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

Abstract

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.