Concatenations of Terms of an Arithmetic Progression

Abstract

Let $(u(n))_{n\in\mathbb{N}}$ be an arithmetic progression of natural integers in base $b\in\mathbb{N}\setminus \{0,1\}$. We consider the following sequences: $s(n)=\overline{u(0)u(1)\cdots u(n) }^b$ formed by concatenating the first $n+1$ terms of $(u(n))_{n\in\mathbb{N}}$ in base $b$ from the right; $s_g(n) = \overline{u(n)u(n-1)\cdots u(0)}^b$; and $(s_*(n))_{n\in\mathbb{N}}$, given by $s_*(0)=u(0)$, $s_*(n)=\overline{s(n)s_g(n-1)}^b, n\geq 1$. We construct explicit formulae for these sequences and use basic concepts of linear difference operators to prove they are not P-recursive (holonomic). We also present an alternative proof that follows directly from their definitions. We implemented $(s(n))_{n\in\mathbb{N}}$ and $(s_g(n))_{n\in\mathbb{N}}$ in the decimal base when $(u(n))_{n\in\mathbb{N}}=\mathbb{N}\setminus \{0\}$.