0
votes
1answer
108 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
172 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 ...
4
votes
2answers
304 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 ...
1
vote
0answers
59 views

Jack symmetric functions and their inner products

I have some questions regarding Jack polynomials. I use the notation of of I.G. Macdonald's book "Symmetric Functions and Hall polynomials". Let $\Lambda$ be the ring of symmetric functions over ...
3
votes
1answer
383 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 $a_k$ ...
1
vote
0answers
128 views

Polynomials satisfying a three-term recurrence

Let ${p_n}(x) = x{p_{n - 1}}(x) - {a_{n - 2}}{p_{n - 2}}(x)$ for some numbers ${a_n}$ with initial values ${p_{ - 1}}(x) = 0$ and ${p_0}(x) = 1.$ By Favard’s theorem about orthogonal polynomials ...
3
votes
1answer
398 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 ...
2
votes
0answers
143 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 $$\lim_{|x|\rightarrow\infty}(V(x)-2\log|x|)=+\infty.$$ ...
0
votes
1answer
296 views

Upper bounds on generalized Laguerre polynomials

I evaluated an integral and obtained an expression with a Laguerre polynomial. I'd like something more explicit and useable. Are there any known simple (e.g. exponential) upper bounds on the ...
4
votes
2answers
422 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 ...
14
votes
2answers
968 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 ...
0
votes
1answer
452 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 ...