The 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.
ASJC Scopus subject areas
- Algebra and Number Theory