Algebraic Independence Results Related to 〈 q , r 〉-Number Systems

We define 〈q, r〉-linear arithmetical functions attached to the 〈q, r〉-number systems and give a necessary and sufficient condition for their generating power series to be algebraically independent over \Bbb C(z){\Bbb C}(z) . We also deduce algebraic independence of the functions values at a nonzero algebraic number in the circle of convergence.