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.

Filter by
Sorted by
Tagged with
3
votes
0answers
32 views

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'...
0
votes
1answer
73 views

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
vote
0answers
31 views

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 ...
2
votes
1answer
112 views

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 ...
4
votes
1answer
200 views

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
vote
0answers
137 views

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 ...
1
vote
1answer
174 views

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
vote
1answer
103 views

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....
2
votes
0answers
42 views

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 ...
0
votes
0answers
55 views

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....
1
vote
0answers
61 views

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)= \...
0
votes
0answers
66 views

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 \...
0
votes
1answer
169 views

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 \...
7
votes
1answer
184 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 ...
3
votes
1answer
210 views

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
vote
0answers
66 views

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
1answer
646 views

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)...
4
votes
2answers
265 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 $...
7
votes
0answers
159 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 ...
3
votes
1answer
535 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 ...
0
votes
1answer
63 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 ...
0
votes
0answers
108 views

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
vote
1answer
271 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 ...
4
votes
1answer
196 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 ...
1
vote
1answer
143 views

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
vote
0answers
99 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}...
2
votes
0answers
60 views

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 ...
3
votes
0answers
166 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 $$ \...
2
votes
0answers
91 views

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 $b_0$). ...
1
vote
0answers
75 views

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 $Z$ ...
4
votes
1answer
237 views

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{...
14
votes
1answer
424 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 $$f(x) = \frac{(x-a)(x-b)}{n(n-1)} f''(x).$$ ...
5
votes
0answers
229 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? ...
3
votes
0answers
117 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}} \...
1
vote
1answer
84 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 ...
4
votes
1answer
101 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}{\...
1
vote
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)=...
0
votes
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 ...
5
votes
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)}...
18
votes
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 ...
1
vote
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 ...
-1
votes
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 ...
3
votes
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 ...
3
votes
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 ...
1
vote
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}...
6
votes
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!} \...
1
vote
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 ...
5
votes
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}...
1
vote
0answers
390 views

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)...
3
votes
0answers
327 views

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}...