Rational Recursive Equations Characterizing Cotangent-tangent and Hyperbolic

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BULLETINof the Malaysian Mathematical Sciences Society

http://math.usm.my/bulletin

Bull. Malays. Math. Sci. Soc. (2)33(3) (2010), 421–428

Rational Recursive Equations Characterizing Cotangent-tangent and Hyperbolic

Cotangent-tangent Functions

1Charinthip Hengkrawit, ²Vichian Laohakosol and

3Patanee Udomkavanich

1,3Department of Mathematics, Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand

2Department of Mathematics and Center for Advanced Studies, Kasetsart University, Bangkok 10900, Thailand

and Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand

1hengkrawit [email protected],²[email protected],³[email protected]

Abstract. Using a technique of Rhouma in 2005, closed form solutions of certain rational recursive equations characterizing the cotangent-tangent and the hyperbolic cotangent-tangent function solutions are derived.

2010 Mathematics Subject Classification: 39B32

Key words and phrases: Rational recursive equations, cotangent-tangent functions, hyperbolic cotangent-tangent functions.

1. Introduction

In 2005, Rhouma, [3], gave a closed form solution to the recursive difference equation

(1.1) yn+2=ynyn+1−1

yn+yn+1

,

which was originated from an open problem in the book [1] (see also [2]) as follows:

Theorem 1.1. Let y0 andy1 be arbitrary real numbers such that yn exists for all n ∈ N∪ {0} and {Fn} is the Fibonacci sequence defined by F0 = F1 = 1 and Fn+2=Fn+1+Fn (n≥0). The solution to equation (1.1)exists for alln∈N∪ {0}

if and only if

F_n−2θ0+F_n−1θ16= 0 (mod 2π), whereθ₀=−2arccot y₀ andθ₁=−2arccot y₁.

Communicated byMohammad Sal Moslehian.

Received:March 25, 2009;Revised: June 11, 2009.

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When it exists, the solution to (1.1)is given by (1.2) yn=−cot

F_n−2θ0+F_n−1θ1

2

= cot(F_n−2arccoty0+F_n−1arccot y1).

Moreover,

(1) if θ₀ and θ₁ are both rational multiples of π, then either {y_n} diverges in finitely many steps or {y_n} is periodic;

(2) if θ₀ is a rational multiple of π andθ₁ is not (or vice versa), then {y_n} is aperiodic and does exist for alln.

It is easily checked that the cotangent function in (1.2) satisfies (1.1) showing that the rational recursive equation (1.1) does indeed characterize the cotangent function. Rhouma’s technique is first to transform (1.1) to an equivalent form of (1.3) yn+2=i(yn+1+i)(yn+i) + (yn+1−i)(yn−i)

(y_n+1+i)(y_n+i)−(y_n+1−i)(y_n−i),

or yn+2−i

y_n+2+i =yn+1−i

y_n+1+i· yn−i y_n+i, which is a difference equation of the shape

(1.4) x_n+2=αx_n+1x_n.

A closed form solution of this last equation is derived without difficulty.

The difference equation (1.1) is interesting at least in two respects. First, it resembles the well-known identity of the cotangent function. Second, putting yn = cotzn, with the values of zn restricted to the open interval (0, π), the difference equation leads to the Fibonacci recurrence modulo π of the form z_n+2 = z_n + z_n+1 modπ. Motivated by the above result of Rhouma, we start by finding a closed form solution for any rational recursive equation extending (1.3), of the form

yn+`=i(yn+`−1+i)Â¹. . .(yn+i)Â^` + (yn+`−1−i)Â¹. . .(yn−i)Â^` (y_n+`−1+i)Â¹. . .(yn+i)Â^` −(y_n+`−1−i)Â¹. . .(yn−i)Â^`, and determine its asymptotic behavior. As the equation (1.1) characterizes the cotangent function, it is natural to consider its counterpart

(1.5) yn+2= y_n+y_n+1

1−ynyn+1

,

or equivalently,

(1.6) yn+2=i(yn+1+i)(yn+i)−(−yn+1+i)(−yn+i) (yn+1+i)(yn+i) + (−yn+1+i)(−yn+i),

which clearly has the tangent function as a solution. Our second objective is to find a closed form solution for any rational recursive equation extending (1.6) of the form

y_n+`=i(y_n+`−1+i)Â¹· · ·(y_n+i)Â^`−(−y_n+`−1+i)Â¹· · ·(−y_n+i)Â^` (yn+`−1+i)Â¹· · ·(yn+i)Â^`+ (−yn+`−1+i)Â¹· · ·(−yn+i)Â^`, and determine its asymptotic behavior. Finally, we solve rational recursive equations which can be used to characterize the hyperbolic tangent and cotangent functions.

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The method adopted here is an extension of the original Rhouma’s technique.

Indeed, in the last section of his paper, Rhouma [3] indicated how to find closed form solutions of other difference equations generalizing (1.4) with some simple examples and our results here may be considered as more elaborate examples illustrating his technique in the direction of characterizing the cotangent-tangent and hyperbolic tangent-cotangent functions.

2. Results

Our first lemma follows easily from a simple calculation whose trivial proof is omit- ted.

Lemma 2.1. Let`∈N, `≥2; b, x1, . . . , x`, zbe complex numbers and letA1, . . . , A`

be nonzero integers such that b(x1+b)^A¹. . .(x`+b)^A^`

(x1+b)Â¹. . .(x`+b)Â^` −(x1−b)Â¹. . .(x`−b)Â^` 6= 0.

Then

z−b z+b =

x1−b x₁+b

^A1

· · ·

x`−b x_`+b

^A`

if and only if

z=b (x1+b)Â¹· · ·(x`+b)Â^`+ (x1−b)Â¹· · ·(x`−b)Â^` (x1+b)Â¹· · ·(x`+b)Â^`−(x1−b)Â¹· · ·(x`−b)Â^`.

The next lemma relates generalized Fibonacci numbers with elements in rational recursive sequences.

Lemma 2.2. Let`∈N, `≥2 and let{Fn}be the(generalized Fibonacci)sequence satisfying a linear recurrence relation of the form

(2.1) Fn+`=A1F_n+`−1+A2F_n+`−2+· · ·+A`Fn,

whereA1, . . . , A`are nonzero integers such thatA1+. . .+A`6= 0, with initial values F₀=A_`, F₁=A_`−1, . . . , F_`−1=A₁.

If

(2.2) xn+`=α^A²¹^+···+A²^`⁻

A2 1 +···+A2

`

A1 +···+A`xÂ_n+`−1¹ xÂ_n+`−2² · · ·xÂ_n^` (n≥0), then

(2.3) xn=α^Fⁿ⁻

A2 1 +···+A2

`

A1 +···+A`x^F₀^n−`x^F₁^n−`+1· · ·x^F_`−1ⁿ⁻¹ for alln≥`.

Proof. For the starting case, the condition (2.2) and the recurrence (2.1) yield x`=α^A

2

1+···+A²_`−^A_A²^{1 +}^...+A²^`

1 +···+A`xÂ_`−1¹ xÂ_`−2² · · ·xÂ₀^` =α^F^`⁻

A2 1 +···+A2

`

A1 +···+A`xÂ_`−1¹ xÂ_`−2² · · ·xÂ₀^`, which agrees with (2.3) whenn=`. Next, suppose that (2.3) is true for all`≤n≤k.

From (2.2), using the induction hypothesis and the recurrence (2.1), we get xk+1=α^A

2

1+···+A²_`−^A_A²^{1 +}^···+A²^`

1 +···+A`x^A_k¹· · ·x^A_k−`+1^`

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=α^A²¹^+···+A²^`⁻

A2 1 +···+A2

` A1 +···+A`

α⁻

A2 1 +···+A2

` A1 +···+A`+F_k

x^F₀^k−`x^F₁^k−`+1· · ·x^F_`−1^k−1 ^A1

× · · ·

×

α⁻

A2 1 +···+A2

`

A1 +···+A`+Fk−`+1

x^F₀^k−`+1−`x^F₁^{k−`+1−`+1}· · ·x^F_`−1^k−`+1−1 ^A`

=α⁻

A2 1 +···+A2

`

A1 +···+A`(−(A₁+···+A_`)+1+A₁+···+A_`)+A₁F_k+···+A_`Fk−`+1

×x^A₀¹^F^k−`^+···+A^`^F^k−`−`+1· · ·x^A_`−1¹^F^k−1^+···+A^`^F^k−1−`+1

=α⁻

A2 1 +···+A2

` A1 +···+A`+F_k+1

x^F₀^(k+1)−`· · ·x^F_`−1^(k+1)−1. Our main result reads:

Theorem 2.1. Let ` ∈ N, `≥ 2; A1, . . . , A` be nonzero integers such that A1+

· · ·+A`6= 0. Lety0, . . . , y_`−1 be real numbers such that those yn which satisfy (2.4) y_n+`=i(y_n+`−1+i)Â¹. . .(y_n+i)Â^`+ (y_n+`−1−i)Â¹. . .(y_n−i)Â^`

(yn+`−1+i)Â¹. . .(yn+i)Â^`−(yn+`−1−i)Â¹. . .(yn−i)Â^`, exist for all n ∈ N∪ {0}. Let {Fn} be the sequence satisfying a linear recurrence relation of the form

Fn+`=A1F_n+`−1+A2F_n+`−2+. . .+A`Fn,

with initial values F0=A`, F1=A`−1, . . . , F`−1=A1. Then the solution to (2.4) exists if and only if

A1F_n−`θ0+. . .+A`F_n−1θ_`−16= 0 (mod 2π) (n∈N∪ {0}), whereθj =_A⁻²

j+1 arccot yj for allj∈ {0, . . . , `−1}. When it exists, the solution is given by

(2.5) y_n=i(y0+i)^F^n−`· · ·(y_`−1+i)^Fⁿ⁻¹+ (y0−i)^F^n−`· · ·(y_`−1−i)^Fⁿ⁻¹ (y₀+i)^F^n−`· · ·(y_`−1+i)^Fⁿ⁻¹−(y₀−i)^F^n−`· · ·(y_`−1−i)^Fⁿ⁻¹, or

y_n= cot

−A1F_n−`θ₀− · · · −A_`F_n−1θ_`−1 2

= cot(F_n−`arccoty0+· · ·+F_n−1arccoty_`−1).

Moreover,

(1) if all theθ_j are rational multiples ofπ, then either{y_n} diverges in finitely many steps or{yn} is periodic;

(2) if θ0, θ1, . . . , θ_`−1, π are linearly independent over Q, and A1, . . . , A` are nonzero integers, then yn exists for all n and the sequence {yn} is never periodic.

Proof. Takingz=yn+`,x1=yn+`−1, . . . , x`=yn,b=iin Lemma 2.1, the rational recursive equation (2.4) is equivalent to

(2.6) y_n+`−i

yn+`+i =

y_n+`−1−i yn+`−1+i

A1

· · ·

y_n−i yn+i

A`

.

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PuttingU_n= ^y_yⁿ⁻ⁱ

n+i, the relation (2.6) becomes

U_n+`=U_n+`−1Â¹ U_n+`−2Â² · · ·U_nÂ^`, whose solution is, by virtue of Lemma 2.2,

Un =U₀^F^n−`U₁^F^n−`+1· · ·U_`−1^Fⁿ⁻¹ (n≥`), and so

(2.7) yn−i y_n+i =

y0−i y₀+i

^Fn−`y1−i y₁+i

^Fn−`+1

· · ·

y_`−1−i y_`−1+i

^Fn−1

,

which, by Lemma 2.1, becomes

(2.8) y_n=i(y0+i)^F^n−`· · ·(y_`−1+i)^Fⁿ⁻¹+ (y0−i)^F^n−`· · ·(y_`−1−i)^Fⁿ⁻¹ (y₀+i)^F^n−`· · ·(y_`−1+i)^Fⁿ⁻¹−(y₀−i)^F^n−`· · ·(y_`−1−i)^Fⁿ⁻¹. Next, settinge^iθ⁰^A¹ = ^y_y⁰⁻ⁱ

0+i, . . . , e^iθ^`−1^A^` =^y_y^`−1⁻ⁱ

`−1+i,we have y_n−i

yn+i =e^i(A¹^F^n−`^θ⁰^+···+A^`^Fⁿ⁻¹^θ^`−1⁾, i.e.,

yn=i1 +e^i(A¹^F^n−`^θ⁰^+···+A^`^Fⁿ⁻¹^θ^`−1⁾ 1−e^i(A¹^F^n−`^θ⁰^+···+A^`^Fⁿ⁻¹^θ^`−1⁾

= cot

−A₁F_n−`θ₀− · · · −A_`F_n−1θ_`−1 2

= cot(Fn−`arccoty0+· · ·+Fn−1arccoty`−1), providedA1F_n−`θ0+· · ·+A`F_n−1θ_`−16= 0 (mod 2π).

If all theθj (j = 0,1, . . . , `−1) are rational multiples ofπ, say, θ_j=m_jπ/t_j with m_j, t_j(>0)∈Z, gcd(m_j, t_j) = 1, then it is easily checked thatP`

k=1A_kF_n−`+k−1θ_k−1mod 2πis equivalent to Gn=

`

X

k=1

AkF_n−`+k−1m_k−1

`−1

Y

j=0 j6=k

tj mod

`−1

Y

j=0

2tj.

Since eachGntakes at mostQ`−1

j=0(2tj) distinct values, each`-tuple (Gt, . . . , G_t+`−1) takes at mostQ`−1

j=0(2tj)^` distinct values. Since the sequence{(Gt, . . . , Gt+`−1)}_t≥0 is infinite, there are integersN16=N2 such that

(G_N₁, . . . , G_N₁_+`−1) = (G_N₂, . . . , G_N₂_+`−1).

Since

Gj+`=Gj+· · ·+G_j+`−1 (j∈N),

we deduce that GN₁+k ≡GN₂+k for all k∈N, i.e., the sequence {Gn} is periodic.

If someGn is zero, then clearly the sequence {yn} diverges.

Ifθ0, θ1, . . . , θ_`−1, π are linearly independent overQ, then A₁F_n−`θ₀+. . .+A_`F_n−1θ_`−16= 2kπ (k∈Z),

showing thatyn exists for eachnand the sequence{yn} is never periodic.

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Theorem 2.1 enables us to solve certain other rational recursive equations characterizing related trigonometric and hyperbolic functions. We begin with the tangent function.

Corollary 2.1. Let ` ∈N, ` ≥2; A1, . . . , A` be nonzero integers such that A1+

· · ·+A`6= 0. Let y0, . . . , y`−1 be real numbers such that those yn which satisfy the rational recursive equation

(2.9) yn+`=i(yn+`−1+i)Â¹· · ·(yn+i)Â^`−(−yn+`−1+i)Â¹· · ·(−yn+i)Â^` (y_n+`−1+i)Â¹· · ·(yn+i)Â^`+ (−y_n+`−1+i)Â¹· · ·(−yn+i)Â^`. exist for all n ∈ N∪ {0}. Let {Fn} be a sequence satisfying a linear recurrence relation of the form

Fn+`=A1Fn+`−1+A2Fn+`−2+. . .+A`Fn,

with initial values F0 =A`, F1 =A`−1, . . . , F`−1 =A1. Then the solution to the equation (2.9) exists if and only if A1Fn−`θ0+A2Fn−`+1θ1+· · ·+A`Fn−1θ`−1 is not an odd multiple ofπ, whereθj =_A⁻²

j+1arctan yj (j = 0,1, . . . , `−1).

When the solution exists, it is given by

(2.10) yn =i(y0+i)Â¹· · ·(y`−1+i)Â^`−(−y0+i)Â¹· · ·(−y`−1+i)Â^` (y0+i)Â¹· · ·(y_`−1+i)Â^`+ (−y0+i)Â¹· · ·(−y_`−1+i)Â^`, or

y_n=−tan

A1F_n−`θ0+· · ·+A`F_n−1θ_`−1 2

= tan (Fn−`arctan y0+· · ·+Fn−1arctan y`−1). Moreover,

(1) if all θj are rational multiples of π, then either the sequence {yn} diverges in finitely many steps or is periodic;

(2) if θ0, θ1, . . . , θ`−1, π are linearly independent over Q, and A1, . . . , A` are nonzero integers, then the valuey_n exists for all nand the sequence{yn} is never periodic.

Proof. Substituting yn by 1/yn turns the equation (2.9) into a rational recursive equation of the form (2.4) and so the corollary follows at once from Theorem 2.1.

Remark 2.1. Although the substitution yn by 1/yn employed in Corollary 2.1 allows us to obtain a closed form solution of the equation (2.9), there remains a difficulty should there exist integer N such thatyN = 0. To overcome this short- coming, 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 Theorem 2.1 to solve the equation (2.9) using auxiliary results analogous to Lemmas 2.1 and 2.2.

Next, we deal with rational recursive equations characterizing the hyperbolic cotangent-tangent functions.

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Corollary 2.2. Let ` ∈N, ` ≥2; A₁, . . . , A_` be nonzero integers such that A₁+

· · ·+A_`6= 0. Lety₀, . . . , y_`−1 be real numbers such that those y_n which satisfy (2.11) yn+`= (y_n+`−1+ 1)Â¹· · ·(yn+ 1)Â^`+ (y_n+`−1−1)Â¹· · ·(yn−1)Â^` (y_n+`−1+ 1)Â¹· · ·(y_n+ 1)Â^`−(y_n+`−1−1)Â¹· · ·(y_n−1)Â^`, exist for all n ∈ N∪ {0}. Let {Fn} be a sequence satisfying a linear recurrence relation of the form

F_n+`=A₁F_n+`−1+A₂F_n+`−2+. . .+A_`F_n,

with initial values F₀ = A_`, F₁ = A_`−1, . . . , F_`−1 = A₁. Then the solution to equation (2.11)exists and is given by

(2.12) yn =(y0+ 1)^F^n−`· · ·(y_`−1+ 1)^Fⁿ⁻¹+ (y0−1)^F^n−`· · ·(y_`−1−1)^Fⁿ⁻¹ (y₀+ 1)^F^n−`· · ·(y_`−1+ 1)^Fⁿ⁻¹−(y₀−1)^F^n−`· · ·(y_`−1−1)^Fⁿ⁻¹, or

y_n= coth (F_n−` arccothy₀+. . .+F_n−1arccoth y_`−1).

Proof. Substitutingynbyiynin the equation (2.11) turns it into a rational recursive equation of the form (2.4) and so Theorem 2.1 yields the desired result.

Corollary 2.3. Let ` ∈N, ` ≥2; A1, . . . , A` be nonzero integers such that A1+

· · ·+A`6= 0. Lety0, . . . , y_`−1 be real numbers such that those yn which satisfy (2.13) y_n+`=(y_n+`−1+ 1)Â¹· · ·(y_n+ 1)Â^`−(−y_n+`−1+ 1)Â¹· · ·(−y_n+ 1)Â^`

(yn+`−1+ 1)Â¹· · ·(yn+ 1)Â^`+ (−yn+`−1+ 1)Â¹· · ·(−yn+ 1)Â^`, exist for all n ∈ N∪ {0}. Let {Fn} be a sequence satisfying a linear recurrence relation of the form

Fn+`=A1F_n+`−1+A2F_n+`−2+· · ·+A`Fn,

with initial values F0 = A`, F1 = A`−1, . . . , F`−1 = A1. Then the solution to equation (2.13)exists and is given by

(2.14) y_n=(y₀+ 1)^F^n−`. . .(y_`−1+ 1)^Fⁿ⁻¹−(−y₀+ 1)^F^n−`. . .(−y_`−1+ 1)^Fⁿ⁻¹ (y0+ 1)^F^n−`. . .(y`−1+ 1)^Fⁿ⁻¹+ (−y0+ 1)^F^n−`. . .(−y`−1+ 1)^Fⁿ⁻¹, or

yn = tanh (arctanhy0F_n−`+· · ·+ arctanh y_`−1F_n−1).

Proof. Replacingyn byiyn in the equation (2.13), we get a rational recursive equation of the form (2.9) and Corollary 2.1 yields the required result.

Acknowledgement. This research is supported by the Thailand Research Funds RTA 5180005, the Centre of Excellence in Mathematics, CHE, and the KU Center for Advanced Studies, Thailand.

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