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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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 $a_k$ ...
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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 ...
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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 ...
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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 ...
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1answer
708 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} ...
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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, ...
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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 ...
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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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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?