# 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.

212 questions
Filter by
Sorted by
Tagged with
1 vote
133 views

### Laguerre polynomial and series

Let $L^\alpha_n(x)$ be Laguerre polynomials of type $n$. Consider the sum $$\sum^\infty_{j=0} \frac{1}{(b-j)}L^{m}_j(x)$$ where $b\not\in\Bbb N,x>0$ and $m\in \Bbb N$. I have found this series ...
74 views

1 vote
133 views

### 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 ...
107 views

### 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 ...
318 views

### Are the “generalized Catalan numbers” of Dumitrescu–Mulase the "moments" of some "multivariate Wigner semicircle distribution"?

The classical Catalan numbers $$C_n = \frac{1}{n+1} \binom{2n}{n},$$ well-known for their numerous combinatorial interpretations (the second volume of Stanley's Enumerative Combinatorics famously ...
Consider the "probabilist's" Hermite polynomials given by $$H_n(x)=(-1)^ne^{\frac{x^2}{2}}\partial_x^ne^{-\frac{x^2}{2}}.$$ These polynomials trivially extend to functions of $w\in\mathbb{C}$...