# Questions tagged [orthogonal-polynomials]

Questions pertaining to certain sets of polynomials that satisfy an orthogonality criterion with respect to some specified inner product.

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### need more explanations on the proof about zeros of orthogonal polynomials

I do not understand why should not the polynomial $P_n(x)(x-x_1)...(x-x_n)$ change sign ?
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### A generalization of the Hermite polynomial: is there a name for this class of polynomials in the literature?

An explicit expression for the `probabilist's' Hermite polynomial is given by $$\sum_{r=0}^{\lfloor n/2\rfloor}(-1)^r\frac{n!}{2^r(n-2r)!r!}x^{n-2r}.$$ In playing around with some combinatorics, I ...
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### Evaluate $\int^1_{-1} \frac{P_{l_1}^{m_1}(x)P_{l_2}^{m_2}(x)}{\sqrt{1-x^2}} dx$

I wish to evaluate $\int^1_{-1} \frac{P_{l_1}^{m_1}(x)P_{l_2}^{m_2}(x)}{\sqrt{1-x^2}} dx$. There is a neat way to show the integral is zero for certain combinations of ms and ls shown here: ...
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### Methods for numerically calculating families of ON polynomials given inner product?

I am aware of some famous families of polynomials, for example Bessel, Jacobi, Chebyshev, Hermite and so on. They have one thing in common : they are all ON bases $\{P_0,P_1,\cdots\}$ with respect to ...
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### How one can reformulate the sentence: $T_{n}(x)/x$ is irreducible over the integers

The motivation to this question can be found in: Chebyshev Polynomials and Primality Testing My question is: How one can reformulate the sentence: $T_{n}(x)/x$ is irreducible over the integers ...
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### The Bernoulli Polynomials

We know that where $B_n(t)$ is Bernoulli polynomials. My question: Can Bernoulli polynomials be orthogonalized with respect to a weight function $\omega$? or I mean what is a weight function under ...
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### Orthogonal polynomials with weight $(1+x^{2m})^{-1}$

Very simple question. Is much known about orthogonal polynomials on the real line with weight function: $$w_m(x)=\frac{1}{1+x^{2m}}\exp(-\alpha\cot^{-1}(x))$$ A reference would be much appreciated. ...
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### Orthogonal Polynomials approximation and $L^2(\mathbb(R))$

I have another basic question, this time about approximation of functions. Given $(p)_{i\in \mathbb{N}}\in \mathcal{A}=\{$all the families of orthogonal polynomials $\}$ and $f\in L^2(\mathbb{R})$, ...
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### Existence, derivation and properties of 2D orthogonal polynomial families?

There exist several famous families of one-dimensional polynomials, which for various different integral inner-products constructed so that every polynomial is orthogonal to every other. Now to the ...
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### Rational functions of orthogonal polynomials

Let $\Omega\in\mathbb{C}$ is defined as \begin{equation} \Omega = s^2\frac{LC}{2} \end{equation} and matrix $\mathbf{A}$ as follows \begin{equation} \mathbf{A} = \begin{bmatrix} T_{n}(1 + \Omega) ...
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### I have a question on Hilbert Spaces

The question: Let $(P_n), n \geq 0$, be the Legendre Polynomials, which is a total orthonormal system in real $L^{2} [-1, 1]$ with the inner product $\langle x,y\rangle = \int_{-1}^{1} x(t)y(t) dt$. ...
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### Orthogonality of Legendre polynomials from generating function

Given the the Legendre polynomials generating function: $$G(x,t)=\frac{1}{\sqrt{1-2xt+t^2}}=\sum_{n=0}^{\infty}P_n(x)t^n$$ prove the relation: $$\int_{-1}^{1} (P_n(x))^2 dx = \frac{2}{2n+1}$$ My ...
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### Why does orthogonalizing the monomials give Legendre polynomials?

It's pretty well known that performing a Gram-Schmidt process on the monomials, $$p_j(x) = x^j - \sum_{i=0}^{j-1} \frac{\langle x^j|p_i\rangle}{\langle p_i|p_i \rangle}p_i(x),$$ gives (scaled) ...
Let $w(x)$ be an even weight function and [a,b] is a symmetric region with respect to $0$. Prove that the orthogonal polynomial satisfies $p_{n}(-x)=(-1)^{n}p_{n}(x)$ for $n=0,1,2..$ It says that an ...
Could someone tell me what is the error when applying the Gauss Quadrature rule in 2 dimensions? I know that for one dimension the error is \frac{(n!)^{4}}{(2n+1)[(2n)!]^{3}} \cdot f(\xi)^{2n}(b-a)...