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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Do you know of orthogonal-polynomial families with complex measure on the square? I'm just looking for family names to read up on
I'm looking for the name(s) of a family or families of polynomials whose normalization and orthogonality are defined by integrals (inner product) over the complex square $\{u+iv\, |\, u, v \in [-1,1]\}...
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Two-term recurrence relation
We consider the following system of recurrence relations for $n \in \mathbb Z$ and $\vert \lambda \vert=1$ with $\lambda \in \mathbb{C}$
$$a_{n+1} = \lambda a_{n-1}+ \lambda^* a_n + \lambda^* n b_n $$
...
5
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2
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Proving that the primitives of the Laguerre functions are uniformly bounded
Let $(L_k)_{k\geq 0}$ be the Laguerre polynomials. These polynmials are orthogonal with respect to the inner product: $$\langle f,g\rangle = \int_0^\infty f(x)g(x)\mathrm e^{-x}\,\mathrm dx.$$
Hence, ...
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1
answer
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Expansion of white noise into infinite series using orthogonal polynomials
Having a white random process $s(t)$, is it possible to write $$s(t)=\sum_{i=0}^\infty\alpha_i\phi_i(t)$$ where the $\alpha_i$ are random variables and the $\phi_i$ orthogonal polynomials (Jacobi ...
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Gaussian quadrature, with no exact result over polynomial, but on inverse functions
Generally, a Gaussian quadrature of degree $n$ over an interval $I$ is defined so that it integrates exactly polynomials up to degree $2n - 1$. The main tool are the orthogonal polynomials.
When $I$ ...
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0
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Recursive formula for integral of Chebyshev-type integral
Define
$$
I_{m,n}(x,y,r) = \int_a^b T_m(x + r \sin(\gamma)) T_n(y-r \cos(\gamma)) d\gamma
$$
where $T_m(x)$ are the Chebyshev polynomials of the first kind, and $a$ and $b$ are constants. Assume that ...
5
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Infinite tridiagonal matrices and a special class of totally positive sequences
Let $\Bbb{y} = \big(y_1, y_2, y_3, \dots \big)$ be an infinite sequence of positive real numbers such that following $\Bbb{N} \times \Bbb{N}$ tridiagonal matrix
\begin{equation}
T(\Bbb{y}) := \,
\...
3
votes
1
answer
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Inequality for generalized Laguerre polynomials
Please. Does anybody know a proof of this inequality
$$\Big|\frac{n!\Gamma(\alpha+1)}{\Gamma(n+\alpha+1)} L^{\alpha}_n(x)\Big|\leq e^{\frac{x}{2}}$$ where $\alpha\geq0$ and $x\geq0$ and
$L^{\alpha}_n$ ...
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answer
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Is there a bijective proof of an identity enumerating independent sets in cycles?
Let $C_m$ be the cycle with $m$ vertices, defined so that $C_1$ has a self-loop on its unique vertex. Let $p_m$ be the generating function enumerating the number of ways to choose $k$ vertices in $C_m$...
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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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1
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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 ...
1
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0
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Higher-order inner products of an orthonormal basis
Let $\pi$ be a probability measure on some space $\mathcal{X}$, and let $\Phi = \{ \phi_k \}_{k \geqslant 0}$ be some (possibly complex-valued) orthonormal basis for $L^2 ( \pi )$, with $\phi_0 \equiv ...
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1
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Integral involving associated Laguerre polynomial and Bessel function
In a quantum mechanics problem I encountered the following integral
\begin{equation*}
\int_0^\infty t^{\nu+1}J_\nu(\beta t)L_{\mu-\nu}^{2\nu}(t)e^{-t/2}dt\,,
\end{equation*}
where $L$ denotes the ...
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votes
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Gegenbauer's addition theorem for Jacobi polynomials
I have the following identity,
$$\int_{-1}^{1} \! dz \, j_0\bigl(\sqrt{x^2 + y^2 - 2xy z}\bigr) \, P_n(z)
= 2 \, j_n(x) \, j_n(y) \;,$$
where $x, y > 0$, $P_n$ is a Legendre polynomial, and $...
1
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1
answer
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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 ...
1
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1
answer
161
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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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Recurrence involving families of orthogonal polynomials
Let $ \forall n \in N, n\geq 1$ $$ R_n(x)=(-1)^n n! \displaystyle \frac{(x-1)...(x-n)}{(x(x+1)..(x+n))^2}$$ thus by decomposition in simple element it's easy to see that
$$ (1): \quad R_n(x)= \...
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About a family of orthogonal polynoms satisfying a recurrence relation
let $P_0(x)=0$;$P_1(x)=1$
Let $\forall n $ integer $ \geq 2$, $\forall x$ real, $$P_n(x)=\displaystyle \sum_{k=0}^{n-1} C_{n+k}^n (-x)^k \alpha_{n,k}$$ and where $\forall k$ such that $0 \leq k \...
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votes
1
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Upper bound over $[0,1] $ for strange family of polynomials
Let $n$ integer $\geq 2,$ $x$ real, and $$P_n(x)=\displaystyle \sum_{k=0}^{n-1} C_{n+k}^n (-x)^k \alpha_{n,k}$$ and where $\forall k$ such that $0 \leq k \leq n-1 $ $$ \alpha_{n,k}= \displaystyle \...
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2
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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 ...
3
votes
1
answer
466
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Evaluating an integral with Jacobi and Legendre polynomials
The following integral came up in one of my studies:
$$\int_{-1}^1 (1-x)^\alpha (1+x)^\beta P_n^{(\alpha,\beta)}(x)\,P_{n+j}(x)\,dx$$
where $P_n^{(\alpha,\beta)}(x)$ is a Jacobi polynomial and $P_m(...
1
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0
answers
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An asymptotic behavior of a sequence of special polynomials
For $n\to\infty$, I would like to know the asymptotic behavior of the polynomials defined in terms of the Gauss hypergeometric series:
$$
p_{n}(z):={}_{2}F_{1}(-n,-nz+\alpha;1;\beta),
$$
where $\alpha,...
4
votes
1
answer
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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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votes
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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 ...
4
votes
1
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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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votes
1
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553
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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 ...
0
votes
0
answers
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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:
$$
<P(x,y),Q(x,y)> = \int_{-1}^{+1}\int_{-1}^{+1}(\frac{\partial P}{\...
1
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1
answer
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Closed form of :$\int_{-1}^1 x^{2k} (\operatorname{erf}(x))^k \,dx $ for $ k$ is even integer and :$\int _{0}^{t}\exp(-x^2 \operatorname{erf}(x))dx$
This question is related to my question here such that i want to find a closed form of $\int_{-1}^1 x^{2k} (\operatorname{erf}(x))^k \,dx $ , for $k$ is even integer because for odd integer is $0$ as ...
4
votes
1
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Gaps between roots of consecutive Hermite polynomials
Let $H_k(x)$ be (probabilists' or physicists', does not matter for this question) Hermite polynomials.
It is well-known that all the gaps between consecutive roots of $H_k(x)$ are at least a multiple ...
1
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1
answer
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Jacobi polynomials with negative integer parameters
Numerical evidence indicates that Jacobi polynomials with negative integer parameters satisfy the identity
$$P_n^{(-m,-k)}(x)=\left(\frac{x-1}{2}\right)^m\left(\frac{1+x}{2}\right)^kP_{n-m-k}^{(m,k)}(...
1
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0
answers
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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}...
2
votes
0
answers
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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 ...
4
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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
$$
\...
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0
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Lower $L^1$ norm estimates of null average trigonometric polynomials depending on the order of the polynomial
Let $p(x)=\sum_{k=1}^m [a_k\cos(n_kx)+b_k\sin(n_kx)]$ be a null average trigonometric polynomial (null average means that is $\int_\mathbb T p =0$ or, equivalently, there are no $a_0$ and $b_0$). ...
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Questions about generalized Polynomial Chaos, book by Dongbin Xiu
I have some questions about Chapter 5 from the book Numerical Methods for Stochastic Computations, by Dongbin Xiu.
Theorem 5.7: Let $Y$ be a random variable and $\mathbb{E}[Y^2]<\infty$. Let $Z$ ...
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1
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Identities for Chebyshev polynomials of the second kind
While calculating an integral in a quantum mechanical problem by two different methods, I came across the following identity
$$\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?
...
3
votes
0
answers
161
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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}} \...
1
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1
answer
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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 ...
4
votes
1
answer
119
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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
$$ w(x)=\frac{1}{\...
1
vote
0
answers
111
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Orthogonal polynomials with quadratic recurrence coefficients
Consider the monic orthogonal polynomials determined by the recurrence
$$p_{n+1}(x)=(x-n(n+b))p_{n}(x)-n(n+a)p_{n-1}(x), \quad n\in\mathbb{N},$$
with the initial conditions $p_{-1}(x)=0$ and $p_{0}(x)=...
0
votes
1
answer
153
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Clenshaw-Curtis integration without Fourier
The Clenshaw-Curtis quadrature rule approximates an integral $I=\int\limits_{-1}^{1} f(x) \, dx$ by $$I\approx I_n = \sum\limits_{j=1}^N f(x_j)w_j \, ,$$
where the $x_j$'s are the roots of the $N$-th ...
6
votes
1
answer
536
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A generalization of binary Krawtchouk polynomials
I am looking for orthogonal polynomials $P_{d,m,n}$ that have their values at integers $i$ specified by the following generating function
$(1-z)^i (1+z+z^2+ \ldots + z^d)^{n-i} = \sum_{m=0}^{i+d(n-i)}...
19
votes
2
answers
1k
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Is there an explicit expression for Chebyshev polynomials modulo $x^r-1$?
This is an immediate successor of Chebyshev polynomials of the first kind and primality testing and does not have any other motivation - although original motivation seems to be huge since a positive ...
1
vote
0
answers
46
views
Determining the Associated Sequence If Sheffer Conditions are not Met
A sheffer sequence $s_n(x)$ is formed by considering the generating function
$$\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 ...
1
vote
1
answer
427
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 ...
3
votes
0
answers
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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 ...