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 $(s_{\ast }(n){)}_{n\in \mathbb{N}}$, given by $s_{\ast }(0)=u(0)$, $s_{\ast }(n)={\overline{s(n)s_g(n-1)}}^b,n\ge 1$. 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 $(u(n){)}_{n\in \mathbb{N}}=\mathbb{N}\setminus \{0\}$.

arXiv commentDefinitions 1, 2, and 3 are new. More details about "holonomicity per range'' have been added