3 problems

Let $A\subset\mathbb{N}$ have positive upper Banach density, and let $k\in\mathbb{N}$. For a finite subset $F$ of $B$, write its sum as $\sum_{n\in F}n$. Higher-order shifted finit…

Let $\mathbb{Q}\setminus\{0\}$ be partitioned into finitely many sets, … For a sequence $\langle x_n\rangle_{n=1}^{\infty}$, let $FS(\langle x_n\rangle_{n=1}^{\infty})$ denote its…

Let $n$ be a natural number, and let $f$ and $g$ be differentiable functions. Differentiable-functions identity. … The preceding identities establish special cases involving linear…