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]\}...
J. M.'s user avatar
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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 $\lambda \in \mathbb{C}$ $$a_{n+1} = \lambda a_{n-1}+ \lambda^* a_n + \lambda^* n b_n $$ ...
Kung Yao's user avatar
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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, ...
F_Dussap's user avatar
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275 views

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 ...
TooManyQuestionsInLife's user avatar
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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 $I$ ...
MathTolliob's user avatar
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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 ...
Oren B.'s user avatar
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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}) := \, \...
Jeanne Scott's user avatar
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3 votes
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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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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'...
Daniel Cramer's user avatar
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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 ...
Luis Giraldo Gonzalez's user avatar
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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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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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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 $...
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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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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)= \...
mamiladi's user avatar
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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 \...
mamiladi's user avatar
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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 \...
mamiladi's user avatar
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8 votes
2 answers
400 views

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 ...
Henri Cohen's user avatar
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3 votes
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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(...
teremok's user avatar
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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,...
Twi's user avatar
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4 votes
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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)...
Zurab Silagadze's user avatar
4 votes
2 answers
578 views

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 $...
Amir Sagiv's user avatar
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7 votes
0 answers
387 views

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 ...
Johann Cigler's user avatar
4 votes
1 answer
987 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 ...
fernando's user avatar
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1 answer
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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 ...
JCM's user avatar
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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}{\...
Jerome Caron's user avatar
1 vote
1 answer
292 views

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 ...
zeraoulia rafik's user avatar
4 votes
1 answer
372 views

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 ...
ilyaraz's user avatar
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1 vote
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)}(...
Zurab Silagadze's user avatar
1 vote
0 answers
114 views

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}...
wzell's user avatar
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2 votes
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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 ...
jum's user avatar
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4 votes
0 answers
260 views

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 $$ \...
Paata Ivanishvili's user avatar
2 votes
0 answers
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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 $b_0$). ...
Arturo Sanjuán's user avatar
1 vote
0 answers
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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 $Z$ ...
jum's user avatar
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4 votes
1 answer
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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{...
Zurab Silagadze's user avatar
14 votes
1 answer
463 views

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 $$f(x) = \frac{(x-a)(x-b)}{n(n-1)} f''(x).$$ ...
David E Speyer's user avatar
5 votes
0 answers
266 views

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? ...
Jia Yiyang's user avatar
3 votes
0 answers
161 views

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}} \...
Edward Lilley's user avatar
1 vote
1 answer
104 views

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 ...
Amir Sagiv's user avatar
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4 votes
1 answer
119 views

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}{\...
S. Willems's user avatar
1 vote
0 answers
111 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)=...
Twi's user avatar
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0 votes
1 answer
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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 ...
Amir Sagiv's user avatar
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6 votes
1 answer
536 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)}...
user117316's user avatar
19 votes
2 answers
1k 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 ...
მამუკა ჯიბლაძე's user avatar
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 ...
AveryJessup's user avatar
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 ...
eyeballfrog's user avatar
3 votes
0 answers
67 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 ...
Chee's user avatar
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