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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.

The Chebyshev polynomials, named after Pafnuty Chebyshev, are a sequence of orthogonal polynomials which are related to de Moivre's formula.

There are two kinds of these polynomials. The first kind $T_n$ is defined by the recurrence $$\begin{align} T_0(x)&=1\\ T_1(x)&=x\\ T_{n+1}(x)&=2xT_n(x)-T_{n-1}(x) \end{align}$$ The second kind $U_n$ is defined by the same recurrence, but with $U_1(x)=2x$.

These polynomials also satisfy the trigonometric identities $$T_n(\cos\theta)=\cos(n\theta)\qquad U_n(\cos\theta)\sin\theta=\sin(n+1)\theta.$$

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Chebyshev expansion of $\log(1 + x)$

I was reading a Wikipedia article on Chebyshev polynomials and got stuck in around the end of the article where the author takes advantage of orthogonality to compute the coefficients of the Chebyshev expansion of $ \log (x+1) $. I will be happy if…
Aleph-null
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How to Change the Interval of Interpolation from [-1,1] to [a,b] for Chebyshev Nodes

(According to this website:http://fac-staff.seattleu.edu/difranco/web/Math_371_W11/Files/Chebyshevnodes.pdf) Between [-1,1], the Chebyshev Nodes are given as: $x_k = \cos\Big((2k-1)\pi/2n)\Big), k=1,......,n$ and over [a,b] it is given as: $x_k=…
user257768
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Maths on the Paris Metro

My brother just saw this on the Paris Metro and asked what it is; unfortunately, it has defeated me. $$ {T_N} ^ {(2)} - {T_{N-1}} ^ {(1)} = {(LRC)}_{N - \frac{1}{2}}, $$ Does anyone know what it is? The $T_N$ look a bit like Chebyshev…
badjohn
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Derive the explicit expression of the Chebyshev polynimial: $T_n(x) =n\sum _{k=0}^{n}(-2)^{k}{\frac {(n+k-1)!}{(n-k)!(2k)!}}(1-x)^{k}$.

The explicit expression for the Chebyshev polynomials of the first kind is given as follows. $$ T_n(x) =n\sum _{k=0}^{n}(-2)^{k}{\frac {(n+k-1)!}{(n-k)!(2k)!}}(1-x)^{k}\qquad n>0 $$ However, no proof of this expression is available in any…
Orkodip Mookherjee
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Chebyshev coefficients- interpolation on [a,b]

My problem is to solve a second order differential equation given two (Dirichlet) boundary conditions. $\frac{d^2y}{dx^2} = M/EI$ Both M and I are functions of x. Owing to complexity of the function, my idea was to generate an interpolation…
speed_of_light
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Are there any alterations for the Chebyshev Differentiation Matrices on an arbitrary domain [a,b]?

I'm implementing the Chebyshev collocation method for solving PDEs, more specifically the shallow water equations. I know the Chebyshev differentiation matrices (or differential operators) are, for default, for functions defined on the domain…
Leonardo de Araujo
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Translation and multiplicative identities to Chebyshev polynomials

I am interested on finding translation and multiplicative identities for Chebyshev polynomials in the way they exist for Bernoulli ones: $$T_n (x+y) = ?$$ $$T_n (ax) = ?$$ Thanks
24th_moonshine
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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 $T_m(x)=0$, $T_{m+j}(x)=T_{m-j}(x), \ j=1,\ldots, \lfloor \frac{n}{2} \rfloor -1$, $T_{m+j}(x) = T_{m+n+1-j}(x), \ j=1,\ldots n.$ Note: condition 3. requires symmetry…
Brad Willms
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What is the coefficient of the power $x^{n-2}$ in a Chebyshev polynomial?

Let us consider the second kind Chebyshev polynomial over the positive integers $U_{n+1}(x) = 2xU_n(x) - U_{n-1}(x)$ with $n>1$ is a positive integer. I know that the leading term of $U_n(x)$ is $2^{n}$ and it's associated with the power $x^{n}$. My…
Safwane
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Chebyshev coefficients upper bounded by sup-norm of function?

Let $T_k(x)$ be the Chebyshev polynomials of the first kind and consider the function $$ f(x) = \sum_{k = 0}^\infty c_k \, T_k(x). $$ Show that $$ |c_k| \leq \|f\|_{[-1,1]}. $$
gTcV
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Square roots modulo p

For Chebyshev-polynomials on finite fields $F_p$ with a prime $p$, I have found the following expression: $$ T_n(x)=\frac{(x+\sqrt{x^2-1})^n+(x-\sqrt{x^2-1})^n}{2} mod\space p. $$ But obviously, $\sqrt{x^2-1}$ is not defined on $F_p$ for some $x$.…
Ada
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How to get the basis of $\cos(nx)$ in Chebyshev polynomials.

Given $\cos(nx) = f_n(\cos x)$ for any positive integer $n$ and polynomial $f_n(x) = 2^{n-1}x^n + a_1 x^{n-1} + ... + a_{n-1}x + a_n$. Show if $n$ is odd, $a_n = 0$ and if $n$ is even, $a_n=(-1)^{\frac n 2}$.
Wei Zhong
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Commutative property of Chebyshev polynomial.

I'm trying to understand a few properties of $T_n$, Chebychev polynomials of the first kind. That is, those of the form: $\cos(kz)=T_k(\cos(z))$. For example, $T_{2}(z)=2z^2-1$. I don't understand why this isn't a trivial fact: for all $m,n$, show…
emka
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Decomposition of function in Chebyshev Polynomials

Let's consider $f(x) = e^{ax}$. I know that there are variants how to decompose function in series. One of such is Fouries series. I found out that there is Chebyshev Polynomials. So, how to decompose this function through Chebyshev Polynomials. I…
Clipper
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Generalization of identities for values of Chebyshev polynomials

Using the theory of Pell equations and the fact that the discriminant $5$ has both the forms $k^2 + 4$ and $k^2 - 4$, I stumbled across a proof of the identities $$ (-1)^n T_{2n} (\tfrac{i}{2}) = T_n (\tfrac{3}{2}) $$ $$ (-i)^{2n+1} U_{2n+1}…
Barry Smith
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