Numeral system with two bases having different signs and related special functions

Abstract

We consider an analytic system of encoding of numbers ($G_2$-representative) from an interval $[0;g_0]$ by means of the two-symbol alphabet $A\equiv\{0;1\}$ with two bases having different signs: $g_0\in(0;1)$ and $g_1=g_0-1$.
The system is based on expansion of the numerical series. Functions with non-homogeneous local properties of structural and differential kind are studied. Inversor of digits of the $G_2$-representation of the numbers and shift operator for the $G_2$-representation are among them.
Properties of these functions are found out being rather surprising: the inversor is not a monotonic function and the shift operator is a continuous function. This implies the fundamental difference between the present and previously studied representations.
We compare properties of the $G_2$- and two-base $Q_2$-representations (both are positive) using the projector of digits of between them.