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Inverse function of $U_{k-1}(\cos(\frac{\pi}{x}))$?
I'm trying to find the inverse function of $$U_{k-1}(\cos(\frac{\pi}{x}))=\sum_{n=0}^{\left\lfloor\frac{k-1}2\right\rfloor}\frac{(-1)^n \Gamma(k-n)}{n!\Gamma(k-2n)} \left(2\cos\left(\frac\pi...
View ArticleAn integral identity:...
I recently came across a parametric definite integral about Chebyshev polynomials:$$ f(a)= \begin{aligned}\begin{gather*}\int_{0}^{\frac{\pi}{2}}\dfrac{\cos^{a}x\sin(a+1)x}{\sin x}\...
View ArticleCoefficients of Chebyshev polynomials
Not long ago, I derived the formula for Chebyshev polynomials$$T_{n}\left( x\right)= \sum_{k=0}^{\lfloor \frac{n}{2} \rfloor}{n \choose 2k}x^{n-2k}\left( x^2-1\right)^{k}$$How to extract the...
View ArticleHow to solve equation involving Chebyshev polynomial ratio?:...
I have the following equation: involving the ratio of two Chebyshev polynomials:$$\frac{\cos\left(\frac{\cos^{-1}(ya)}{2N}\right)}{\cos\left(\frac{\cos^{-1}(y)}{2N}\right)}=x$$(for some reason, I am...
View ArticleShow that for any positive integers $i$ and $j$ with $ i > j$, we have $T_i...
Guys can you explain this demo to me step by step, I don't understand it at all.Show that for any positive integers $i$ and $j$ with $ i > j$, we have$$T_i (x)T_j(x)=...
View ArticleComplex argument in Chebyshev polynomials of second kind?
I am looking at Chebyshev polynomials of second kind in order to characterize the spectra of $2$-Toeplitz perturbed matrices (I am not a mathematician myself, just a control theoretician). In all the...
View ArticleSolving product of two cosine terms [closed]
I have an equation of $\cos Ax \cos Bx = c$ where $A$,$B$ and $c$ are known constants - how to solve for unknown $x$?
View ArticleExpansion of cosine function by shifted chebyshev polynomials
$f(t)=f_1(t)+f_2(t) $$w_1=1$, $w_2=6$$f_1(t)=A \cos(w_1 t)$ $f_2(t)=B \cos(w_2 t)$$A$ and $B$ are constant $2 \times 2$ matrixAfter normalizing the above equation by $t=T*y$ where $T$ is the principal...
View ArticleShow that $\max _{x \in[-1,1]}|P(x)| \geq \frac{1}{2^{n-1}}$ without...
For every monic polynomial $P$ of degree $n$ (with leading coefficient 1), it is well-known that$$\max _{x \in[-1,1]}|P(x)| \geq \frac{1}{2^{n-1}}.$$A standard proof uses Chebyshev polynomials.Is there...
View ArticleChebyhev polynomials and Primality Testing
It is a well known Theorem that an odd positive integer $n$ is prime if and only if$T_{n}(x) \equiv x^n \pmod{n}$, where $T_{n}(x)$ is the $n^{th}$ Chebyshev polynomial of the first kind.Do we deduce...
View ArticleCoefficient of $x^n$ in Legendre series expansion
Suppose we are approximating a function $f$ with a Legendre series of order $N$, namely$$f(x) \approx \sum_{n=0}^N c_n P_n(x) \equiv f_N(x)$$where $P_n(x)$ is the $n^{th}$ Legendre polynomial and$$c_n...
View ArticleCan the coefficients for the Fourier transform of the Chebyshev polynomials...
In an article by Fokas and Smitheman, it is shown, among others things, how the finite Fourier transform of the Chebyshev polynomials can be computed. They show that, when $T_{m}(x)$ is the $m$'th...
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How to derive the sum $ \sum_{k=1}^{n-1} \frac{1}{\cosh^2\left(\frac{\pi...
\begin{align*}\sum_{k=1}^{n-1} \frac{1}{\cosh^2\left(\frac{\pi k}{n}\right)}\end{align*}I tried to solve with mathematica that shows Does anyone know how to derive this and does it is possible for...
View ArticleA closed form for the coefficients of Chebyshev polynomials
The Chebyshev polynomials are defined recursively:$T_0(x)=1$;$T_1(x)=x$;$T_n(x)=2xT_{n-1}(x)-T_{n-2}(x)$I have been trying to find a closed form for the coefficient on the monomial $x^j$ of the $k$th...
View ArticleHow is the Chebyshev polynomial approximation defined when using the spectral...
I am studying spectral methods for numerical solutions for PDEs. Currently, I am on a chapter that explains how to use Chebyshev polynomials to solve non-periodic boundary value problems.I understand...
View ArticleWhat should the $\mathfrak{sl}(3)$ Chebyshev polynomials be?
Consider $\mathfrak{sl}_2$ and its fundamental weight $\lambda_1$. The character of the simple representation $L(n\lambda_1)$ with highest weight $n\lambda_1$ is given by a polynomial in...
View ArticleFinding the Chebyshev polynomials $T_n$ by elementary means
Suppose that one person called the Student—virtually, an advanced schoolchild—obtained a tip that the Chebyshev polynomials of the first kind exist and unique for each $n$. By the Chebyshev polynomial...
View ArticleChebyshev differential equations
Consider the Chebyshev polynomial of the first kind$$ (1-x^2)y'' - xy'+ n^2y = 0 , n \in \mathbb{N}. $$Use the substitution $ x=\cos\theta $ and show that the transformed ODE has solutions $y_1 = \cos...
View ArticleConvergence rate for Chebyshev polynomials to approximate $\text{erf}(x)$ on...
Let $[-\alpha, \alpha] \subset \mathbb{R}$, and let\begin{equation}\text{erf}(x) = \frac{2}{\sqrt{\pi}}\int_{0}^x e^{-t^2}dt.\end{equation}given projections of $\text{erf}(x)$ onto the first $k$...
View ArticleChebyshev approximation for bivariate function
I read the paper.I am a litte bit confused regarding formulation of Chebyshev approximation for bivariate function(See photo). There is only one integral over variable x. Should it be in formula one...
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