# Tag Info

Accepted

### Integer Polynomial solutions to functional equation

Note that $L(f):=(2x+1)^2f(x+1)-4x(x+n+1)f(x)$ is a linear operator which maps the $\mathbb{Q}$-linear space $\pi_n$ of polynomials $h(x)\in \mathbb{Q}[x]$ of degree at most $n$ to itself. Assume that ...
• 91k
Accepted

### Combinatorial Identity with Connection Coefficients and Falling Factorial $\langle i x\rangle_n$

It is very probable that what is written below is the simplification of Darij's argument. I use the notation $x^{\underline{n}}=x(x-1)\dots(x-n+1)$ [as in Knuth's books] for the falling factorial, and ...
• 91k
Accepted

### for which values of $\theta$ does this equation $x_{n+1}=\cos(\theta)x^2_{n}-\sin(\theta)x^2_{n-1}$ have bounded solutions?

I ran a small computation: The following plot is created as follows: For each point $r(\cos \theta, \sin \theta)$, I use $x_{-1} = 0$, $x_0 = r$ as initial values, and $\theta$ as the parameter. The ...
• 14.8k

### Combinatorial Identity with Connection Coefficients and Falling Factorial $\langle i x\rangle_n$

This is correct. Let me prove a more general fact: Theorem 1. Let $i$, $j$ and $n$ be three nonnegative integers such that $i\leq n$ and $j\leq n$. Let $P\in\mathbb{Q}\left[ X\right]$ be a ...
• 31.5k
Accepted

• 85.1k