6 problems

For natural numbers $k,n$, define $P^a_b=b!/(b-a)!$ whenever the indices are defined. Let $k,m,n$ be positive integers satisfying $1<2k\leq n-m$ and $1<2k\leq m$. Permutation ident…

General Hankel-transform conjecture. The Hankel transform is given by

Let $\Lambda=(\lambda_1,\dots,\lambda_d)$ be a finite sequence with sum … Define the polynomial $F_\Lambda(x)$ by $F_\emptyset(x)=1$ and, for $1\leq d=\sharp(\Lambda)<\infty$, … Fo…

Mixed cubic identity. The identity

Generalized Prouhet–Thue–Morse divisibility conjecture. If $k=2$, then

The generalized Prouhet–Thue–Morse identity. One has