Representations for real numbers

A. Ghiyasi^a, I. P. Mikhailov^b, V. N. Chubarikov<sup>c

a</sup> Allameh Tabataba’i University (Iran)
^b Kazan Aviation Institute (Leninogorsk)
^c Lomonosov Moscow State University (Moscow)

Abstract: In this paper theorems on the representations of real numbers $\alpha$ by using infinite iteration of a sequence of positive monotonic functions $\alpha_n=f_n(x_n)$ in the form
$$ \alpha=\lambda_0+f_1(\lambda_1+f_2(\lambda_2+f_3(\lambda_3+\dots))), $$
where “digits” $\lambda_n, n\geq 0,$ and “remainders”
$$ r_n=r_n(\alpha)=f_{n+1}(\lambda_{n+1}+f_{n+2}(\lambda_{n+2}+f_{n+3}(\lambda_{n+3}+\dots))), n\geq 0, $$
are defined by the following recurrent formulas
$$ \lambda_0=[\alpha], r_0=\{\alpha\}, $$

$$ \lambda_n=[\varphi_n(r_{n-1}(\alpha))], r_n=\{\varphi(r_{n-1})\}, $$
moreover $\{z\}$ and $[z]$ denote accordingly the fractional and the integral parts of the real number $z,$ and $x_n=\varphi_n(\alpha_n), n\geq 1,$ are inverse functions of $\alpha_n=f_n(x_n).$
In particular, the representation of the number $\alpha$ by using function $f(x)=\frac 1x$ leads to the continued fraction of the number $\alpha.$ The general case when $f(x)$ is decreasing function have been considered by B.H. Bissinger (1944) and A. Rényi (1957). For the function $f(x)=\frac xq$ as $q\geq 2$ is the natural number, is obtained $q$-adic the representation of the form $\alpha=\sum\lambda_nq^{-n},$ where digits $\lambda_n, n\geq 1,$ can to receive all integral values from $0$ to $q-1.$ The case when $f(x)$ is increasing function have been investigated by C.I. Everett (1946) and A. Rényi (1957). The representation $\alpha$ for $f(x)=\frac x\theta$ is nonintegral number $\theta>1$ have been studied A. Rényi (1957) and A.O. Gelfond (1959). In the present paper for the sequence of functions $f_n(x)=\frac x{q_n}, q_n\geq 2,$ are integer, has been investigated the representation of $\alpha$ on the multiplicative system of numbers as $n\geq 1$ in the form
$$ \alpha=\lambda_0+\frac{\lambda_1}{q_1}+\dots +\frac{\lambda_n}{q_1\dots q_n}+\frac{x_n}{q_1\dots q_n}, $$
where digits $\lambda_n$ can to receive integral values from $0$ to $q_n-1.$
A. Kh. Ghyasi (2007) has been generalized Gelfond theorem concerning the multiplicative system of numbers. Let $\theta_n, n\geq 1, $ be a sequence of real numbers, each of which greater than $1$. Then any real number $\alpha, 0<\alpha<1,$ can be represented in the form $\alpha=\sum\limits_{k=1}^n\frac{\lambda_k}{\theta_1\dots\theta_k}+\frac{x_n }{\theta_1\dots\theta_n}, n\geq 1,$ where the sequence $x_n$ of error terms is defined by recurrence
$$ x_0=\{\alpha\}, x_1=\{\theta_1x_0\}, x_n=\{\theta_nx_{n-1}\},\dots, $$
and the sequence of integers $\lambda_n$ is defined by the rule
$$ \lambda_0=[\alpha], \lambda_1=[\theta_1x_0],\dots,\lambda_n=[\theta_nx_{n-1}],\dots. $$


Keywords: $q$-adic expansion, continued fraction, multiplicative number system.

UDC: 511.3

Received: 25.01.2025
Accepted: 07.04.2025

DOI: 10.22405/2226-8383-2025-26-2-61-70