All Questions
Tagged with chebyshev-polynomials roots
13
questions
4
votes
2
answers
73
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Roots and extrema of the polynomial $P_n(x)=\sum_{k=0}^n\binom{n+k}{2k}(-x)^k$.
Answering a recent question I came across the family of polynomials:
$$P_n(x)=\sum_{k=0}^n\binom{n+k}{2k}(-x)^k$$
with numerical evidence of the following interesting properties:
$P_n(2)=\begin{cases}...
- 26.7k
0
votes
1
answer
57
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Largest root of a linear combination of Chebyshev polynomials
I wonder if we can say something about roots of a linear combination of Chebyshev polynomials of the first kind. I have an example in my hand: $$(m+1)T_n(x)+(m-3)T_{n-2}(x)=0$$ for some $m>0$. I ...
- 433
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1
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363
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Complex argument in Chebyshev polynomials of second kind?
I am looking at Chebyshev polynomials of second kind in order to characterize the spectra of $2$-Toeplitz perturbed matrices (I am not a mathematician myself, just a control theoretician). In all the ...
3
votes
1
answer
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Relationship between Chebyshev polynomials and square roots ($\sqrt{3}+\sqrt{2}=\frac{1}{\sqrt{T_1(5)-\sqrt{T_1(5)^2-1}}}$ etc.)
(If my English is strange, I would appreciate it if you could correct it.)
There seems to be a property about the sum of square roots. (This is almost self-explanatory.)
let $ a,\ b,\ t \in \mathbb{N}^...
- 426
3
votes
3
answers
121
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Proving that $\sec\frac\pi{30}=\sqrt{2-\sqrt{5}+\sqrt{15-6\sqrt{5}}}$
I recently saw on this site, the identity
$$\sec\frac\pi{30}=\sqrt{2-\sqrt{5}+\sqrt{15-6\sqrt{5}}}$$
which I instantly wanted to prove.
I know that I can "reduce" the problem to the evaluation of $\...
- 17.3k
2
votes
0
answers
95
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Vieta's Formula for Chebyshev basis
Let $p(x)=x^d+\sum_{i=0}^{d-1} a_ix^i$. Then Vieta's formula tells us that the $a_i$ can be expressed as signed elementary symmetric polynomials of the roots $\{\alpha_1,\ldots,\alpha_d\}$ of $p(x)$: $...
- 161
3
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0
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101
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$\cos\frac\pi{n}$ Analytic expression
I recently found out that $$\sin\frac\pi5=\frac12\sqrt{\frac{5-\sqrt5}2}$$
Which means that $$\cos\frac\pi5=\frac{1+\sqrt5}4$$
I also recently found that if $n\in\Bbb N$,
$$\sin nx=\sin x\,U_{n-1}(\...
- 17.3k
2
votes
0
answers
40
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Numerical analysis: Chebyshev coefficient representation error.
I am unsure if numerical analysis questions are suitable for this forum, but I have nowhere else to ask, so if this question is inappropriate, tell me and I will delete it.
If $x_k$ are the Chebyshev ...
- 1,077
0
votes
1
answer
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Solving $x+y+z+u+t=0$, $x^3+y^3+z^3+u^3+t^3=0$, $x^5+y^5+z^5+u^5+t^5=-10$
Find all real numbers $x$, $y$, $z$, $t$, $u$ that
$x+y+z+u+t=0$
$x^3+y^3+z^3+u^3+t^3=0$
$x^5+y^5+z^5+u^5+t^5=-10$
I'm learning about Chebyshev polynominals but in this case, I still haven't got ...
- 155
7
votes
3
answers
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Roots of the Chebyshev polynomials of the second kind.
It is known that the roots of the Chebyshev polynomials of the second kind, denote it by $U_n(x)$, are in the interval $(-1,1)$ and they are simple (of multiplicity one). I have noticed that the roots ...
- 1,849
2
votes
1
answer
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Building a cubic function with integer coefficients and trigonometric roots
I want to find the answer to the following problem: Construct a cubic polynomial with integer coefficients, whose roots - $\cos{\frac{2 \pi}{7}}$, $\cos{\frac{4 \pi}{7}}$ and $\cos{\frac{6 \pi}{7}}$.
...
- 21
4
votes
1
answer
120
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Is it possible to calculate the roots of the difference between successive terms of this polynomial series $\rm{P}_n (x)=x\rm{P}_{n-1}-r\rm{P}_{n-2}$
Consider the polynomial series defined by the following recursion formula:
$$
\begin{align}
&\mathrm{P}_0 = 1 \\
&\mathrm{P}_1 = x-r \\
&\mathrm{P}_n = x\mathrm{P}_{n-1} - r\mathrm{P}_{n-...
- 493
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votes
1
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Calculation of Chebyshev coefficients
The Chebyshev polynomials can be defined recursively as:
$T_0(x)=1$;
$T_1(x)=x$;
$T_{n+1}(x)=2xT_n(x) + T_{n-1}(x)$
The coefficients of these polynomails for a function, $\space f(x)$, under ...
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