The maximal order of hyper-($b$-ary)-expansions

TL;DR

This paper provides a new proof for the maximal number of hyper-($b$-ary)-expansions of a nonnegative integer across various bases, extending previous results with novel methods.

Contribution

It introduces a new proof technique for determining the maximal order of hyper-($b$-ary)-expansions, generalizing prior findings for all integer bases $b \\geq 2$.

Findings

Determined the maximal order of hyper-($b$-ary)-expansions for all bases $b \\geq 2$

Extended previous results using new proof methods

Confirmed the bounds for the number of expansions in general bases

Abstract

Using methods developed by Coons and Tyler, we give a new proof of a recent result of Defant, by determining the maximal order of the number of hyper-($b$-ary)-expansions of a nonnegative integer $n$ for general integral bases $b\geqslant 2$.