# Questions tagged [orthogonal-polynomials]

A familly of orthogonal polynomials is a sequence of polynomials in one variable, one in each degree, such that any two of them are orthogonal with respect to some fixed scalar product on the space of polynomials. They are closely related to continued fractions and useful in harmonic analysis. There are many different families of orthogonal polynomials, among which one can cite Hermite polynomials, Laguerre polynomials, and Jacobi polynomials.

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### Are there extensions of Hilb's and Laplace's formulas to Jacobi polynomials with $\alpha,\beta\le-1$?

In Szegő's Orthogonal Polynomials book, he gives two interesting asymptotic formulas for Jacobi polynomials with $\alpha,\beta>-1$. The first (Theorem 8.21.12, page 197 is a generalization of Hilb'...
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### Transformation which “opens up” an arc

I am reading Harold Widom's paper "Extremal Polynomials Associated with a System of Curves in the Complex Plane". At the beginning of section 11 he states that: [There is] a simple transformation ...
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### Chebyshev polynomials from roots [closed]

Given a polynomial $P_n(x)$ with roots at $\{a_i\}$: $P_n(x)=\prod\limits_{i=1}^{i=n} (x-a_i)$ one can obtain the Chebyshev coefficients from the roots. It is implemented eg in python. Can we ...
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### Integration on sphere $\mathbb{S}^{d-1}$ for $d$ large — Change of variables

I'm trying to integrate a function over two vectors which lie on the surface of the unit sphere in D dimensions. The function depends only on the difference between the two vectors, and their dot ...
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### mollifier satisfying moment conditions

I wish to find a mollifier $\psi\in C_0^{d+1}(-1,1)$ such that $$\int_{-1}^1 x^k \psi(x)dx = \begin{cases} 1, & k=0;\\ 0, & k=1,\dots,d. \end{cases}$$ This paper (https://home.cscamm....
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### Convergence of gPC expansions for random variables in the total variation distance

Suppose that a random variable $Y$ can be written as $Y=g(Z)$, where $g$ is a function and $Z$ is a random variable. When $Z$ is a continuous random variable with finite absolute moments, we consider ...
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### Convergence of Gaussian quadrature rules for integration

I would like to discuss some issues about convergence of Gaussian quadrature rules for integration. I asked this question in Mathematics Stack Exchange here with a bounty period but received no answer....
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### Riemann-Hilbert and Orthogonal polynomials

Sorry for perhaps naive questions, I am not at all a specialist in the subject but I need it for my research. I know that there are close relations between Riemann-Hilbert problems and orthogonal ...
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### Determinants associated to orthogonal polynomials

Let $${p_n}(x) = \sum\limits_{j = 0}^{n } {{{( - 1)}^{n - j}}p(n,j){x^j}}$$ be orthogonal polynomials satisfying $${p_n}(x) = (x - {s_{n - 1}}){p_{n - 1}}(x) - {t_{n - 2}}{p_{n - 2}}(x)$$ with ...
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### Orthogonal basis of polynomials?

Let us define the basis of polynomials given by: $$\begin{array}\ P_0=1, \\ P_1=x, \\ P_2=x(x-1), \\ P_3=x(x-1)(x-2), \\ P_4=x(x-1)(x-2)(x-3), \ldots\\ \end{array}$$ I would like to know if this ...
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### Function approximation via an orthonormal basis (with singular weight)

If you don't mind, please consider the eigenvalue problem $$(1-x^2)u''+ \lambda u=0 \ \ \ \forall x\in (-1,1),$$ $$u(\pm 1) = 0.$$ Observe that for suitable values of $\lambda$, the ODE resembles ...
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### Proof Reference - Polynomial interpolation at quadrature points

If $\left( p_n \right)_{n=0}^{\infty}$ is a family of orthogonal polynoamials with respect to a measure $\mu$ on $[-1,1]$, and $\left( x_j, w_j \right)$ are the quadrature points and weights for the ...
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Consider the Hilbert space $L^2_w$ with scalar product $\langle f,g\rangle_w =\int_0^\infty f(x)g(x)w(x)dx$ where the weight $w$ is the density function of a log-normal distribution $$w(x)=\frac{1}{\... 0answers 85 views ### Orthogonal polynomials with quadratic recurrence coefficients Consider the monic orthogonal polynomials determined by the recurrence$$p_{n+1}(x)=(x-n(n+b))p_{n}(x)-n(n+a)p_{n-1}(x), \quad n\in\mathbb{N},$$with the initial conditions p_{-1}(x)=0 and p_{0}(x)=... 1answer 109 views ### Clenshaw-Curtis integration without Fourier The Clenshaw-Curtis quadrature rule approximates an integral I=\int\limits_{-1}^{1} f(x) \, dx by$$I\approx I_n = \sum\limits_{j=1}^N f(x_j)w_j \, ,$$where the x_j's are the roots of the N-th ... 1answer 435 views ### A generalization of binary Krawtchouk polynomials I am looking for orthogonal polynomials P_{d,m,n} that have their values at integers i specified by the following generating function (1-z)^i (1+z+z^2+ \ldots + z^d)^{n-i} = \sum_{m=0}^{i+d(n-i)}... 2answers 960 views ### Is there an explicit expression for Chebyshev polynomials modulo x^r-1? This is an immediate successor of Chebyshev polynomials of the first kind and primality testing and does not have any other motivation - although original motivation seems to be huge since a positive ... 0answers 43 views ### Determining the Associated Sequence If Sheffer Conditions are not Met A sheffer sequence s_n(x) is formed by considering the generating function$$\sum_{k=0}^\infty s_k(x)\frac{t^k}{k!}=A(t)e^{xB(t)}where A is an invertible power series, and B is a delta ... 1answer 231 views ### Completeness of the solutions to the Schrödinger Hydrogen Atom I once did some work on using orthogonal function expansions for fitting 3D distribution functions. To ensure completeness over L^2 (which was considered sufficient even though technically a ... 0answers 43 views ### Orthogonal polynomial expansion for bivariate noncentral chi-square and bi-variate noncentral student t distribution This is a research question for which I am not able to find any existing reference. So, I am reaching out for help. The project is related to studying the sequence of rejections in multiple hypothesis ... 1answer 389 views ### Bounds on Legendre polynomials on the complex plane Let P_n be the Legendre polynomial of degree n. Could you suggest me some references to bound the polynomials on the complex plane (near the real line in particular) ? More specifically, I need to ... 0answers 123 views ### Holonomic generating function Let \lambda denote a hook of size d and c(\Box) the content of  \Box \in \lambda . Let  \text{Hooks}(d)  be the set of hooks with d boxes. Define \begin{align} B(d)&= \frac{1}{d!h^{d-1}... 1answer 316 views ### Rational generating function and recursion Let \lambda denote a hook of size d and c(\Box) the content of  \Box \in \lambda . Let  \text{Hooks}(d)  be the set of hooks with d boxes. Define \begin{align} B(d)&= \frac{1}{d!} \... 1answer 159 views ### Expand the pdf of Wishart distribution into power series via orthogonal polynomials In the univariate case (\chi^2 distribution), I know we can expand the pdf into power series of the variance \sigma^2 with Laguerre polynomials. Indeed, since the Laguerre polynomials are related ... 2answers 1k views ### Relation between Legendre and Chebyshev polynomials Where I could find relationships between Legendre and Chebyshev polynomials? For example I found with maple P_n(\cos\theta)=\sum_{k=0}^n(-1)^{n+k}\frac{2-\delta_{k0}}{4^n} \binom{n-k}{\frac{n-k}{2}...
According to the so-called formula of Laplace (see, e.g., [1, Theorem 8.21.2]), Legendre polynomials admit the following asymptotic expansion:  P_n(\cos\theta) = \sqrt{2} (\pi n\sin\theta)^{-\frac12}...