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 ...

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47 views

Spectrum of Kernel - Discrete orthogonal polynomials

Trying to solve a problem, I encounter a Kernel of the form $$K(n,m)=\frac{2^{m+n} q^{(m+n+1)}}{n-m} \left[\Gamma(\frac{n}{2}+1)\sin(\pi \frac{n}{2})\Gamma(\frac{m-1}{2}+1) \sin(\pi \frac{m-1}{2}) - ...
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1answer
84 views

Various limits of the Christoffel Darboux Kernel

In a different thread, we stumbled upon the following question: Given a continuous finite measure $w(x)dx$ on some interval $(a,b)$, $-\infty \leq a <b \leq \infty$, and the set of respective ...
12
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5answers
494 views

Applications of space filling curves

I am seeking articles where a space filling curve has been used as a theoretical application, such as in the study of general orthogonal polynomials.
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1answer
140 views

What are the best known bounds on the Hermite polynomials?

The best I could find on the net is this paper, http://arxiv.org/pdf/math/0401310.pdf Has this been improved?
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1answer
86 views

Symmetric matrix formula for Gaus-Legendre quadrature

While searching the web, I came across the following algorithm for the Gaus-Legendre quadrature. I wasn't able to find a reference or a proof of my own as to why it works. I'll present it, and the ...
7
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1answer
90 views

Fast Fourier Transforms for non-trigonometric bases

The fast Fourier transform allows decomposition into a sin/cos basis in $N \log(N)$ complexity. Can one generalize the algorithm (or the ideas used) to other bases, e.g. orthogonal polynomial bases ...
4
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0answers
102 views

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 ...
2
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1answer
84 views

Reference for the exponential decay of Legendre coefficients

In Short: I look for a reference to the proof that the spectral coefficients in the Legendre (or Jacobi) expansion are of exponential decay rate. Longer: If $p_n$ is the $n$-th Legendre polynomial, ...
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0answers
39 views

hypergeometric representation of Hermite $H_n(x)$

The DLMF has a hypergeometric representation for the Hermite polynomial $H_n(x)$ for real $x$, apparently. $$H_n(x)=(2x)^n{}_2F_0\left(\frac{-\tfrac12n;-\tfrac12n+\tfrac12}{};-\frac{1}{x^2}\right)$$ ...
2
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2answers
160 views

Do we have Karhunen–Loève expansion for White Noise?

Let $W$ be a random process (my White Noise) on $[-1,1]$ such that: $W(t)$ is a normal random variable with mean $0$ and standard deviation $1$ for all $t \in [-1,1]$ $E(W(t)W(s)) = 0$ for all $t, s ...
1
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1answer
47 views

Upper-bounding the value of a generalized Laguerre polynomial (using recurrence relation?)

I would like to produce an easily-interpretable explicit upper bound (i.e. no unspecified constants) for the function $$ f(n) := L_n^{\left(-n-\frac{d}{2}\right)}\left(-\frac{1}{2}\right), \quad n,d ...
2
votes
2answers
291 views

Why decompose a function with eigenvectors of Laplace operator? [closed]

On periodic domain, people always use Fourier basis, which eigenvectors of Laplace operator. On sphere, people use spherical harmonics, which also are eigenvectors of Laplace operator. In applied ...
2
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1answer
76 views

Octahedron and System of trigonometric equations

Could somebody help me to prove the following? $$\sum_{k=1}^6 \cos(2 \theta_k) (\cos(2\phi_k)-1))=0$$ $$\sum_{k=1}^6 \sin(2 \theta_k) (\cos(2\phi_k)-1))=0$$ $$\sum_{k=1}^6 \cos (\phi_k)=0$$ ...
3
votes
2answers
163 views

Roots of the Chebyshev polynomials of the second kind

It is known that the roots of Chebyshev polynomials of the second kind, denote it by $U_n(x)$, are in the interval $(-1,1)$. I have noticed that, by looking at the low values of $n$, the roots of ...
28
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1answer
1k views

Proving the irrationality of $\pi e$ and $\pi / e$

Rather than relying on the consequences of Schanuel's conjecture, I set about using the same ideas Apery had used to construct integer arguments converging fast enough to show $\zeta(3)$ is irrational ...
4
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0answers
62 views

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, ...
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0answers
223 views

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 ...
2
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0answers
70 views

Summation of an integral involving Laguerre polynomial and Bessel function

In an engineering setting, I reduced my problem to calculating the following sum: $$\sum_{n=0}^\infty \frac{n!}{(k+n)!}\left[\int_0^a ...
2
votes
0answers
58 views

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 ...
2
votes
1answer
121 views

Orthogonal polynomials with respect to the lognormal distribution

I am currently doing some inspection on the orthogonal polynomials with respect to the lognormal distribution. Does anyone already work on that or know some cool references? All the best, Pierre-O.
3
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0answers
66 views

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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0answers
86 views

Bounds on the smallest eigenvalue of a Hankel matrix

Let $H=H_n$ be a positive definite Hankel matrix of size $n$ with $\lambda_n$ is it's smallest eigenvalue. What bounds are known on $\lambda_n$ in terms of the entries on $H$. I can see some results ...
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0answers
114 views

Evaluation of Macdonald Polynomials at $x_1=x_2=…=x_n = h$

This question is related to my previous question (here). Let $P_\lambda$(q,t) be the Macdonald polynomials with partition $\lambda$. Let $\Lambda$ denote the ring of symmetric functions over the ...
3
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0answers
115 views

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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2answers
220 views

How do I Calculate :$\int_{0}^{1}x^{k}\psi(x)dx$ where $k\geq 3$ is an integer?

How do I Calculate, if possible, in terms of well-known constants the integral : $\int_{0}^{1}x^{k}\psi(x)dx$ , where $k\geq 3$ is an integer ? note: $\psi(x)$ is digamma function. Any help would ...
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0answers
45 views

Growth of average first derivative of orthogonal polynomials

Let $L_k(t)$ be the Legendre polynomials normalized so that $$\int_{-1}^1 L_k(t)^2\,\frac{1}{2}\,dt = 1.$$ With a few identities (http://en.wikipedia.org/wiki/Legendre_polynomials), one can show that ...
10
votes
2answers
359 views

Computing Gauss Legendre Quadrature for Large N

I've been scanning across the web, and haven't found a good method to compute the Gauss Legendre abscisas and weights $\{ x_j, w^j \} _{j=1}^N$ for large $N\in\mathbb{N}$. My question is how to do it, ...
1
vote
1answer
68 views

Generating function for products of laguerre polynomials?

In a quantum physics context, I would like to evaluate $S=\sum_{n=0}^\infty z^n\cos(\pi L_n(x))$ for $z<1$. I found generating functions for squares of Laguerre polynomials but not for any higher ...
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vote
0answers
82 views

q-Hermite polynomials

It is well known that the q-Hermite polynomials defined by $$H_n(\theta; q)= \sum\limits_{k=0}^n \frac{(q;q)_n}{(q;q)_k(q;q)_{n-k}}e^{i(n-2k)\theta}$$ are orthogonal in $\theta \in [0, \pi]$ with ...
0
votes
1answer
118 views

Equality cannot hold unless $x \in \{-1,1\}$ and/or Wronskian is not zero [closed]

By playing around with assoc. Legendre polynomials, I arrived at $$((l+1)+m) (P_l^m(x))^2+((l+1)-m)(P_{l+1}^m(x))^2 = 2(l+1)x P_l^m(x)P_{l+1}^m(x).$$ Now, I want to show that we don't have equality ...
2
votes
1answer
193 views

Chebyshev polynomials factoring uniformly modulo all primes

Consider the Chebyshev polynomial of the first kind $T_n(x)$ and its factorization in $\mathbb F_p$ for a given prime $p$. Most often, this factorization is not uniform (meaning that the irreducible ...
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0answers
59 views

How to find out if a given sequence of orthogonal polynomials belongs to the Askey scheme?

I am studying some classes of orthogonal polynomials and want to find out which of them belong to the Askey scheme. To give a simple example consider the polynomials $${p_n}(x,r) = \sum\limits_{k = ...
11
votes
2answers
427 views

Orthogonal polynomial under linear transformation

Let $M_n(x) = x^n$ be the standard monomials. The binomial formula allows one to expand $M_n(ax+b)$ as a linear combination of $M_k(x)$, for $k \leq n$, giving $$ M_n(ax+b) = (ax+b)^n = \sum_{k=0}^n ...
11
votes
0answers
252 views

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 - ...
2
votes
1answer
169 views

Polynomials orthogonal w.r.t. the logarithmic weight

Recently, I have encountered the family of orthogonal polynomials $p_{n}(x)$ which is orthogonal w.r.t. the function $-\ln(x)$ on $(0,1)$. This means we have ...
7
votes
4answers
726 views

Numerical integration of legendre polynomials

I hope that numerical questions are also permitted here. I want to expand a smooth functions $f \in C^{\infty}$in terms of Legendre polynomials. Thus I need to calculate integrals of the form ...
4
votes
1answer
124 views

Legendre Q(n,x) function coefficients in terms of P(n,x) coefficients

Empirically, the Legendre functions of second kind, $Q_n(x)$, appear to be of form $$ Q_n(x)=\frac{P_n(x)}{2} \cdot\ln(\frac{1+x}{1-x})+p_n(x), $$ with $P_n(x)$ the Legendre polynomials of first kind ...
3
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0answers
59 views

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 ...
20
votes
1answer
1k views

A conjectured formula for Apéry numbers

A conjecture by the late Romanian mathematician Alexandru Lupas. Posted in sci.math in 2005, but no proof was found. Physicist Alan Sokal just reminded me of it, saying it was related to something he ...
4
votes
1answer
151 views

Differential Operator Simplification

Does anyone know the explicit formulation for the $q_k$'s in, $$(x+D)^n=\sum_{k=0}^n q_k(x)D^k\ \ \ \ ?$$ I know that $e^{-x^2/2+x}$ is a fixed point of $(x+D)$. I also, know that ...
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203 views

Littlewood-Richardson coefficients for Jack symmetric functions

Let $\Lambda$ be the algebra of symmetric functions over $\mathbb{Q}(\alpha)$. We define a scalar product $\langle \cdot,\cdot\rangle_\alpha$ on $\Lambda$ by setting $\langle ...
2
votes
1answer
248 views

What does this ODE have to do with the associated Legendre polynomials?

I am currently struggeling with the following differential equation: $$(t^2-1)f''(t)+tf'(t)(1-8a+8at^2)-4(a+a^2-2at^2+\phi (-a+2at^2))f(t)= 4\lambda f(t),$$ where $a \in \mathbb{R}$ constant, $\phi ...
5
votes
1answer
177 views

Orthogonal basis for the multilinear polynomials with zero “trace”

We say that a multilinear polynomial $P(x_1,\ldots,x_n)$ in $n$ commuting variables over $\mathbb{R}$ has zero trace if $$ \frac{d}{dt} P(t,\ldots,t) = 0. $$ Equivalently, $$ \left(\sum_{i=1}^n ...
1
vote
1answer
365 views

The Largest Root of Associated Laguerre Polynomial

The Laguerre polynomial $L_n(x)$ is the solution to the Laguerre differential equation \begin{equation*} x\,y'' + (1 - x)\,y' + n\,y = 0. \end{equation*} The associated Laguerre polynomial ...
2
votes
3answers
516 views

Multivariate Hermite Polynomials

Let $h_0, h_1, \dots$ be the classical univariate Hermite polynomials, renormalized to have constant norm. Is $$x\mapsto\prod_{j=1}^n h_{l_j}(x_j), \quad l_j\in \mathbb N$$ a complete orthogonal ...
3
votes
1answer
277 views

The asymptotic behavior of hypergeometric function around -1

Recently, in studing some specific orthogonal polynomials on unit circle, I was lead to study the asymptotic behavior of the following hypergeometric function at the neighberhood of $-1$: $$ ...
1
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1answer
278 views

Accurate bounds for derivatives of Legendre polynomials

Let $P_n(x)$ denote the $n$th Legendre polynomial. What bounds can one give for $d_{n,m}(x) = |\frac{d^m}{dt^m}P_n(t)|_{t=x}$ assuming that $|x| \le 1$? Clearly $$d_{n,m}(x) \le d_{n,m}(1) = ...
3
votes
0answers
162 views

Generalization of Frobenius formula involving Macdonald polynomials

Given a vector $\vec k=(k_1,k_2,\cdots)$ with $k_i$ are non-negative integers, the Newton polynomial $p_{\vec k}(x)$ is defined as \begin{equation} p_{\vec k}(x)=\prod_{j=1}^n p_j^{k_j}(x)~, ...
5
votes
1answer
591 views

Integral of a product of Laguerre polynomials

In order to estimate the non linear term in a particular PDE, I have to decompose $L_k^\alpha(x)^3\cdot x^{-\delta}$ (with $0<\delta<\alpha+1$) into a basis consisting of Laguerre polynomials ...
3
votes
0answers
157 views

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 ...