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

• Datum: 2020
• Автори та афіліації: I.M. Lysentko — National Pedagogical Dragomanov University; Yu.P. Maslova — National Pedagogical Dragomanov University; M.V. Pratsevytyi — National Pedagogical Dragomanov University
• Ключові слова: keywords
• Hauptverfasser: Lysentko, I.M., Maslova, Yu.P., Pratsevytyi, M.V., Лысенко, И.М., Маслова, Ю.П., Працевитый, Н.В., Лисенко, І.М., Працьовитий, М.В.
• Format: Artikel
• Sprache: Ukrainisch; Englisch
• Veröffentlicht: Інститут математики НАН України 2020
• Online Zugang: https://trim.imath.kiev.ua/index.php/trim/article/view/395
• Назва журналу: Transactions of Institute of Mathematics of NAS of Ukraine

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.