An extension of algebraic independence of special values for non-lacunary power series

We study the algebraic independence of special values of power series $f({\beta }^{-1})$, where $\beta$ is a fixed Pisot or Salem number. In particular, we consider the case where $f(X)={\sum }_{n\ge 0}t(n)X^{w(n)}$ is not a lacunary series and is not assumed to satisfy any special functional equation, such as a Mahler-type functional equation. In our main results, we give a new criterion of the algebraic independence of three values. Applying our main results, we prove that the following three values are algebraically independent: $$$$ where $y$ is an arbitrary positive real number. Since our criterion is flexible, we have considerable freedom in choosing the coefficients $(t(n){)}_{n\ge 0}$ and the exponents $(w(n){)}_{n\ge 0}$.