Difference between revisions of "Chebyshev polynomials of the first kind"
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===Rational roots=== | ===Rational roots=== | ||
Other than <math>1</math> and <math>-1</math>, any root of a Chebyshev polynomial of the first kind must be a root of either <math>W_n(x)</math> or <math>U_n(x)</math> for some <math>n</math>. | Other than <math>1</math> and <math>-1</math>, any root of a Chebyshev polynomial of the first kind must be a root of either <math>W_n(x)</math> or <math>U_n(x)</math> for some <math>n</math>. | ||
− | The polynomials <math>w_n(x) = W_n \left( \frac{x}{2} \right)</math> and <math>u_n(x) = U_n \left( \frac{x}{2} \right)</math> satisfy the recursive formulas <cmath>\begin{align*} w_0(x) &= 1 \\ w_1(x) &= x + 1 \\ w_{n+1}(x) &= xw_n(x) - w_{n-1}(x), \end{align*}</cmath> and <cmath>\begin{align*} u_0(x) &= 1 \\ u_1(x) &= x \\ u_{n+1}(x) &= xu_n(x) - u_{n-1}(x), \end{align*}</cmath> | + | The polynomials <math>w_n(x) = W_n \left( \frac{x}{2} \right)</math> and <math>u_n(x) = U_n \left( \frac{x}{2} \right)</math> satisfy the recursive formulas <cmath>\begin{align*} w_0(x) &= 1 \\ w_1(x) &= x + 1 \\ w_{n+1}(x) &= xw_n(x) - w_{n-1}(x), \end{align*}</cmath> and <cmath>\begin{align*} u_0(x) &= 1 \\ u_1(x) &= x \\ u_{n+1}(x) &= xu_n(x) - u_{n-1}(x), \end{align*}</cmath> ensuring that they have integer coefficients and are monic for all <math>n</math>. Thus, by the [[Rational Root Theorem]], any rational roots of <math>w_n(x)</math> or <math>u_n(x)</math> must be integers. The only possible integer roots of <math>w_n(x)</math> or <math>u_n(x)</math> are <math>-1</math>, <math>0</math>, and <math>1</math>, so the only possible rational roots of <math>W_n(x)</math> or <math>U_n(x)</math> are <math>-\frac{1}{2}</math>, <math>0</math>, or <math>\frac{1}{2}</math>. Thus, the only rational roots of <math>T_n(x) - 1</math> across all choices of <math>n</math> are <math>-1</math>, <math>-\frac{1}{2}</math>, <math>0</math>, <math>\frac{1}{2}</math>, and <math>1</math>. |
− | Because any value <math>\cos {a\pi}{b}</math> for integers <math>a</math> and <math>b</math> is a root of <math>T_b(x)</math>, these five are the only rational values of the cosine of a rational multiple of <math>\pi</math>. This result is known as <b>Niven's Theorem</b>. | + | Because any value <math>\cos \frac{a\pi}{b}</math> for integers <math>a</math> and <math>b</math> is a root of <math>T_b(x)</math>, these five are the only rational values of the cosine of a rational multiple of <math>\pi</math>. This result is known as <b>Niven's Theorem</b>. |
Revision as of 00:39, 3 March 2022
The Chebyshev polynomials of the first kind are defined recursively by or equivalently by
Contents
Proof of equivalence of the two definitions
In the proof below, will refer to the recursive definition.
For the base case,
for the
base case,
Now for the inductive step, let , so that
. We then assume that
and
, and we wish to prove that
.
From the cosine sum and difference identities we have and
The sum of these equations is
rearranging,
Substituting our assumptions yields
as desired.
Composition identity
For nonnegative integers and
, the identity
holds.
First proof
By the trigonometric definition, .
As before, let . We have
for some integer
. Multiplying by
and distributing gives
; taking the cosine gives
.
For now this proof only applies where the trigonometric definition is defined; that is, for . However,
is a degree-
polynomial, and so is
, so the fact that
for some
distinct
is sufficient to guarantee that the two polynomials are equal over all real numbers.
Second proof (Induction)
First we prove a lemma: that for all
. To prove this lemma, we fix
and induct on
.
For all , we have
and for all
,
proving the lemma for
and
respectively.
Suppose the lemma holds for and
; that is,
and
. Then multiplying the first equation by
and subtracting the second equation gives
which simplifies to
using the original recursive definition, as long as
. Thus, the lemma holds for
(as long as
), completing the inductive step.
To prove the claim, we now fix and induct on
.
For all , we have
and
proving the claim for
and
respectively.
Suppose the claim holds for and
; that is,
and
. We may also assume
, since the smaller cases have already been proven, in order to ensure that
. Then by the lemma (with
) and the original recursive definition,
completing the inductive step.
Roots
Since , and the values of
for which
are
for integers
, the roots of
are of the form
These roots are also called Chebyshev nodes.
Connection to roots of unity
Because cosine is only at integer multiples of
, the roots of the polynomial
follow the simpler formula
. The
th roots of unity have arguments of
and magnitude
, so the roots of
are the real parts of the
th roots of unity. This lends intuition to several patterns.
All roots of are also roots of
, since all
th roots of unity are also
th roots of unity. This can also be shown algebraically as follows: Suppose
. Then
using the composition identity and the fact that
for all
.
Particular cases include that , being a root of
, is a root of
for all positive
, and
, being a root of
, is a root of
for all positive even
. All other roots of
correspond to roots of unity which fall into conjugate pairs with the same real part. One might therefore suspect that all remaining roots of
are double roots. In fact,
and
for polynomials
and
satisfying the same recurrence relation as
, but with the different base cases
and
. The polynomials
are known as Chebyshev polynomials of the second kind.
Rational roots
Other than and
, any root of a Chebyshev polynomial of the first kind must be a root of either
or
for some
.
The polynomials
and
satisfy the recursive formulas
and
ensuring that they have integer coefficients and are monic for all
. Thus, by the Rational Root Theorem, any rational roots of
or
must be integers. The only possible integer roots of
or
are
,
, and
, so the only possible rational roots of
or
are
,
, or
. Thus, the only rational roots of
across all choices of
are
,
,
,
, and
.
Because any value for integers
and
is a root of
, these five are the only rational values of the cosine of a rational multiple of
. This result is known as Niven's Theorem.