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.
81
questions with no upvoted or accepted answers
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representation theoretic interpretation of Jack polynomials
Monomial symmetric polynomials on $n$ variables $x_1, \ldots x_n$ form a natural basis of the space $\mathcal{S}_n$ of symmetric polynomials on $n$ variables and are defined by additive symmetrization ...
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312
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Ratio of entries of A and log A where A is a triangular matrix
Consider triangular matrices $A = \left( {a(n,k)} \right)$ of arbitrary order with $a(n,k) = 0$ if $n + k$ is odd and $a(n,n - 2k) = \frac{{n!}}{{k!(n - 2k)!}}\frac{{(m + n - k - 1)!}}{{(m + n - 1)!}}$...
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Generating functions of real-rooted polynomials
Suppose $f_n(t)$ is a degree $n$ polynomial. Let $F(t,u) = \sum_n f_n(t)u^n$.
What conditions on $F(t,u)$ tell us that the $f_n(t)$ are real-rooted? Similarly, are there any conditions on $F(t,u)$ ...
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7
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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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6
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170
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3-term recurrence relation including integral or differential operator for polynomials
Sequences of polynomials with a 3-term recurrence relations are well known for orthogonal polynomials. Do recurrence relations using differential or integral operators also appear in some theories?
I ...
6
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183
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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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6
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0
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390
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Recurrence Formula for Zernike polynomials
I'm not sure if this is research level, so if this result is known, please excuse the intrusion. I am trying to find a relation between solutions of the Laplacian equation in $4$ dimensions and those ...
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5
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280
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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}) := \,
\...
- 1,847
5
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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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5
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156
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Kostka polynomials in root systems other than A
The q, t - Kostka polynomials $K_{\lambda\mu}(q, t)$ are defined as follows (all notations I do not explain here come from the classical book by Macdonald: Symmetric Functions and Hall polynomials, ...
user38762
4
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260
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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
$$
\...
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4
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Spectrum of Kernel - Discrete orthogonal polynomials
Trying to solve a problem, I encounter a Kernel of the form
$$K(m,n)= e^{-\frac{\beta}{4} (m+n+1)} \frac{2^{2+\frac{m+n}{2}}}{\sqrt{m! n!}} \frac{\sqrt{\pi}}{n-m} \left[ \frac{1}{\Gamma(-m/2)\Gamma(...
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Christoffel-Darboux type identity
The classical Christoffel-Darboux identity for Hermite polynomials reads
$$\sum_{k=0}^n\frac{H_k(x)H_k(y)}{2^k k!}=\frac{1}{2^{n+1} n!}\frac{H_{n+1}(x)H_n(y)-H_n(x)H_{n+1}(y)}{x-y}.$$
I am ...
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Closed formula for $\sum\limits^\infty_{k=0}\frac1{(k+a)(k+b)} L^1_k(x)L^1_k(y) $
Let $ L^{\alpha}_{n}(x)=\sum^{n}_{k=0} \binom{n+\alpha}{n-k}\big(-1\big)^{k}\frac{x^{k}}{k!},\alpha>-1$ be Laguerre polynomials of type $ n$.
Is there a closed formula for $$\sum^{\infty}_{k=0}\...
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3
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105
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Tight bounds on the iterates of the Clenshaw algorithm for Chebyshev polynomials
I'm trying to bound the iterates of the Clenshaw algorithm when applied to the Chebyshev series, related to a question I'm running into related to the stability of this algorithm. Recall that, for $p(...
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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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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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3
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67
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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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3
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472
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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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3
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113
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Fourier coefficients of positive polynomials
Let $p(x) \geq 0$ be a positive polynomial on the hypersphere ($x \in S^{n-1}$) satisfying $\int_{S^{n-1}} p(x) = 1$. Writing $p(x) = \sum_{j=0}^s p_j(x)$ where $p_j(x) = \sum_m p_{jm} s_{jm}(x)$ with ...
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3
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208
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A "nice" Orthogonal Basis for Translation Invariant Symmetric Polynomials
It is going to be a rather long question, so I will first state it and then try to explain and motivate it.
Take $\Lambda_n $ as the graded ring of symmetric polynomials of a field $F$ in $n$ ...
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67
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Polynomials connected with Gale's condition and cyclic polytopes
I was reading about cyclic polytopes and (I think naturally) became interested in polynomials $f(X) \in \mathbb{R}[X]$ which have a factorization of the form $f(X)=g(X)g(X+1)$ for some $g(X) \in \...
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197
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A generalization of Macdonald functions?
I am interested in finding a set of functions $f(z_1,\cdots ,z_k;q,\,t)$, conjecturally polynomials, which depend on two parameters $(q,t)$ and an integer $k$, and are orthogonal under the following ...
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3
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Integral involving Laguerre polynomial
In an engineering application, I've been trying to calculate the following integral involving Laguerre polynomials:
$\int_{-\infty}^\infty dx L_n(x^2+\beta^2) e^{-x^2/2+i x \alpha}$,
where $n=0,1,2,3,\...
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2
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61
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Iterated chaos expansion
Using the notations from Normal Approximations with Malliavin Calculus, Chapter 2
random variables $F$ in the probability space generated by an iso-normal Gaussian family $X(h)$,
$$E[X(h)X(g)] = \...
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2
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63
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The $n$-th reproducing kernel of orthogonal polynomial
Let $N$ be a non negative integer. Define the sequence of monic orthogonal polynomials $\{P_n(x)\}_{n}$ with respect to the inner product
$$
\langle f , g\rangle =\sum^{N}_{k=0}{f(k)g(k)\rho(k)}
$$
...
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measure corresponding to certain orthogonal polynomials
Given a scalar $\lambda\in (0,1)$, consider a sequence of monic polynomials $\{p_n(x)\}_{n\geq 0}$ over real variables satisfying the following recurrence relations:
$xp_n(x)=p_{n+1}(x)+\lambda p_{n}(...
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2
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Explicit error bounds for orthogonal polynomials with exponential weights
Let $\rho > -1$, and define the weight function $W_{\rho}(x) = |x|^{\rho} \exp(-2|x|)$. Associated with this weight is the sequence of orthogonal polynomials $\{ p_{n}(x) \}_{0}^{\infty}$, where $...
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Finite version of Mehlers formula?
This is a crosspost from Math Stack Exchange, please let me know if this is not an appropriate use of crossposting, and I will delete.
Mehler's formula is the following identity for Hermite ...
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2
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Deduce Sheffer's classification of orthogonal polynomials of A-type 0
Theorem 1.9 in Daniel Galiffa and Tanya Riston's paper, An elementary approach to characterizing Sheffer A-type 0 orthogonal polynomial sequences, 2015, presents without proof Isador Sheffer's ...
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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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2
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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 ...
user85801
2
votes
0
answers
110
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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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2
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110
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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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2
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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)};$$
where we set $x=1$ and both ...
2
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0
answers
52
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Function expansion in high partition Jack polynomials?
For Jack polynomials $J_\lambda^\alpha(x_1,x_2,...,x_n)$ there exists i.e. the following relation
$$\sum_\lambda J_\lambda^\alpha(x_1,x_2,...,x_n)J_{\lambda'}^{\frac{1}{\alpha}}(y_1,y_2,...,y_m)=\...
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answers
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Octahedron and System of trigonometric equations
Could somebody help me to prove the following?
$$\sum_{k=1}^6 \cos(2 \theta_k) (\cos(2\phi_k)-1))=0$$
$$\sum_{k=1}^6 \sin(2 \theta_k) (\cos(2\phi_k)-1))=0$$
$$\sum_{k=1}^6 \cos (\phi_k)=0$$
$$\sum_{k=...
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2
votes
0
answers
129
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Hermite interpolation
I need a help to my problem, I would be grateful if anyone could help.
Let $\epsilon \in [0,1]$ and for an integer $n$ we consider a set of nodes $T_n={t_0,t_1,....t_n}$.
We define the function $f(x)...
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2
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0
answers
124
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Perturbation analysis for three term recurrences
Jacobi polynomials, denoted by $J^{(\alpha,\beta)}_n$, on $[-1,1]$ satisfy a three term recurrence
$$ J_{n+1}^{(\alpha,\beta)}(x) = (A_n+B_nx)J^{(\alpha,\beta)}_n + C_nJ_{n-1}^{(\alpha,\beta)}(x), \...
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2
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133
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Lower asymptotic bounds for the derivative of Laguerre polynomials
Let $ L_{d}^{(1)}(x)$ denote the generalized Laguerre polynomial of degree $d$ and order $\alpha=1$. Clearly, since all the roots $r_1,\dots,r_d$ of $L_{d}^{(1)}$ are simple, there exists a strictly ...
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2
votes
0
answers
187
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Growth of the recurrence coefficients of orthogonal polynomials
Consider the sequence of measures $$d\mu_N(x)=e^{-NV(x)}dx$$ on the real axis, where $V$ is continuous and satisfies the growth assumption $$\lim_{|x|\rightarrow\infty}(V(x)-2\log|x|)=+\infty.$$
...
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2
votes
0
answers
500
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About a Christoffel-Darboux-type sum
I've been using the Christoffel-Darboux identity for the Hermite polynomials,
$$\sum_{k=0}^n\frac{H_k(x)H_k(y)}{2^k k!}=\frac{1}{2^{n+1} n!}\frac{H_{n+1}(x)H_n(y)-H_n(x)H_{n+1}(y)}{x-y},$$
for some ...
- 2,318
2
votes
0
answers
171
views
Orthogonality of Pfaffian polynomials in $SO(2m)$
I've been struggling here to invert some integral equation involving Pfaffians, and it would be very nice if you could shed some light on the problem.
Let's go to it. Let $V=\{-1,1\}^{m}$ and $S = \...
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1
vote
0
answers
61
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Product of d-dimensional Legendre polynomials
Let $P_n:\mathbb{R}\rightarrow\mathbb{R}$ be the $d$-dimensional Legendre polynomials, that is they are orthonormal polynomials w.r.t the probability measure $\mu_d$ on $[-1,1]$ given by $\mu_d= \...
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1
vote
0
answers
116
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Algorithm for converting from 2D Legendre basis to 2D Monomial basis
I am dealing with function written in a 2D Legendre polynomial basis and I need to convert it so that it's written in a 2D monomial basis. I've found of an algorithm that allows for change of basis ...
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1
vote
0
answers
201
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Proving that quotient of orthogonal polynomials is a Padé approximant of Stieltjes transform
This question is reposted from Math Stack Exchange (you can see the original post here). The motivation for reposting is that I feel like the question isn't getting much attention in MSE - if there is ...
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1
vote
0
answers
71
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Geometric series involving the Laguerre polynomials
Let put $\alpha=5$ and $x=3$. Consider the following set given by
$$M=\lbrace \; n \in N, \; \; 0 < |L_{n}^{5}(3)| < 1 \; \rbrace$$
Where $L_{n}^{\alpha}(x)$ is the generalized Laguerre ...
1
vote
0
answers
133
views
Generating Hermite polynomial with coefficient recurrance relation algorithm
I am writing a math paper for my numerical analysis class about orthogonal Hermite polynomials. I want to implement the algorithm for generating the "probabilist's Hermite polynomials":
$$
\...
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1
vote
0
answers
137
views
A holonomic function and its singularity
The following series where $q_i , h$ are constant parameters. $G(z)$ is a rational function.
$$F(x):=\sum_{d= 1}^\infty \sum_{k=1}^d (-1)^{d-k} \, s_{(k, 1^{d-k})}(\tfrac{q_1}{h}, \tfrac{q_2}{h}, \...
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vote
0
answers
194
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Bounds on coefficients $c_i$ of Chebyshev expansion $f(x) = \sum_{k=0}^{n} c_kT_k(x) : [-1,1] \mapsto [-1,1]$
Let $n$ be a given positive integer and let $f(x) = \sum_{k=0}^{n} c_kT_k(x)$, where $c_i \in \mathbb{R}$, $0 \leq i \leq n$. If
$|f(x)| \leq 1$, for $|x| \leq 1$, is it possible to get the maximum ...
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