A new family of expansions of real numbers

For $\alpha >1$ we represent a real number in $(0,1]$ in the form $$\sum _{i=1}^{\mathrm{\infty }}(\alpha -1{)}^{i-1}{\alpha }^{-(d_1+\cdots +d_i)}$$ with $d_i\in \mathbb{N}$. We discuss ergodic theoretical and dimension theoretical aspects of this expansion. Furthermore we study their base-change-transformation.