Conditions for the algebraic independence of certain series involving continued fractions and generated by linear recurrences

AbstractThe main theorem of this paper, proved using Mahler's method, gives a necessary and sufficient condition for the values Θ(x,a,q) at any distinct algebraic points to be algebraically independent, where Θ(x,a,q) is an analogue of a certain q-hypergeometric series and generated by a linear recurrence whose typical example is the sequence of Fibonacci numbers. Corollary 1 gives Θ(x,a,q) taking algebraically independent values for any distinct triplets (x,a,q) of nonzero algebraic numbers. Moreover, Θ(a,a,q) is expressed as an irregular continued fraction and Θ(x,1,q) is an analogue of q-exponential function as stated in Corollaries 3 and 4, respectively