On schizophrenic patterns in $b$-ary expansions of some irrational numbers

Abstract:

In this paper we study the $b$-ary expansions of the square roots of the function defined by the recurrence $f_b(n)=b f_b(n-1)+n$ with initial value $f(0)=0$ taken at odd positive integers $n$, of which the special case $b=10$ is often referred to as the “schizophrenic” or “mock-rational” numbers. Defined by Darling in $2004$ and studied in more detail by Brown in $2009$, these irrational numbers have the peculiarity of containing long strings of repeating digits within their decimal expansion. The main contribution of this paper is the extension of schizophrenic numbers to all integer bases $b\geq 2$ by formally defining the schizophrenic pattern present in the $b$-ary expansion of these numbers and the study of the lengths of the non-repeating and repeating digit sequences that appear within.