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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How does this relationship between the Catalan numbers and SU(2) generalize?

This is a question, or really more like a cloud of questions, I wanted to ask awhile ago based on this SBS post and this post I wrote inspired by it, except that Math Overflow didn't exist then. As ...
20
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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 ...
14
votes
2answers
982 views

Minimal polynomial with a given maximum in the unit interval

Find the lowest degree polynomial that satisfies the following constraints: i) $F(0)=0$ ii) $F(1)=0$ iii)The maximum of $F$ on the interval $(0,1)$ occurs at point $c$ iv) $F(x)$ is positive ...
14
votes
1answer
548 views

Can you name these orthogonal polynomials?

I have a collection of orthogonal polynomials in infinitely commuting variables $x_1, x_2, x_3, \ldots$. I think they must be well known (perhaps Schur or Hermite polynomials or some variant ...
13
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0answers
672 views

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 ...
11
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1answer
330 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
238 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 - ...
10
votes
3answers
601 views

MacWilliams Identity for Asymptotic Weight Spectrum of a Code

Introduction Let $C$ be a code of block length $n$ having $A_i^C$ words of Hamming weight $i$, for $i\in [n]$, where $[n]:=\{0,\ldots,n\}$. Then, the sequence $\{ A_i^C \}_{i \in [n]} $ is called the ...
9
votes
2answers
226 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 $\{ x_j, w^j \} _j$ for large N. My question is how to do it, and why should it ...
8
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0answers
171 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 ...
7
votes
2answers
913 views

Multi-dimensional moment problem

Let $\mu$ be a measure on $\def\r{\mathbb{R}}\r^n$, $1\le n \le \infty$. Given a (finite) multi-index $\bar{i} = (i_1, i_2, \ldots)$, one can define the moment $$ m_{\bar i} = \int x_i^{i_1} ...
7
votes
4answers
481 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 ...
6
votes
4answers
3k views

Visualizing Orthogonal Polynomials

Recently I was introduced to the concept of Orthogonal Polynomials through the poly() function in the R programming language. These were introduced to me in the concept of polynomial transformations ...
6
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2answers
1k views

Spread polynomials

Norman Wildberger's "rational trigonometry" has been viewed by some mathematicians as a clever new take on an ancient topic. Wildberger's "spread polynomials" $S_n$ are characterized by the identity ...
5
votes
1answer
324 views

Lower bound of integral involving Laguerre polynomials

I want to lower bound the expected value of the square root of a randomly chosen eigenvalue of a Wishart matrix. To get the bound I want I need a lower bound on $$T_n = ...
5
votes
1answer
701 views

Integral identity for Legendre polynomials

How does one prove the following integral identity, where $P_n(x)$ is the $n$th Legendre polynomial? $$ \int_0^1 P_n(2t^2-1) dt = \frac{(-1)^n}{2n+1} $$ Notes & Background A variant of this ...
5
votes
1answer
438 views

Numerically finding a Mercer expansion for a given covariance kernel

Let $c(r)$ be a nice, continuous function with compact support. For example, $c(r) = \tfrac 1 5 (1-r)^{11} \big( 5 + 55r + 239 r^2 + 429 r^3 \big)$ for $r \in [0,1]$, and $c(r) = 0$ otherwise. On ...
5
votes
1answer
517 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 ...
5
votes
1answer
1k views

Two-variable generating functions for Laguerre polynomials

Where can I find generating functions for orthogonal polynomials in two variables? Lebedev's book (Special Functions and their Applications, Dover, 1972) gives a closed form for $$ \sum_{n=0}^\infty ...
4
votes
3answers
2k views

Finding a recursion for a sum of Legendre polynomials

The polynomial $a_n(x):=P_n(x)-\frac{n-1}{n}P_{n-2}(x)$ where $P_n(x)$ is a Legendre polynomial came up while I was investigating methods for estimating the error in Gaussian quadrature. I am ...
4
votes
4answers
1k views

Orthogonal polynomials/functions on the interval [0,1] but with same weight as Gegenbauer polynomials

I am looking for an othogonal basis of functions over the interval $[0,1]$ with weight function $(1-x^2)^{\alpha-1/2}$. Gegenbauer polynomials are frustratingly close to what I need, but they are ...
4
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3answers
1k views

Convergence of orthogonal polynomial expansions

"Everyone" knows that for a general $f\in L^2[0,1]$, the Fourier series of $f$ converges to $f$ in the $L^2$ norm but not necessarily in most other senses one might be interested in; but if $f$ is ...
4
votes
1answer
571 views

Why are all these families of polynomials finally log-concave?

This started when I was examining certain families of unimodal polynomials, i.e. $\sum_{k=0}^n a_kx^k$ where $a_0\le a_1\le\cdots \le a_k\ge\cdots \ge a_n$. (Notation: in the following, the $a_k$ ...
4
votes
2answers
443 views

Polynomials from a Recurrence Relation

Hi guys, I have recently started looking at polynomials $q_n$ generated by initial choices $q_0=1$, $q_1=x$ with, for $n\geq 0$, some recurrence formula $$q_{n+2}=xq_{n+1}+c_n q_n$$ where $c_n$ is ...
4
votes
2answers
336 views

On $a^4+nb^4 = c^4+nd^4$ and Chebyshev polynomials

In a 1995 paper, Choudhry gave a table of solutions to the quartic Diophantine equation, $a^4+nb^4 = c^4+nd^4\tag{1}$ for $n\leq101$. Seiji Tomita recently extended this to $n<1000$ and solved ...
4
votes
1answer
100 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 ...
4
votes
1answer
141 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 ...
4
votes
1answer
151 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 ...
3
votes
1answer
650 views

Where does the Chebyshev polynomial notation come from?

The $k$th Chebyshev polynomial is denoted by $T_k$ where $T_k(x) = \cos(k\cos^{-1}(x))$ I was wondering where this notation came from. It has been suggested that it comes from Tschebyscheff (the ...
3
votes
1answer
455 views

When can a family of polynomials get a weight function to be made orthogonal?

Let $\lbrace P_n(z)\rbrace_{n\in\mathbb N_0}$ be a family of polynomials defined by a generating function $g(t,z)=\sum\limits_{n=0}^\infty P_n(z)t^n$ or by a contour integral $P_n(z)=\frac1{2\pi ...
3
votes
1answer
420 views

asymptotic behaviour of a sum

I'd like to know the asymptotic behaviour as $N\to\infty$ of the following sum $$ Z_N(x) := 2^{-N/2} \sum_{k=0}^{N/2} \frac{N!}{k! (N-2k)!} (N-1)^{-k} (\sqrt{2} x)^{N-2k} $$ in order to compute ...
3
votes
1answer
561 views

Pochhammer symbol of a differential, and hypergeometric polynomials

I have a minor result which I'm sure has come up somewhere before but I can't seem to find it. Consider a confluent hypergeometric function of the form $$\newcommand{\ff}{{}_1F_1} ...
3
votes
0answers
54 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 ...
3
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0answers
85 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 $n$ ...
3
votes
0answers
57 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 ...
3
votes
0answers
156 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)~, ...
3
votes
0answers
631 views

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 ...
2
votes
3answers
311 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 ...
2
votes
1answer
684 views

Closed form for an orthogonal polynomial integral?

The following integral came up in one of my applications: $\int_{-1}^1P_n(x)T_j(x)T_k(x)\mathrm{d}x$ where $P_n(x)$ is a Legendre polynomial, $T_k(x)$ is a Chebyshev polynomial, and $j$, $k$, and ...
2
votes
1answer
180 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 ...
2
votes
1answer
127 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 ...
2
votes
1answer
240 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 ...
2
votes
0answers
150 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 ...
2
votes
0answers
95 views

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), ...
2
votes
0answers
88 views

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 ...
2
votes
1answer
629 views

Three term recurrence relation.

For given $n,\ell\in\mathbb N_0$, I am interested in studying the following recursion relation for some $\mu\in\mathbb R$: $$\sqrt{-1} \tfrac{j(\ell-j+1)(n-j+1)(n+j+1)}{2(2j-1)(2j+1)} a_{j-1} - ...
2
votes
0answers
153 views

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 $$\lim_{|x|\rightarrow\infty}(V(x)-2\log|x|)=+\infty.$$ ...
2
votes
0answers
318 views

About a Christoffel-Darboux-type sum

Hi! I've been using the Christoffel-Darboux identity for the Hermite polynomials, $$\sum_{k=0}^n\frac{H_k(x)H_k(y)}{2^k k!}=\frac{1}{2^n n!}\frac{H_{n+1}(x)H_n(y)-H_n(x)H_{n+1}(y)}{x-y},$$ for some ...
2
votes
0answers
127 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 = ...
1
vote
4answers
1k views

Functions orthogonal to x^n

I would like to ask if there are any set of functions $u_n(x)$ which is orthogonal to $x^n$? i.e.: $\int_0^1 x^n u_m(x) dx = \delta_{n,m}$ Edit: For clarification, this question asked for all ...