$Q_S$ representation of numbers$ x ∈ [0,1]$ is a generalization of classic ternary representation and is determined by the parameters $q_0,q_1,q_2,...q_{s-1}  (q_i>0, \sum_{i=0}^{s-1} q_i=1$ and the following equality 

$x=β_{α_1(x)} + \sum_∞^{k=2} [β_{α_k(x)} \prod_{j =1}^{k-1} q_{a_j(x)}] ≡ \Delta_{a_1(x) a_2(x)...a_n(x)...}^{Q_S}$,

where $a_k(x) ∈ A_S ≡$ {$0,1,2,...,s-1$}, $β_0=0, β_i=\sum_{j=0}^{i-1} q_j$. For a given $Q_S$-representation, we study functions 

$ω_n(\Delta_{a_1 a_2...a_n...}^{Q_S}) = \Delta_{a_1 a_2...a_{n-2} a_{n-1} a_{n+1} a_{n+2}...}^{Q_S}$

and

$f_{ni}(\Delta_{a_1 a_2...a_n...}^{Q_S}) = \Delta_{a_1 a_2...a_{n-1} i a_n a_{n+1} ...}^{Q_S}$,where $n,i ∈ N$.

We prove that all functions are piecewise continuous and have a finite number of points of discontinuity of the first kind. Their analytic expression is found. The Gauss-Kuzmin problem is also solved.

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