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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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Is this Hermite polynomial identity known?

In some physics related problem, I found out the curious identity $$\sum\limits_{n_1+n_2+n_3=n}\frac{n!}{n_1!\,n_2!\,n_3!}\,H_{2n_1}(x)\,H_{2n_2}(y)\,H_{2n_3}(z)=\frac{H_{2n+1}(r)}{2r},$$ where $H_n(x)...
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Orthogonal Polynomials and Sturm Liouville operators

Classical Orthogonal polynomials (e.g., Hermite, Legendre) are eigenfunctions of Sturm Liouville operators. For example, define $L[u]=u''-xu'$, then the $n$-th order Hermite polynomial satisfies $...
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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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229 views

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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55 views

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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building set of 2D orthogonal polynomials with minimum crossed terms

I am working with 2D orthogonal polynomials. They are function of (x,y) and are gradient-orthogonal on the unit square: $$ <P(x,y),Q(x,y)> = \int_{-1}^{+1}\int_{-1}^{+1}(\frac{\partial P}{\...
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Closed form of :$\int_{-1}^1 x^{2k} (\operatorname{erf}(x))^k \,dx $ for $ k$ is even integer and :$\int _{0}^{t}\exp(-x^2 \operatorname{erf}(x))dx$

This question is related to my question here such that i want to find a closed form of $\int_{-1}^1 x^{2k} (\operatorname{erf}(x))^k \,dx $ , for $k$ is even integer because for odd integer is $0$ as ...
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Gaps between roots of consecutive Hermite polynomials

Let $H_k(x)$ be (probabilists' or physicists', does not matter for this question) Hermite polynomials. It is well-known that all the gaps between consecutive roots of $H_k(x)$ are at least a multiple ...
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Jacobi polynomials with negative integer parameters

Numerical evidence indicates that Jacobi polynomials with negative integer parameters satisfy the identity $$P_n^{(-m,-k)}(x)=\left(\frac{x-1}{2}\right)^m\left(\frac{1+x}{2}\right)^kP_{n-m-k}^{(m,k)}(...
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Existence of moment-constrained maximum entropy distribution with support $[0,1]^n$

Given a finite set of moment values $\{\mu_1,\ldots,\mu_N\}$, for which the multi-dimensional finite Hausdorff moment problem is determined. That is, we know that at least one distribution $\mathcal{D}...
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Rate of convergence of generalized polynomial chaos

Let $\eta=g(\xi_1,\ldots,\xi_M)$ be a random variable expressed as a function of the random vector $\xi=(\xi_1,\ldots,\xi_M)$. Assume that $\xi_1,\ldots,\xi_M$ are absolutely continuous and ...
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Reverse Markov-Bernstein inequality for trigonometric polynomials

Let $r(t)$ be a real trigonometric polynomial of degree $n>1$. Assume it has zero at $t=0$ of multiplicity $k>0$. What can be said about the lower bound of the constant $c(k,n)$ such that $$ \...
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Lower $L^1$ norm estimates of null average trigonometric polynomials depending on the order of the polynomial

Let $p(x)=\sum_{k=1}^m [a_k\cos(n_kx)+b_k\sin(n_kx)]$ be a null average trigonometric polynomial (null average means that is $\int_\mathbb T p =0$ or, equivalently, there are no $a_0$ and $b_0$). ...
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Questions about generalized Polynomial Chaos, book by Dongbin Xiu

I have some questions about Chapter 5 from the book Numerical Methods for Stochastic Computations, by Dongbin Xiu. Theorem 5.7: Let $Y$ be a random variable and $\mathbb{E}[Y^2]<\infty$. Let $Z$ ...
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Using empirical expectation of orthogonal polynomials for density estimation

Estimating the probability density function using empirical moments is quite popular. Is there any advantage to using a different polynomial basis than the usual $1$,$x$, $x^2$ ... etc? For the moment ...
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Identities for Chebyshev polynomials of the second kind

While calculating an integral in a quantum mechanical problem by two different methods, I came across the following identity $$\sum_{k=0}^n\sum_{m=0}^{2k}(-2)^m\binom{2(n-k)}{n-k}\binom{2k}{k}\binom{...
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Polynomials for which $f''$ divides $f$

Let $n \geq 2$ and let $a < b$ be real numbers. Then it is easy to see that there is a unique up to scale polynomial $f(x)$ of degree $n$ such that $$f(x) = \frac{(x-a)(x-b)}{n(n-1)} f''(x).$$ ...
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What are the orthogonal polynomials with respect to the weight $2\cosh(\beta x)e^{-x^2}$?

In the study of a statistical physics problem, I need to know the orthogonal polynomials with respect to the weight $$2\cosh(\beta x)e^{-x^2},$$ where $\beta \in \mathbb{R}^+$. Is this already known? ...
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How to use this generalised 'generating function' for the Gegenbauer polynomials

Cohl (2011) gives a generalisation of the standard generating function for the Gegenbauer polynomials $C_n^\mu(x)$: $(1 -2tx + t^2)^{-\nu} = A_{\mu,\nu} \frac{(1-t^2)^{-\nu+\mu+1/2}}{t^{\mu+1/2}} \...
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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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Closure of polynomials in $L^2_w$ with log-normal weight function

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}{\...
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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)=...
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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 ...
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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)}...
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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 ...
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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 ...
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151 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 ...
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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 ...
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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 ...
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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}...
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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!} \...
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1answer
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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 ...
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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}...
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need help on the proof of the Christoffel–Darboux formula of Laguerre Polynomial

I found the Christoffel–Darboux formula of the Laguerre polynomials on wikipedia, https://en.wikipedia.org/wiki/Laguerre_polynomials \begin{align} K_n^{(\alpha)}(x,y) &:= \frac{1}{\Gamma(\alpha+1)...
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Formula of Laplace for the asymptotic expansion of Legendre polynomials

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}...
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Generalized polynomial chaos with 2D independent uniform variables for first order equation

I'm trying to transform the first order differential equation $\dot{x} = k x(t)$ into the corresponding set of stochastic differential equations. I have two independent uniformly distributed ...
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1answer
310 views

maximum of orthogonal vectors

$$v_1=(x_1,x_2,x_3\cdot\cdot\cdot,x_n)$$is such a vector. By changing its signs and positions of each component $x_i$, we can get different vectors. When n is odd, it's impossible for any of ...
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Sum of squared hypergeometric polynomials

$\sum_{m=1}^\infty \frac{1}{m} \bigg[{}_2F_1(-m,m,2,u)\bigg]^2 = \frac 1 4 -\frac 1 2 \log u$ I have very strong numerical support that this is true when $0<u\le1$. Can anyone help proving or ...
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Are there graphs whose matching polynomials are Legendre?

It is well-known (at least well-known enough to be on Wikipedia) that there are quite simple graphs whose matching polynomials $$M(G;x) = \sum_{m\geq 0} (-1)^m \#\{\text{matchings with $m$ edges}\}\,...
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Combinatoric formulas of Generalized Laguerre polynomials

I came up with the following quantity: $$\sum_{r=0}^p \binom{n}{n-p+r} \binom{2r}{2r-p} L^{(n-p-r)}_{2r}(z) \, ,$$ where $n\ge p$ are integers and $L^{(\alpha)}_m(z)$ is the generalized Laguerre ...
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Fourier sine and cosine transforms and Laguerre polynomials

Let $S$ and $C$ denote the Fourier sine transform and the Fourier cosine transform, respectively, i.e., \begin{align*} S f(k) &= \sqrt{\frac2\pi} \, \int_0^\infty f(x) \sin(kx)\,dx, \\ C f(k)...
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Generating function for products of complex Hermite polynomials

By making use of the generating function $$\sum_{m=0}^\infty \frac{H_m(x)}{m!} t^m=e^{-t^2 + 2xt} $$ for the real Hermite polynomials $H_m$, we get easily that $$(*)\quad \sum_{m,n=0}^\infty \frac{u^...
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An inverse spectral problem for Jacobi matrices (or orthogonal polynomials)

I will formulate this question in the language of Jacobi operators and spectral measures although it could be entirely rewritten in terms of orthogonal polynomials and measures of orthogonality. ...
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190 views

Polynomial interpolants in quadrature points and L2 convergence spectral rate

We recall that the Lagrange Interpolation Polynomial $p_n(x)$ of a function $f\in C^n(\Omega )$ for some $\Omega \subseteq \mathbb{R}$ and $n\in \mathbb{N}$, has a pointwise error term of the form $$|...
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1answer
151 views

q-versions of the geometric distribution and their names

I'm trying to set straight various $q$-deformations of the standard geometric distribution. The geometric distribution on $\left\{ 0,1,\ldots \right\}$ is well-known, it has $$ \mu_1(X=j)=(1-p)p^j,\...
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2answers
116 views

Chebyshev coefficient of $1/(z-x)$

In his paper "The Evaluation and Estimation of the Coefficients in the Chebyshev Series Expansion of a Function", David Elliott writes: Writing $x = \cos(\theta)$, it can easily be shown that $$ \...
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1answer
176 views

Sturm Liouville problems for non-classical orthogonal polynomials

It is known that for the classical orthogonal-polynomials there exist a set of Sturm Liouville problems. E.g. , the Hermite polynomial of order $n$ is a solution of $$y''(x) -xy'(x)+ny(x)=0 \, .$$ My ...
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150 views

Numerical Computation of Orthogonal Polynomials Recurrence Relations

Background and notations: Given an interval $I\subseteq \mathbb{R}$ and a continuous finite measure $d\mu = w(x)dx$, and denote $p_n(x)$ the orthogonal polynomials with respect to $d\mu$. We have the ...
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60 views

Special value of generalized Laguerre polynomials or of their quotient

Let $L_n^{(\alpha)}(x)$ denote a generalized Laguerre polynomial. I would like to have a closed form for the expression $$\frac{L_n^{(\alpha)}(1)}{L_{n-1}^{(\alpha)}(1)};$$ where we set $x=1$ and both ...
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2answers
456 views

Zeroes of Laguerre polynomials

The simplest Laguerre polynomials are $$ L_k(x)=(\frac{d}{dx}-1)^k\left(\frac{x^k}{k!}\right). $$ I would like to find a simple reference for proving or disproving the following assertions. (1) All ...