that it is singular is that equation cannot be used, in this case, to define all the variables $Q_{{{\alpha}},k}$ with $k<0$ because, for example, $Q_{{{\alpha}},-1}=0$ in this case.

More general initial conditions, of the form $$\label{deformedinitial}
Q_{{{\alpha}},0}=b_{{\alpha}}, \ Q_{{{\alpha}},1}=a_{{\alpha}}$$ for some formal variables $a_{{\alpha}}, b_{{\alpha}}$ give a solution of (a special case of) the “deformed Q-system” introduced in [@DFK]. We will explain this point in Section \[discussion\].

### Normalized $Q$-systems

In this paper we deal with cluster algebras without coefficients. To see that it is possible to reformulate the $Q$-system in this way, a version of the following Lemma was given in [@Ke07]:

\[renormalize\] There exist complex numbers $\epsilon_1,...,\epsilon_r$ (all of them roots of 1) such that the variables ${{R}}_{{{\alpha}},i}=\epsilon_{{\alpha}}Q_{{{\alpha}},i}$ satisfy the normalized $Q$-system $$\label{tQsystem}
{{R}}_{{{\alpha}},k+1} {{R}}_{{{\alpha}},k-1} = {{R}}_{{{\alpha}},k}^2 + \prod_{\beta\sim{{\alpha}}}
\widetilde{{\mathcal T}}^{({{\alpha}},\beta)}_k,$$ where $$\widetilde{{\mathcal T}}^{({{\alpha}},\beta)}_k = \prod_{i=0}^{|C_{{{\alpha}},\beta}|-1}
{{R}}_{\beta,\lfloor\frac{t_\beta(k+i)}{t_{{\alpha}}} \rfloor}.$$

From Equation we note that the total degree of $Q_{\beta,j'}$ (for various $j'$) in ${\mathcal T}^{({{\alpha}},\beta)}_k$ is always $|C_{{{\alpha}},\beta}|$, where $C$ is the Cartan matrix of ${{\mathfrak{g}}}$. Therefore ${{R}}_{{{\alpha}},k}=\epsilon_{{\alpha}}Q_{{{\alpha}},k}$ satisfy the following system: $${{R}}_{{{\alpha}},k+1}{{R}}_{{{\alpha}},k-1}={{R}}_{{{\alpha}},k}^2 - (\prod_{\beta}
\epsilon_\beta^{C_{{{\alpha}},\beta}})\prod_{\beta\sim{{\alpha}}}
\widetilde{{\mathcal T}}^{({{\alpha}},\beta)}_k,$$ where ${{\widetilde{\mathcal T}}}$ is just ${\mathcal T}$ with the variables $Q$ replaced by ${{R}}$.

Thus, if the variables $\epsilon_{{\alpha}}$ satisfy $$\prod_{\beta=1}^r 