Questions tagged [chebyshev-polynomials]
In mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev, are two sequences of orthogonal polynomials which are related to de Moivre's formula. These polynomials are also known for their elegant Trigonometric properties, and can also be defined recursively. They are very helpful in Trigonometry, Complex Analysis, and other branches of Algebra.
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Chebyshev polynomials to hypergeometric function?
I am trying to derive this hypergeometric version of the Chebyshev polynomials
https://en.wikipedia.org/wiki/Chebyshev_polynomials#Explicit_expressions
$$U_n(x) = \frac{\left (x+\sqrt{x^2-1} \right )^...
- 310
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Can the Chebyshev polynomials evaluated at a zero for index m, form a sequence of fixed length (in the index) with even parity?
For $n \in \mathbb{Z},n\geq 3$ find $x\in \mathbb{R}$ and index $m$ such that
$T_m(x)=0$,
$T_{m+j}(x)=T_{m-j}(x), \ j=1,\ldots, \lfloor \frac{n}{2} \rfloor -1$,
$T_{m+j}(x) = T_{m+n+1-j}(x), \ j=1,\...
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Question on convergence of Chebyshev series
I have written a script that plots some function and its truncated Chebyshev expansion in $[-1,1]$, which is given by
$$
\sum a_nT_n(x) \quad \text{with} \quad a_n = \int_{-1}^{1}T_n(x)f(x)/\sqrt{1-x^...
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Product Formula for Real Cyclotomic Polynomials
Let $n$ be a natural number, $\zeta_{n}$ be a primitive $n^{th}$ root of unity and $\Phi_{n}(x)$ be the $n^{th}$ cyclotomic polynomial. Let $\Psi_{n}(x)$ be the $n^{th}$ real cyclotomic polynomial (...
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Closed Form of the Chebyshev Polynomials of the First Kind [Proof Request]
$$T_n(x) = \frac{n}{2} \sum_{r=0}^{\lfloor \frac{n}{2} \rfloor} \frac{(-1)^r}{n-r} \binom{n-r}{r} (2x)^{n-2r}$$
Searching on the web yielded no results, and the result is given without proof on OEIS ...
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Find the spectral solution to ODE
I am trying to find a spectral solution for the for the boundary value problem of the ODE $$u^{\prime\prime}(x)-u(x)=f, \quad \quad u^{\prime}(0)=u^{\prime}(\pi)=0$$
for any $f: [o,\pi]\rightarrow \...
- 139
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Approximating Chebyshev Polynomials by Truncated Fourier Series
It is known (see e.g. https://www.math.ucdavis.edu/~bremer/classes/fall2018/MAT128a/lecture9.pdf) that any continuous function $f: [-1,1] \to \mathbb{C}$ admits a Chebyshev expansion
$$
f(x) = \sum_{n=...
- 191
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Sum of Chebyshev Polynomials of the 2nd kind
I am attempting to simplify the following summation and would appreciate being pointed in the correct direction on how to do this, or if even possible. When I say simplify, I specifically mean ...
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Doubt about evaluating an integral
I need to prove that
\begin{equation}
I_n=\displaystyle\int_{-1}^{1} \ln T_n^2(x) \frac{dx}{\sqrt{1-x^2}} = -2\pi\ln2
\end{equation}
where $\ln$ denotes the natural logarithm and $T_n(x)$ are the ...
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Proving that $\sum_{i=0}^{n/2} (-1)^i \frac{n}{n-i} {n-i \choose i} = 2\cos(\frac{\pi n}{3})$
I was investigating the Girard-Waring identity, specifically for two variables:
$$x_1^n + x_2^n = \sum_{i=0}^{\frac{n}{2}} (-1)^i \frac{n}{n-i} {n-i \choose i}(x_1+x_2)^{n-2i}(x_1x_2)^i $$
This lead ...
- 195
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Mathematical function in the form $f(\cos \theta)$
I was reading about Chebyshev polynomials and came across this interesting notation that defines a Chebyshev polynomial of the first kind:
$$ F_n(\cos \theta) = \cos (n\theta) $$
What threw me was $\...
- 103
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Reference for the exponential generating function for the Chebyshev polynomials of the second kind?
Does anyone have a reference for the exponential generating function for the Chebyshev polynomials of the second kind?
$$
\sum_{n=0}^{\infty}\frac{t^n U_n(x)}{n!}= etc
$$
I know what it is from the ...
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Bounds for Chebyshev polynomials outside [-1,1]
It is well known that, for all $x\in [-1,1]$ and for all $j>0$, the Chebyshev polynomials of the first and second kind satisfy
$$
|T_j(x)|\leq 1
,\qquad
|U_j(x)|\leq j+1.
$$
I am wondering about ...
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Interpolation program not graphing
I'm using this program for my Elementary Numerical Analysis class. This creates an interpolant of degree n to the function fcn(x) on $[-1,1]$, which is given below ...
- 377
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Solutions to general Chebyshev function $T_n(x)=\pm1$
Find a general formula for the points $x$ at which $T_n=\pm 1$. How many such points are there on $[-1,1]$? Hint: begin with a special case such as $n=3$.
We know that $|T_n(x)|\leq 1$ on $[-1,1]$ and ...
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