On an expansion real numbers on some sequences

In this paper theorems on the expression of real numbers on multiplicative number system, Fibonacci sequence and integral valued sequences satisfiing recurrent correlations and connected with Pisot–Vidgajraghavan, are proven. It pay a special attention to “explicit formulas” and conditions of the uniqueness of such representations. We note that unifiing of an expression of a real number over inverse values of a multiplicaticative system permits to get the estimation of the form
$$ e-\sum_{k=0}^n\frac 1{k!}=\frac{x_n}{n!}, \frac 1{n+1}\leq x_n<\frac 1n. $$
Expressions of numbers over the sequence of inverse of Fibonacci numbers essentially uses these representation throw powers of “the gold section” $\varphi=\frac{1+\sqrt 5}{2}.$
Systems numbers connected with Pisot–Vidgajraghavana were considered less than in details, as demands to make a properties of examinated numbers more concrete.