Abstract
We prove, using Mahler's method, the following results: Theorem 1 asserts that the series Θ(x,a,q) are algebraically independent for any distinct triplets (x,a,q) of nonzero algebraic numbers, where Θ(x,a,q) has the property shown in Corollary 1 that Θ(a,a,q) is expressed as a continued fraction. Theorem 2 asserts, under the weaker condition than that of Theorem 1, that the values Θ(x,1,q) are algebraically independent for any distinct pairs (x,q) of nonzero algebraic numbers. Typical examples of these results are generated by Fibonacci numbers.
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