On expansions of numbers in alternating s-adic series and Ostrogradskii series of the first and second kind

Abstract

We present expansions of real numbers in alternating $s$-adic series $(1 < s ∈ N)$, in particular, $s$-adic Ostrogradskii series of the first and second kind. We study the “geometry” of this representation of numbers and solve metric and probability problems, including the problem of structure and metric-topological and fractal properties of the distribution of the random variable $$ξ = \frac1{s^{τ_1−1}} + ∑^{∞}_{k=2}\frac{(−1)^{k−1}}{s^{τ_1+τ_2+...+τ_k−1}},$$ where $τ_k$ are independent random variables that take natural values.