Arithmetical properties of the values of certain C-fractions at rational points

Let $\psi:\mathbb{Z}_{\geq 1}\rightarrow$ $\mathbb{Z}_{\geq 1}$ be increasing.We study the arithmetical properties of the values of the C-fraction \[\xi:=1+\frac{c_{1}q^{\psi\left( 1\right) }}{1}%%TCIMACRO{\QATOP{{}}{+}}%%BeginExpansion\genfrac{}{}{0pt}{}{{}}{+}%%EndExpansion\frac{c_{2}q^{\psi\left( 2\right) }}{1}%%TCIMACRO{\QATOP{{}}{+\cdots+}}%%BeginExpansion\genfrac{}{}{0pt}{}{{}}{+\cdots+}%%EndExpansion\frac{c_{n}q^{\psi\left( n\right) }}{1}%%TCIMACRO{\QATOP{{}}{+\cdots}}%%BeginExpansion\genfrac{}{}{0pt}{}{{}}{+\cdots}%%EndExpansion\]where $\left( c_{n}\right) _{n\geq1}$ and $q$ are non zero rational numbers satisfying suitable conditions. Under these conditions, we prove that $\xi$ is irrational and give lower and upper bounds for its irrationality exponent. Inthe most favorable cases, we deduce from these bounds and Roth's theorem that $\xi$ is transcendental.