Algebraic independence of the values of certain series by Mahler's method

Suppose thatf ₁(z), ...f _m(z) are algebraically independent functions of a complex variable satisfying $$f_i (z) = a_i (z)f_i (Tz) + b_i (z),$$ wherea _i (z),b _i (z) are rational functions andTz=p(z ^?1)^?1 for a polynomialp(z) of degree larger than 1. We show thatf ₁(a), ...,f _m (a) are algebraically independent under suitable conditions onf anda. As an application of our main result, we deduce three corollaries, which generalize earlier work by Davison and Shallit and by Tamura.