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6
votes
4answers
2k 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 ...
13
votes
0answers
630 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 ...
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 ...
0
votes
1answer
457 views

modular arithmetic of Hermite polynomials

I wonder if there is anything known (formula, asymptotics, etc) of computing the remainder $R_{k,m} \equiv H_{k} ~ \mod H_m$ for $k > m$, where $H_m$ denotes the $m$th Hermite polynomial ...
2
votes
1answer
654 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 ...
1
vote
4answers
927 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 ...
3
votes
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 ...
23
votes
4answers
2k views

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 ...