Journal of Mathematical Physics, Analysis, Geometry 2022, vol. 18, No 1, pp. 105-117   https://doi.org/10.15407/mag18.01.105     ( to contents , go back ) ### Implicit Linear Nonhomogeneous Difference Equation over ℤ with a Random Right-Hand Side

S.L. Gefter

V. N. Karazin Kharkiv National University, 4 Svobody Sq., Kharkiv, 61022, Ukraine
E-mail: gefter@karazin.ua

A.L. Piven'

V. N. Karazin Kharkiv National University, 4 Svobody Sq., Kharkiv, 61022, Ukraine
E-mail: aleksei.piven@karazin.ua

Received January 13, 2021, revised February 22, 2021.

Abstract

Let $\{f_n\}_{n=0}^{\infty}$ be a sequence of independent identically distributed integer- valued random variables which are defined on a probability space ($(\Omega,{\cal F},P)$. We assume that these variables have a non-degenerate distribution. Let $a$ and $b$ be integers, $b\not=0,\pm 1$, and let $a$ be not divisible by $b$. For every $\omega\in\Omega$, we consider the implicit first-order linear nonhomogeneous difference equation $bx_{n+1}+ax_{n}=f_n(\omega)$, $n=0,1,2,\ldots$. It is proved that the probability that there exists an integer solution of this implicit difference equation is equal to zero. Hence, under the random choice of integers $f_0,f_1,f_2,\ldots$, the implicit linear difference equation $bx_{n+1}+ax_{n}=f_n$, $n=0,1,2,\ldots$, has no solutions in integers. We also prove that if $a$a and $b$ are co-prime integers, then the solvability set for this difference equation is an uncountable dense meagre set in the space of all sequences of integers.

Mathematics Subject Classification 2010: 39A06, 39A10, 39A50
Key words: difference equation, independent random variables, solvability set