PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 97(111) (2015), 57–67 DOI: 10.2298/PIM150220001S

ON LINEAR COMBINATIONS OF CHEBYSHEV POLYNOMIALS

Dragan Stankov

Abstract. We investigate an infinite sequence of polynomials of the form: a_{0}T_{n} (x) + a_{1}T_{n−1} (x) + ⋯+ a_{m} T_{n−m} (x) where (a_{0}, a_{1}, …, a_{m} ) is a fixed m-tuple of real numbers, a_{0}, a_{m} ≠ 0, T_{i}(x) are Chebyshev polynomials of the first kind, n = m, m + 1, m + 2, … Here we analyze the structure of the set of zeros of such polynomial, depending on A and its limit points when n tends to infinity. Also the expression of envelope of the polynomial is given. An application in number theory, more precise, in the theory of Pisot and Salem numbers, is presented.

1. Introduction

It is well known [6] that the Chebyshev polynomial T_{n}(x) of the first kind is a polynomial in x of degree n, defined by T_{n}(x) = cosnθ where x = cosθ. Let A = (a_{0}, a_{1}, …, a_{m}) be a (m + 1)-tuple of real numbers, a_{0}, a_{m} ≠ 0, m ⩾ 1. We introduce an infinite sequence of polynomials

T_{n,A} (x) = a_{0}T_{n}(x) + a_{1}T_{n−1} (x) + ⋯+ a_{m}T_{n−m} (x) (n ⩾ m).

We will refer to T_{n,A} (x) as an A-Chebyshev polynomial. We can naturally extend this definition to the case m = 0 and A = a_{0} ≠ 0 by T_{n,a0} (x) = a_{0}T_{n}(x). Also, it will be useful to introduce the polynomial P_{A}(x) = a_{0}x^{m} + a_{1}x^{m−1} + ⋯+ a_{m}. We will refer to P_{A}(t) as the characteristic polynomial of the A-Chebyshev polynomial.

2. Roots of A-Chebyshev polynomial

Let Z_{n,A} denote the set of zeros of the A-Chebyshev polynomial. We will analyse the structure of the set lim inf Z_{n,A} , when n tends to infinity, depending on A. lim inf Z_{n,A} consists of those elements which are limits of points in Z_{n,A} for all n. That is, x ∈ lim inf Z_{n,A} if and only if there exists a sequence of points {x_{k}} such that x_{k} ∈ Z_{k,A} and x_{k} → x as k → ∞.

2010 Mathematics Subject Classification: Primary 11B83; Secondary 11R09, 12D10.

Key words and phrases: Chebyshev polynomials, envelope, Pisot numbers, Salem numbers.

Partially supported by Serbian Ministry of Education and Science, Project 174032.

Communicated by Žarko Mijajlović.

Example 2.1. Using our notation, the Chebyshev polynomial T_{n}(x) is A-Chebyshev polynomial with A = 1. It is well known [5] that the zeros of T_{n}(x) are x_{n,k} = cos \frac{(n−k+1/2)π}{n} (k = 1, 2, …, n). It is obviously that x_{n,1} approaches to −1 and x_{n,n} approaches to 1 when n → ∞. Since x_{n,k} are equispaced, it is clear that lim inf Z_{n,1} = [−1, 1].

Example 2.2. What can we say for A-Chebyshev polynomial if A = (2, −5, 2)? It is well known [5] that T_{n}(x) = 2xT_{n−1} (x) − T_{n−2} (x), n = 2, 3, . . .. So

T_{n,(2,−5,2)} (x)= 2T_{n}(x) − 5T_{n−1} (x) + 2T_{n−2} (x)
= 4xT_{n−1} (x) − 2T_{n−2} (x) − 5T_{n−1} (x) + 2T_{n−2} (x),
T_{n,(2,−5,2)} (x)= (4x − 5)T_{n−1} (x).

Now it is obvious that x = \frac{5}{4} is a zero of T_{n,(2,−5,2)} (x), for all n = 2, 3, . . .. So x = \frac{5}{4} ∈ Z_{n,(2,−5,2)} for all n = 2, 3, . . .. Taking into account the previous example we conclude that lim inf Z_{n,(2,−5,2)} = [−1, 1] ∪ \{ \frac{5}{4}\}.

Lemma 2.1. We have T_{n,A} (x) = \frac{1}{2}(P_{A}(w)w^{n−m} + P_{A}(w^{−1} )w^{−(n−m)} ) where w = x + \sqrt{x^{2} − 1} .

Proof. Starting from the definition of A-Chebyshev polynomial and using the well known formula T_{n}(x) = \frac{1}{2}(w^{n} + w^{−n} ) [5] we have

T_{n,A} (x)= ∑_{i=0}^{m} a_{i}T_{n−i} (x) = ∑_{i=0}^{m} a_{i} \frac{1}{2}(w^{n−i} + w^{−n+i} )

= \frac{1}{2}( ∑_{i=0}^{m} a_{i}w^{n−i} + ∑_{i=0}^{m} a_{i}w^{−n+i} )

= \frac{1}{2}( w^{n−m} ∑_{i=0}^{m} a_{i}w^{m−i} + w^{−n+m} ∑_{i=0}^{m} a_{i}w^{−m+i} )

= \frac{1}{2}(w^{n−m} P_{A}(w) + w^{−n+m} P_{A}(w^{−1} )).  □

One can calculate that if w = x + \sqrt{x^{2} − 1} , then x = \frac{1}{2}(w + w^{−1} ). So, we can deduce from the previous lemma the following

Corollary 2.1. If there is w such that P_{A}(w) = P_{A}(w^{−1} )= 0, then T_{n,A} (x)= 0

for x = \frac{1}{2}(w + w^{−1} ) and for all n ⩾ m.

From the previous example we can see that 2 and \frac{1}{2} are the roots of P_{A}(x) = 2x^{2} − 5x + 2, therefore x = \frac{1}{2}(2 + \frac{1}{2}) = \frac{5}{4} is a zero of T_{n,A} (x) for all n ⩾ 2.

For the next corollary we need the following definition: the set T of Salem numbers is the set of real algebraic integers τ greater than 1, such that all its conjugate roots have modules at most 1, at least one of them having modulus equal to 1.

Corollary 2.2. If τ is a Salem number and P_{A}(x) is its minimal polynomial, then T_{n,A} (x) = 0 for x = \frac{1}{2}(τ + τ^{−1} ) and for all n ⩾ m.

The claim is a direct consequence of a well known property of a Salem number [2] that P_{A}(τ) = P_{A}(τ^{−1} ) = 0.

Theorem 2.1. If there is a root ω, out of the unit circle, of the polynomial P_{A}, that is P_{A}(ω) = 0, |ω| > 1, then for every real number ε > 0, there exists a natural number n_{0} such that for all n > n_{0}, there is a root ξ of the A-Chebyshev polynomial T_{n,A} (x) such that |ξ − \frac{1}{2}(ω + ω^{−1} )| < ε.

Proof. It is convenient to use Lemma 2.1 to express T_{n,A} (x)= \frac{1}{2}P_{A}(w)w^{n−m} + \frac{1}{2}P_{A}(w^{−1} )w^{−(n−m)} where w = x+\sqrt{x^{2} − 1} , or equivalently x = x(w) = \frac{1}{2}(w+w^{−1} ). Since x(w) is continuous for w > 0, there is δ_{1} > 0 such that if |w − ω| < δ_{1}, then |\frac{1}{2}(w + w^{−1} ) − \frac{1}{2}(ω + ω^{−1} )| < ε. We can take a δ_{2} < |ω| − 1 such that, in the circle {z : |z − ω| ⩽ δ_{2}}, there is no root of P_{A}(w) which is different from ω. Let δ = min(δ_{1}, δ_{2}) and C = {z : |z − ω| ⩽ δ}. Since ∂C, the boundary of C is a compact set, |P_{A}(w)|, |P_{A}(w^{−1} )| are continuous on ∂C, there is w_{min} where |P_{A}(w)| attains its minimum, and w_{max} where |P_{A}(w^{−1} )| attains its maximum on ∂C. Since \frac{1}{2}|P_{A}(w_{max}^{−1} )| is constant and |ω| − δ > 1, there is n_{0} such that \frac{1}{2}|P_{A}(w_{min} )|(|ω| − δ)^{n_{0} −m} > \frac{1}{2}|P_{A}(w_{max}^{−1} )|. For n ⩾ n_{0}, let us denote f(w) = \frac{1}{2}P_{A}(w)w^{n−m} , g(w) = \frac{1}{2}P_{A}(w^{−1} )w^{−(n−m)} . This notation corresponds to Rouché’s theorem which we intend to use. We have to prove that |f(w)| > |g(w)| on ∂C. Since |w| ⩾ |ω| − δ > 1 we have on ∂C

|f(w)| = \frac{1}{2}|P_{A}(w)||w|^{n−m} ⩾ \frac{1}{2}|P_{A}(w_{min} )|(|ω| − δ)^{n_{0} −m}

> \frac{1}{2}|P_{A}(w_{max}^{−1} )| ⩾ \frac{1}{2}|P_{A}(w^{−1} )||w|^{−(n−m)} | = |g(w)|.

The conditions in Rouché’s theorem are thus satisfied. Consequently, since f(w) has root ω, we conclude that f(w)+g(w) has a root, say ω_{1}, inside the circle C. Clearly, since |ω_{1} −ω| < δ_{1}, if we denote ξ = \frac{1}{2}(ω_{1} +ω_{1}^{−1} ), we conclude |ξ − \frac{1}{2}(ω +ω^{−1} )| < ε. Finally, we conclude that

T_{n,A} (ξ) = \frac{1}{2}P_{A}(ω_{1})ω_{1}^{n−m} + \frac{1}{2}P_{A}(ω_{1}^{−1} )ω_{1}^{−(n−m)} = f(ω_{1}) + g(ω_{1}) = 0.  □

Theorem 2.2. If x ∈ [−1, 1], then for every real number ε > 0, there exists a natural number n_{0} such that for all n > n_{0}, there is a root ξ of the A-Chebyshev polynomial T_{n,A} (x) such that |x − ξ| < ε.

Proof. Directly from the definitions of Chebyshev and A-Chebyshev polynomials, we can show that

(2.1) T_{n,A} (x) = a_{0} cosnθ + a_{1} cos(n − 1)θ + ⋯+ a_{m} cos(n − m)θ, n ⩾ m,

when x = cosθ. Since

a_{k} cos(n − k)θ = a_{k} cos(n − m + m − k)θ
= a_{k}(cos(n − m)θ cos(m − k)θ − sin(n − m)θ sin(m − k)θ)

the equation T_{n,A} (x) = 0 is equivalent with

cos(n − m)θ ∑_{k=0}^{m} a_{k} cos(m − k)θ = sin(n − m)θ ∑_{k=0}^{m} a_{k} sin(m − k)θ.

Finally we get

tan(n − m)θ = \frac{∑ _{k=0}^{m} a_{k} cos(m − k)θ}{∑ _{k=0}^{m} a_{k} sin(m − k)θ} .

The function on the right-hand side, denote it R(θ), does not depend on n. The graph of tan(n − m)θ consists of parallel equispaced tangents branches. So if we take n := 2n−m, we double n−m and get a new graph which is actually the union of the old one with branches settled in the middle of each pair of neighbouring branches of the old graph. We conclude that all roots of tan(n − m)θ = R(θ), remain to be roots of tan2(n − m)θ = R(θ), and new roots interlace with old. Finally, changing variables θ = arccosx will preserve order and denseness of the roots. □

3. Envelope of an A-Chebyshev polynomial

Let us observe the Chebishev polynomial T_{n}(x) again. It is well known that, for any n, the graph of the polynomial oscillates between −1 and 1 when x ∈ [−1, 1]. As n increases, we have more and more oscillations. Something like that we have in the case of an A-Chebyshev polynomial.

Example 3.1. Let A = (1, 0, 1), so

T_{n,A} (x) = T_{n}(x) + T_{n−2} (x) = 2xT_{n−1} (x) − T_{n−2} (x) + T_{n−2} (x) = 2xT_{n−1} (x).

Now it is obvious that T_{n,A} (x) oscillates between the lines y = ±2x, for x ∈ [−1, 1]. We will refer to these lines as an envelope of the A-Chebyshev polynomial.

Using (2.1) we can study the following

Example 3.2. Let A = (1, 0, −1), so

T_{n,A} (x)= cosnθ − cos((n − 2)θ) = −2 sin((n − 1)θ) sinθ
= −2 sin((n − 1)θ)\sqrt{1 − cos^{2} θ} = −2 sin((n − 1)θ)\sqrt{1 − x^{2}} .

Now it is obvious that T_{n,A} (x) oscillates between the upper and lower halves of the ellipse y = ±2\sqrt{1 − x^{2}} , for x ∈ [−1, 1]. These halves constitute the envelope of the A-Chebyshev polynomial in this case.

With the same technique we can find the envelope in the next

Example 3.3. Let A = (1, −1), so

T_{n,A} (x)= cosnθ − cos((n − 1)θ) = −2 sin((n − \frac{1}{2})θ) sin \frac{θ}{2}
= −2 sin((n − \frac{1}{2})θ)\sqrt{(1 − cosθ)/2} = −\sqrt{2}sin((n − \frac{1}{2})θ)\sqrt{1 − x} .

Now it is obvious that the envelope of the A-Chebyshev polynomial is a parabola y = ±\sqrt{2}\sqrt{1 − x} , for x ∈ [−1, 1].

Using previous examples, we can formulate the characteristics that an envelope of the A-Chebyshev polynomial must have.

(Env1) The envelope depends only on A. If A is fixed, it is unique for T_{n,A} (x),

n ∈ ℕ. (Env2) The envelope is a non negative function. (Env3) The A-Chebyshev polynomial is not greater in modulus than the envelope,

x ∈ [−1, 1]. (Env4) The envelope is a smooth function except at its zeros. (Env5) If the envelope and the A-Chebyshev polynomial have equal positive value

in x, they have also equal the first derivative in x.

We define the envelope of an A-Chebyshev polynomial as a function satisfying (Env1)–(Env5). It is naturally to ask how we can find the envelope for an A-Chebyshev polynomial. The next lemma will be useful.

Lemma 3.1. Let R(t), I(t) be real differentiable functions of real argument, with R′(t), I′(t) continuous, E(t) = \sqrt{R^{2}(t) + I^{2}(t)} , t ∈ ℝ. Then

(i) |R(t)| ⩽ E(t), (ii) |R(t)| = E(t) if and only if I(t) = 0, (iii) if I(t) = 0 and R(t) > 0, then R(t) = E(t) and R′(t) = E′(t).

Proof. The first and the second statements are straightforward. To demonstrate that R′(t) = E′(t), we need to determinate E′(t) = \frac{2R(t)R′(t)+2I(t)I′(t)}{2\sqrt{R^{2} (t)+I^{2} (t)}} . Using

I(t) = 0, R(t) > 0 we get the claim. □

Theorem 3.1. The envelope E_{A}(x) for an A-Chebyshev polynomial T_{n,A} (x) is the square root of the modulus of

∑_{i=0}^{m} a_{i}^{2} + 2 ∑_{i=0}^{m−1} a_{i}a_{i+1} T_{1}(x) + 2 ∑_{i=0}^{m−2} a_{i}a_{i+2} T_{2}(x) + ⋯

⋯+ 2 ∑_{i=0}^{m−k} a_{i}a_{i+k} T_{k}(x) + ⋯+ 2a_{0}a_{m}T_{m}(x),

or in more compact form

E_{A}(x) = | ∑_{i=0}^{m} ∑_{k=0}^{m} a_{i}a_{k}T_{|i−k|} (x)|^{1/2} .

Proof. Let z_{A}(t) = a_{0} cos(nt) + a_{1} cos((n − 1)t) + ⋯+ a_{m} cos((n − m)t) +

i(a_{0} sin(nt) + a_{1} sin((n − 1)t) + ⋯+ a_{m} sin((n − m)t)) be an auxiliary function on t ∈ ℝ. We can see that T_{n,A} (x) = Re(z_{A}(t)) so |T_{n,A} (x)|^{2} ⩽ |z_{A}(t)|^{2}, x = cos(t). We will show that E_{A}(x) = |z(t)|.

|z_{A}(t)|^{2} = ( ∑_{k=0}^{m} a_{k} cos((n − k)t)) ^{2} + ( ∑_{k=0}^{m} a_{k} sin((n − k)t)) ^{2}

= ∑_{i=0}^{m} ∑_{k=0}^{m} a_{i}a_{k} cos((n − i)t) cos((n − k)t)

+ ∑_{i=0}^{m} ∑_{k=0}^{m} a_{i}a_{k} sin((n − i)t) sin((n − k)t)

= ∑_{i=0}^{m} ∑_{k=0}^{m} a_{i}a_{k}(cos((n − i)t) cos((n − k)t) + sin((n − i)t) sin((n − k)t))

= ∑_{i=0}^{m} ∑_{k=0}^{m} a_{i}a_{k} cos((i − k)t).

Substituting x = cos(t) in cos((i − k)t) we get T_{|i−k|} (x). Since E_{A}(x) does not depend on n, (Env1) is fulfilled. (Env2), (Env3), (Env4) are straightforward. Using the previous lemma if R(t) = Re(z(t)), I(t) = Im(z(t)), we can easily obtain (Env5). □

It is useful to calculate the envelope of the A-Chebyshev polynomial for m = 1, 2, 3, 4. Actually, we give the calculation of the square of the envelope, to avoid cumbersome square roots. Using the previous formula we get (m =1) a_{0}^{2} + a_{1}^{2} + 2a_{0}a_{1}x; (m = 2) a_{0}^{2} + a_{1}^{2} + a_{2}^{2} + 2(a_{0}a_{1} + a_{1}a_{2})x + 2a_{0}a_{2}(2x^{2} − 1) =

a_{0}^{2} + a_{1}^{2} + a_{2}^{2} − 2a_{0}a_{2} + (2a_{0}a_{1} + 2a_{1}a_{2})x + 4a_{0}a_{2}x^{2}; (m =3) a_{0}^{2} + a_{1}^{2} + a_{2}^{2} + a_{3}^{2} + 2(a_{0}a_{1} + a_{1}a_{2} + a_{2}a_{3})x

+2(a_{0}a_{2} + a_{1}a_{3})(2x^{2} − 1) + 2a_{0}a_{3}(4x^{3} − 3x) =

a_{0}^{2} + a_{1}^{2} + a_{2}^{2} + a_{3}^{2} − 2a_{0}a_{2} − 2a_{1}a_{3} + (2a_{0}a_{1} + 2a_{1}a_{2} + 2a_{2}a_{3} − 6a_{0}a_{3})x +
(4a_{0}a_{2} + 4a_{1}a_{3})x^{2} + 8a_{0}a_{3}x^{3};

(m = 4) a_{0}^{2} + a_{1}^{2} + a_{2}^{2} + a_{3}^{2} + a_{4}^{2} + 2(a_{0}a_{1} + a_{1}a_{2} + a_{2}a_{3} + a_{3}a_{4})x + 2(a_{0}a_{2} + a_{1}a_{3} + a_{2}a_{4})(2x^{2} − 1) + 2(a_{0}a_{3} + a_{1}a_{4})(4x^{3} − 3x) + 2a_{0}a_{4}(8x^{4} − 8x^{2} + 1) =

a_{0}^{2} + a_{1}^{2} + a_{2}^{2} + a_{3}^{2} + a_{4}^{2} − 2a_{0}a_{2} − 2a_{1}a_{3} − 2a_{2}a_{4} + (2a_{0}a_{1} + 2a_{1}a_{2} + 2a_{2}a_{3} +
a_{3}a_{4} − 6a_{0}a_{3} − 6a_{1}a_{4})x + (4a_{0}a_{2} + 4a_{1}a_{3} + 4a_{2}a_{4} − 16a_{0}a_{4})x^{2} + (8a_{0}a_{3} +
8a_{1}a_{4})x^{3} + 16a_{0}a_{4}x^{4}.

Remark 3.1. There is a relation with the theory of signal processing. The analytic signal z(t) can be expressed in terms of complex polar coordinates, z(t) = f(t) + if̂(t) = A(t)e^{iϕ(t)} where A(t) = (f^{2}(t) + f̂^{2} (t))^{1/2} , and ϕ(t) = arctan \frac{f̂(t)}{f(t)} . These functions are respectively called the amplitude envelope and instantaneous phase of the signal, f̂(t) is the Hilbert transform of f(t).

4. The relation between the envelope and the characteristic polynomial

Until now we have used the envelope to describe the graph of the A-Chebyshev polynomial if x is of modulus not greater than 1. If |x| > 1 we have preferred the characteristic polynomial P_{A}(x). It is natural to ask, if there is any relation between the envelope and the characteristic polynomial of the A-Chebyshev polynomial. The following theorem shows that the answer is affirmative.

Theorem 4.1. The envelope of the A-Chebyshev polynomial is the function

E_{A}(x) = |P_{A}(x + \sqrt{x^{2} − 1} )P_{A}(x − \sqrt{x^{2} − 1} )|^{1/2} .

Proof. We shall start from the compact form of the envelope given in Theorem 3.1 and use well known [5] formula T_{n}(x) = \frac{1}{2}(w^{n} + w^{−n} ) where w = x + \sqrt{x^{2} − 1} .

E_{A}(x) = | ∑_{i=0}^{m} ∑_{k=0}^{m} a_{i}a_{k}T_{|i−k|} (x)|^{1/2} = | ∑_{i=0}^{m} ∑_{k=0}^{m} a_{i}a_{k} \frac{1}{2}(w^{i−k} + w^{−(i−k)} )|^{1/2}

= | ∑_{K=0}^{m} ∑_{I=0}^{m} \frac{1}{2}a_{K} a_{I} w^{K−I} + ∑_{i=0}^{m} ∑_{k=0}^{m} \frac{1}{2}a_{i}a_{k}w^{k−i} |^{1/2}

(Here we renamed i with K and k with I in the first double sum. Now we shall switch the order of summing in the first double sum and apply obvious a_{K} a_{I} = a_{I} a_{K}.)

= | ∑_{I=0}^{m} ∑_{K=0}^{m} \frac{1}{2}a_{I} a_{K} w^{K−I} + ∑_{i=0}^{m} ∑_{k=0}^{m} \frac{1}{2}a_{i}a_{k}w^{k−i} |^{1/2}

= |2 ∑_{i=0}^{m} ∑_{k=0}^{m} \frac{1}{2}a_{i}a_{k}w^{k−i} |^{1/2} = | ∑_{i=0}^{m} ∑_{k=0}^{m} a_{i}a_{k}w^{m−i} w^{−m+k} |^{1/2}

= | ∑_{i=0}^{m} a_{i}w^{m−i} ∑_{k=0}^{m} a_{k}w^{−m+k} |^{1/2}

= |P_{A}(w)P_{A}(w^{−1} )|^{1/2} = |P_{A}(x + \sqrt{x^{2} − 1} )P_{A}(x − \sqrt{x^{2} − 1} )|^{1/2} .  □

Graphs of A-Chebyshev polynomials T_{14,(1,0,0,1)} (x), T_{44,(1,0,0,1)} (x) are showed on Figure 1 together with their common envelope E(x) = \sqrt{|2 + 6x − 8x^{3}|} .

Figure 1. Graphs of A-Chebyshev polynomials T_{14,(1,0,0,1)} (x), T_{44,(1,0,0,1)} (x) and their common envelope ±|2 + 6x − 8x^{3}|^{1/2} .

5. A-Chebyshev polynomial of the second kind

It is well known [6] that the Chebyshev polynomial U_{n}(x) of the second kind is a polynomial in x of degree n, defined by

U_{n}(x) = \frac{sin(n + 1)θ}{sinθ} when x = cosθ.

Let A = (a_{0}, a_{1}, …, a_{m}) be a (m+1)-tuple of real numbers, a_{0}, a_{m} ≠ 0, m ⩾ 1. We introduce an infinite sequence of polynomials

U_{n,A} (x) = a_{0}U_{n}(x) + a_{1}U_{n−1} (x) + ⋯+ a_{m}U_{n−m} (x) (n ⩾ m).

We will refer to U_{n,A} (x) as an A-Chebyshev polynomial of the second kind. We can naturally extend this definition in the case m = 0 and A = a_{0} ≠ 0:

U_{n,a0} (x) = a_{0}U_{n}(x).

We will refer to the polynomial

P_{A}(x) = a_{0}x^{m} + a_{1}x^{m−1} + ⋯+ a_{m}

as the characteristic polynomial of the A-Chebyshev polynomial.

Lemma 5.1. U_{n,A} (x) = \frac{1}{w−w^{−1}} (w^{n+1−m} P_{A}(w) − w^{−n−1+m} P_{A}(w^{−1} )) where w = x + \sqrt{x^{2} − 1} .

Proof. Starting from the definition of A-Chebyshev polynomial and using well known [5] formula U_{n}(x) = \frac{w^{n+1} −w^{−n−1}}{w−w^{−1}} we have

U_{n,A} (x) = ∑_{i=0}^{m} a_{i}U_{n−i} (x) = ∑_{i=0}^{m} a_{i} \frac{w^{n+1−i} − w^{−n−1+i}}{w − w^{−1}}

= \frac{1}{w − w^{−1}} ( ∑_{i=0}^{m} a_{i}w^{n+1−i} − ∑_{i=0}^{m} a_{i}w^{−n−1+i} )

= \frac{1}{w − w^{−1}} ( w^{n+1−m} ∑_{i=0}^{m} a_{i}w^{m−i} − w^{−n−1+m} ∑_{i=0}^{m} a_{i}w^{−m+i} )

= \frac{1}{w − w^{−1}} (w^{n+1−m} P_{A}(w) − w^{−n−1+m} P_{A}(w^{−1} )).  □

Theorem 5.1. If there is a root ω, out of the unit circle, of the polynomial P_{A}, that is P_{A}(ω) = 0, |ω| > 1, then for every real number ε > 0, there exists a natural number n_{0} such that for all n > n_{0}, there is a root ξ of the A-Chebyshev polynomial of the second kind U_{n,A} (x) such that |ξ − \frac{1}{2}(ω + ω^{−1} )| < ε.

Proof. It is convenient to use the previous lemma to express U_{n,A} (x) = \frac{1}{w−w^{−1}} (w^{n+1−m} P_{A}(w) − w^{−n−1+m} P_{A}(w^{−1} )) where w = x + \sqrt{x^{2} − 1} or equivalently x = x(w) = \frac{1}{2}(w + w^{−1} ). Since x(w) is continuous for w > 0, there is δ_{1} > 0 such that if |w − ω| < δ_{1}, then |\frac{1}{2}(w + w^{−1} ) − \frac{1}{2}(ω + ω^{−1} )| < ε. We can take an δ_{2} < |ω| − 1 such that, in the circle {z : |z − ω| ⩽ δ_{2}}, there is no root of P_{A}(w) which is different from ω. Let δ = min(δ_{1}, δ_{2}) and C = {z : |z − ω| ⩽ δ}. Since ∂C, the boundary of C, is a compact set, |P_{A}(w)|, |P_{A}(w^{−1} )| are continuous on ∂C, there is w_{min} where |P_{A}(w)| gets its minimum and w_{max} where |P_{A}(w^{−1} )| gets its maximum on ∂C. Since \frac{1}{w−w^{−1}} |P_{A}(w_{max}^{−1} )| is constant and |ω| − δ > 1, there is n_{0} such that

\frac{1}{w − w^{−1}} |P_{A}(w_{min} )|(|ω| − δ)^{n_{0} +1−m} > \frac{1}{w − w^{−1}} |P_{A}(w_{max}^{−1} )|.

For n ⩾ n_{0} let us denote

f(w) = \frac{1}{w − w^{−1}} w^{n+1−m} P_{A}(w), g(w) = \frac{−1}{w − w^{−1}} w^{−n−1+m} P_{A}(w^{−1} ).

This notation corresponds to Rouché’s theorem which we intend to use. We have to prove that |f(w)| > |g(w)| on ∂C. Since |w| ⩾ |ω| − δ > 1 we have on ∂C:

|f(w)| = \frac{1}{w − w^{−1}} |P_{A}(w)||w|^{n+1−m} ⩾ \frac{1}{w − w^{−1}} |P_{A}(w_{min} )|(|ω| − δ)^{n_{0} +1−m}

> \frac{1}{w − w^{−1}} |P_{A}(w_{max}^{−1} )| ⩾ \frac{1}{w − w^{−1}} |P_{A}(w^{−1} )||w|^{−(n+1−m)} | = |g(w)|.

The conditions in Rouché’s theorem are thus satisfied. Consequently, since f(w) has root ω, we conclude that f(w) + g(w) has a root, let it be ω_{1}, inside the circle C. Clearly, since |ω_{1} − ω| < δ_{1}, if we denote ξ = \frac{1}{2}(ω_{1} + ω_{1}^{−1} ), we conclude |ξ − \frac{1}{2}(ω + ω^{−1} )| < ε. Finally, we conclude that

U_{n,A} (ξ) = \frac{1}{w − w^{−1}} P_{A}(ω_{1})ω_{1}^{n+1−m} − \frac{1}{w − w^{−1}} P_{A}(ω_{1}^{−1} )ω_{1}^{−(n+1−m)}

= f(ω_{1}) + g(ω_{1}) = 0 .  □

6. An application in number theory

Recall that q > 1 is a Pisot number if q is an algebraic integer, whose other conjugates are of modulus strictly less than 1. Salem proved that every Pisot number is a limit point of the set T of Salem numbers. Let Q(x) = x^{m}P (\frac{1}{x}) = a_{0} + a_{1}x + ⋯+ a_{m}x^{m} be the reciprocal polynomial of the polynomial P (x) = a_{0}x^{m} + a_{1}x^{m−1} + ⋯+ a_{m}. Salem showed that if P (x) is the minimal polynomial of a Pisot number q, then R_{k}(x) = x^{k}P (x) + Q(x) is a polynomial with a root τ_{k} that is a Salem number, and the limit of the sequence τ_{k} is q, k → ∞. There is a relation between the Salem sequence R_{k}(x) and the A-Chebyshev polynomials T_{n,A} (x), the characteristic polynomial of which is P (x). We have seen that

T_{n,A} (\frac{1}{2}(w + w^{−1} )) = T_{n,A} (x) = ∑_{i=0}^{m} a_{i}T_{n−i} (x) = ∑_{i=0}^{m} a_{i} \frac{1}{2}(w^{n−i} + w^{−n+i} ).

Now we can show that 2w^{n}T_{n,A} (\frac{1}{2}(w + w^{−1} )) = R_{2n−m} (w). Really, we obtain

2w^{n}T_{n,A} (\frac{1}{2}(w + w^{−1} )) = ∑_{i=0}^{m} a_{i}(w^{2n−i} + w^{i})

= w^{2n−m} ∑_{i=0}^{m} a_{i}(w^{m−i} ) + ∑_{i=0}^{m} a_{i}(w^{i}) = w^{2n−m} P (w) + Q(w).

The question is what will happen if we use A-Chebyshev polynomial of the second kind instead of T_{n,A} (x). We will demonstrate that one more sequence of Salem numbers, which converges to the Pisot number q, appears. It is obvious that

U_{n,A} (\frac{1}{2}(w + w^{−1} )) = U_{n,A} (x) =∑_{i=0}^{m} a_{i}U_{n−i} (x) =∑_{i=0}^{m} \frac{a_{i}}{w − w^{−1}} (w^{n+1−i} − w^{−n−1+i} )

= ∑_{i=0}^{m} a_{i}(w^{n−i} + w^{n−i−2} + w^{n−i−4} + ⋯+ w^{−n+i+2} + w^{−n+i} ).

We claim that S_{2n} (w) = w^{n}U_{n,A} (\frac{1}{2}(w + w^{−1} )) is a polynomial of degree 2n with a root τ_{2n} that is a Salem number, and the limit of the sequence τ_{2n} is q, n → ∞. Using the previous theorem it is clear that there is a root τ_{2n} of S_{2n} (w) such that τ_{2n} → q. It is obvious that S_{2n} (w) is a reciprocal polynomial. It remains to be proved that all other roots of S_{2n} (w) are in the unit circle. We shall apply the method Salem (communicated by Hirschman) used to prove the same property of his sequence R_{k}(x) [7]. Using Lemma 5.1 we have

S_{2n} (w) = w^{n}U_{n,A} (\frac{1}{2}(w + w^{−1} ))

= \frac{w^{n}}{w − w^{−1}} (w^{n+1−m} P_{A}(w) − w^{−n−1+m} P_{A}(w^{−1} ))

= \frac{w^{n+1}}{w^{2} − 1} (w^{n+1−m} P_{A}(w) − w^{−n−1+m} P_{A}(w^{−1} ))

= \frac{1}{w^{2} − 1} (w^{2n+2−m} P_{A}(w) − w^{m}P_{A}(w^{−1} ))

= \frac{1}{w^{2} − 1} (w^{2n+2−m} P_{A}(w) − Q(w)).

We denote by ϵ a positive number and consider the equation

(1 + ϵ)w^{2n+2−m} P_{A}(w) − Q(w) = 0.

Since for |w| = 1, we have |P (w)| = |Q(w)|, it follows by Rouché’s theorem that inside the circle |w| = 1 the number of roots of the last equation is equal to the number of roots of w^{2n+2−m} P (w), that is, (2n + 2 − m) + m − 1. As ϵ → 0, these roots vary continuously. Hence, for ϵ = 0 we have 2n + 1 roots with modulus ⩽ 1. It is obvious that two roots are 1, −1 so the fraction can be reduced with w^{2} − 1. Finally we conclude that at most one root of S_{2n} (w) is outside the unit circle.

Example 6.1. Let q be the golden ratio, the greater root of P (x) = x^{2} − x − 1. Then R_{k}(w) = w^{k+2} −w^{k+1} −w^{k}−w^{2}−w+1 is the Salem’s sequence. Our sequence of Salem numbers τ_{2m} which converge to q consists of the greatest in modulus roots of the polynomials

S_{2n} (w) = \frac{1}{w^{2}−1} (w^{2n+2−m} P_{A}(w) − Q(w)) = \frac{1}{w^{2}−1} (w^{2n} (w^{2}− w −1) + w^{2}+ w −1)

= (w^{2n} + w^{2n−2} + w^{2n−4} +⋯ + w^{2}+1) − (w^{2n−1} + w^{2n−3} + w^{2n−5} + ⋯ +w^{3}+ w)

− (w^{2n−2} + w^{2n−4} + w^{2n−6} +⋯ + w^{4}+ w^{2}) .

Finally

S_{2n} (w) = w^{2n} − (w^{2n−1} + w^{2n−3} + w^{2n−5} ⋯+ w^{3} + w) + 1.

References

1. H. Bavinck, On the zeros of certain linear combinations of Chebyshev polynomials, J. Comput. Appl. Math. 65 (1995), 19–26.

2. M.-J. Bertin, A. Decomps-Guilloux, M. Grandet-Hugot, M. Pathiaux-Delefosse, J.-P. Schreiber, Pisot and Salem Numbers, Birkhäuser, Basel, 1992.

3. J. Cigler, A simple approach to q-Chebyshev polynomials, arXiv:1201.4703.

4. A. Dubickas, Totally real algebraic integers in small intervals, Lith. Math. J. 40 (2000), 236– 240.

5. J. C. Mason, D. C. Handscomb, Chebyshev Polynomials, Rhapman and Hall – CRC, Boca Raton, London, New York, Washington D.C., 2003.

6. T. J. Rivlin, The Chebyshev Polynomials, Wiley, New York, 1974.

7. R. Salem, Algebraic Numbers and Fourier Analysis, Heath, Boston, 1963.

8. D. Stankov, A class of discrete spectra of non-Pisot numbers, Publ. Inst. Math., Nouv. Sér. 83(97) (2008), 9–14.

9. , On spectra of neither Pisot nor Salem algebraic integers, Monatsh. Math. 159 (2010), 115–131.

Katedra Matematike RGF-a (Received 18 03 2014) University of Belgrade (Revised 25 02 2015) Belgrade, Ðušina 7 Serbia

    dstankov@rgf.bg.ac.rs