R E S E A R C H
Some properties of Chebyshev polynomials
firstname.lastname@example.org Department of Mathematics, Sookmyung Women’s University, Seoul, 140-742, Korea
In this paper we obtain some new bounds for Chebyshev polynomials and their analogues. They lead to the results about zero distributions of certain sums of Chebyshev polynomials and their analogues. Also we get an interesting property about the integrals of certain sums of Chebyshev polynomials.
Keywords: Chebyshev polynomials; bounds; sums; zeros
LetTn(x) andUn(x) be the Chebyshev polynomials of the ﬁrst kind and of the second kind, respectively. These polynomials satisfy the recurrence relations
T(x) = , T(x) =x, Tn+(x) = xTn(x) –Tn–(x) (n≥)
U(x) = , U(x) = x, Un+(x) = xUn(x) –Un–(x) (n≥).
Chebyshev polynomials are of great importance in many areas of mathematics, particu-larly approximation theory. Many papers and books [, ] have been written about these polynomials. Chebyshev polynomials deﬁned on [–, ] are well understood, but the poly-nomials of complex arguments are less so. Reported here are several bounds for Cheby-shev polynomials deﬁned onCincluding zero distributions of certain sums of Chebyshev polynomials. Moreover, we will introduce certain analogues of Chebyshev polynomials and study their properties. Also we get an interesting property about the integrals of cer-tain sums of Chebyshev polynomials.
Other generalized Chebyshev polynomials (known as Shabat polynomials) have been introduced in  and they are studied in the theory of graphs on surfaces and curves over number ﬁelds. For a survey in this area, see .
Forn≥ and –> +≥,i.e., ≤< /, we let
Tn,(z) := ( +)
Un,(z) := ( –)
Tn() =Un() =
⎧ ⎨ ⎩
(–)n/ ifnis even,
T,(z) =U,(z) =T(z) =U(z) = ,
and it is easy to show that for an odd integern,
Tn+,(z) = zTn,(z) –Tn–,(z),
Un+,(z) = zUn,(z) –Un–,(z).
SinceTn,(z) =Tn(z), we may apply results aboutTn,(z) to those aboutTn(z). ButUn,(z) =
Un(z) andwas a nonnegative real number less than /, and so properties ofUn(z) will be investigated separately from those ofUn,(z).
2 New results
In this section we list some new results related to the bounds of Chebyshev polynomials
Tn(z),Un(z) and their analoguesTn,(z),Un,(z) deﬁned onCincluding zero distributions
of certain sums of Chebyshev polynomials and their analogues. And we will get an inter-esting property about the integrals of certain sums of Chebyshev polynomials. We ﬁrst begin with properties about bounds ofTn(z),Un(z),Tn,(z) andUn,(z). We may compute
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
( +)Tn(z) ifnis odd,
( +)Tn(z) – ifnis even and |n,
( +)Tn(z) + ifnis even and n,
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
( –)Un(z) ifnis odd,
( –)Un(z) – + ifnis even and |n,
( –)Un(z) + – ifnis even and n.
Proposition Suppose that z is a complex number satisfying|z| ≥. Then for n≥,
⎧ ⎨ ⎩
+ if n is odd,
if n is even. ()
Remarks For a complex numberzwith|z| ≥, we may follow the procedure of the proof of () to obtain
that is best possible since|Un()|=n+ . There seem to be larger lower bounds than for
|Un(z)|–|Tn(z)|in (). First, we observe that
because|Un()|–|Tn()|= (n+ ) – =n. Deciding in some general situations exactly where
the minimum occurs seems to be extremely diﬃcult. For example, machine calculation suggests that forn= ,|U(z)|–|T(z)|takes its minimum . . . . in|z| ≥ at four modulus roots±. . . .±i. . . . of the polynomial
x– x+ x– x+ x
– x+ x– x+ .
But we may conjecture that, by numerical computations, the value
occurs in|z|= and lies betweenn– / andn, wheren– / can be replaced by something larger. We now ask naturally what the minimum is for|z|=t> . If one simply looks at the casez=t, it seems that|Un(z)|–|Tn(z)|is close to its minimum atz=t. But
is the coeﬃcient ofxn–in the power series expansion oft/( – tx+t). In fact,
– tx+x = – tx+x –
For|z| ≥,|Un(z)| ≥ |Tn(z)|+ by (). In the following proposition, we obtain an upper bound for arbitraryz=cosθ,θ∈/R.
Proposition Let z=cosθ, whereθ=α+iβandβ= . Then, for n≥,
Remarks With the same notations with Proposition , it follows from
that, for|z|large,|β|is large. Butcothβcothnβ> and
These imply that the upper bound
is greater than |Tn(z)|, but for|z|large, it is close to |Tn(z)|. Also by machine computa-tions (e.g.,n= andz= +i), we may check that the inequality () is sharp.
It is natural to ask about the bounds on the unit circle.
Proposition For|z|= , we have
n+ Un(z)≤Tn(z)<Un(z). ()
Remarks The right inequality in () will be shown in Section by using (). So obtaining a better lower bound than in () can improve this inequality. The left inequality in () is best possible in the sense that
n+ Un(±)=Tn(±)= .
For|z|= , it is easy to see
by (), and it seems to be true that
n+ Un,(z)≤Tn,(z). ()
The proof of () will be given in Section by using a well-known identityUn(z) =Tn(z) + zUn–(z). ButUn,(z) =Tn,(z) +zUn–,(z) does not hold. So we cannot use this to prove ()
if it is true.
All zeros of the polynomialTn(z) +Un(z) lie in (–, ). More generally the convex com-bination of Tn(z) andUn(z) has all its zeros in (–, ). This will be proved in Proposi-tion below. So one might ask: where are the zeros of polynomials likeTn(z) +zkUn(z)
orUn(z) +zkTn(z) around|z|= ? The next theorem answers this forTn(z) +zkUn(z).
Theorem Let P(z) :=Tn(z) +zkUn(z)for positive integers n and k. Then P
(z)has all its
zeros in|z|< . Furthermore, for k even, P(z)has at least n real zeros, and for k odd, P(z)
Remarks LetP(z) :=Un(z) +zkTn(z) for positive integersnandk. We can use the same
method as in the proof of Theorem to show thatP(z) has at leastnreal zeros in (–, ). Furthermore, forkeven, there is no real zero outside [–, ], and forkodd, there is one more real zero onz< –.
Proposition The polynomial
( –λ)Tn(z) +λUn(z) (≤λ≤)
has all zeros in(–, ).
Using analogues ofTn(z) andUn(z), we consider analogues ofP(z) and investigate their zero distributions. Deﬁne
P,(z) :=Tn,(z) +zkUn,(z).
Theorem P,(z)has all its zeros in|z|< .
Finally, we get an interesting property about the integrals of sums of Chebyshev polyno-mials. Observe that
⎧ ⎨ ⎩
(–)n/ ifnis even, ifnis odd
π· sum of the squares of all coeﬃcients ofTn(z) (orUn(z)).
For example, fromT(z) = z– z+ andU(z) = z– z+ z– we can calculate π
eiθdθ= π+ + = π, π
eiθdθ= π+ + = π
and we see that these two integrals are diﬀerent. But forP(z) =Tn(z) +zkUn(z) andP(z) =
Un(z) +zkTn(z), the integrals have the same value.
Proposition For z=eiθ,
Remark It seems to be true that forklarge,z=eiθ,
These remain open problems.
Proof of Proposition Suppose thatzis a complex number satisfying|z| ≥. Using (), for
n≥, we have
Then by recurrence,
Un(z)–Un–(z)≥U(z)–U(z)=|z|– ≥. ()
By () and the identity
=Un(z)– . ()
Next we prove the results aboutUn,(z) andTn,(z). Fornodd and |n– , it follows from
the deﬁnition ofUn,(z) and () that
Un,(z)–Un–,(z)=( –)Un(z)–( –)Un–(z) – ( –)
≥( –)Un(z)–( –)Un–(z)– ( –)
= ( –)Un(z)–Un–(z)– ( –)
for other three cases (i.e.,nodd and n– ,neven and |n– ,neven and n– ) can be proved in the same way. Finally, fornodd, by –> +and (), we have
Un,(z)–Tn,(z)= ( –)Un(z)– ( +)Tn(z)
> ( +)Un(z)–Tn(z)≥ +.
In the same way, we can check that forneven,
Proof of Proposition Using the identityUn(z) =Tn(z) +zUn–(z), forz=cosθ we have
UnTn((zz))= +zUn–(z) Tn(z)
=| +cotθtannθ| ≤ +|cotθtannθ|.
If we setθ=α+iβ,α,β∈R, then
Since¯z=cosθ¯, it suﬃces to consider the caseβ> . The above implies
αcosα+sinhβcoshβ (cosα+sinhβ) ≤
|tannθ| ≤cothnβ. ()
|cosθ| =cosαcoshβ+sinαsinhβ ≤sinα+cosαcoshβ,
Now we see with () and () that
Proof of Proposition Suppose thatzis a complex number satisfying|z|= . First, it follows from () that
Un(z) < .
Using the identityUn(z) =Tn(z) +zUn–(z), we have
UnTn((zz))= –zUn–(z) Un(z)
and so it is enough to show that
n n+ .
We use induction onn. Forn= ,UU((zz))=z=. Assume the result holds fork. Then
≥ – k
Proof of Theorem All zeros ofTn(z) andUn(z) are real and lie in (–, ) and for ≤k≤n,
cos(k– )π n
= , Un
For convenience, by removing ‘cos’ and the constantπ,cos(k–)π
n andcos kπ
n+can be iden-tiﬁed with the ascending chain of rational numbers (k–)n and k
n+, respectively. We may calculate that fornodd
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
k– n <
n , ≤k< n+
n = k+
n– , k–
n < k+
n , n–
<k<n– , k–
n < k+
and forneven ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
k– n <
n , ≤k< n
n < k n+<
n , k= n
n < k+
<k<n– , k–
n < k+
By using the above and denotinga zero ofTn(z),a zero ofzkUn(z), we see that, forn
even, all zeros between – and listed in increasing order are of the form
· · · · · · ,
where the center is the that is the zero ofzk. Fornodd, all zeros listed in increasing
order are of the form
· · · · · · ,
where the center means that all those three numbers,,arecosπ = and one comes from the zero ofzk. Now consider sign changes using the above two chains in
increasing order so that fornodd or even we may check that, ifkis even,P(z) has at least
nreal zeros in (–, ), and ifkis odd,P(z) has at leastn– real zeros in (–, ). On the other hand, the zeroszofP(z) satisfy
If|z| ≥, then|Tn(z)| ≥ |Un(z)|, which contradicts (). Thus all zeros ofP(z) lie in|z|< .
‘Bad pairs’ of polynomial zeros were deﬁned in . It is an easy consequence of Fell  that, if the all zeros ofTn(z) andUn(z) form ‘good pairs’, their convex combination has all its zeros real.
Proof of Proposition Following the proof of Theorem , we may see that forneven, all zerosTn(z) andUn(z) between – and listed in increasing order are of the form
· · · · · · .
Fornodd, all zeros listed in increasing order are of the form
· · · · · · ,
where the center means that both numbers, arecosπ = . Thus we can see that forneven, all zeros ofTn(z) andUn(z) form good pairs, and fornodd, all pairs from integral polynomialsTn(z)/zandUn(z)/zare good. It follows that, by Fell , all zeros of the convex combination are real and in (–, ).
Proof of Theorem The zeroszofP,(z) satisfy
If|z| ≥, then|Tn,(z)| ≥ |Un,(z)|, which contradicts ().
Proof of Proposition Using|z|=zz¯, we have
So we only need to show that
But this equality follows from just replacing the variableθ by –θ.
The author declares that they have no competing interests.
The author wishes to thank Professor Kenneth B. Stolarsky who let the author know some questions in this paper. The author is grateful to the referee of this paper for useful comments and suggestions that led to further development of an earlier version. This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2010-0011010).
Received: 26 October 2011 Accepted: 16 July 2012 Published: 31 July 2012 References
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