξ€Έπœƒβ„“βˆ’ξ€·π‘˜1ξ€Έπœƒ(π‘βˆ’1)β„“ξ€·βˆ’β‹―βˆ’π‘˜π‘˜ξ€Έπœƒ(π‘βˆ’1)π‘˜β„“2ξƒͺ.(4.10) This explicit form shows that, for each fixed β„“β‰’0(mod𝑏), the behavior of π‘₯π‘π‘˜β„“ considered as a function of π‘˜βˆˆβ„• depends on all πœƒβ„“,…,πœƒ(π‘βˆ’1)π‘˜β„“, and we can merely infer that the values π‘₯π‘π‘˜β„“ are well defined (i.e., finite) if and only if ξ‚΅π‘˜0ξ‚Άπœƒβ„“+ξ‚΅π‘˜1ξ‚Άπœƒ(π‘βˆ’1)β„“ξ‚΅π‘˜π‘˜ξ‚Άπœƒ+β‹―+(π‘βˆ’1)π‘˜β„“β‰’(mod2πœ‹).(4.11)

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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