Taka Aki Tanaka

Algebraic independence results related to linear recurrences

One of the techniques used to prove the algebraic independence of numbers is Mahlers method, which deals with the values of so-called Mahler functions satisfying a certain type of functional equation. In order to apply the method, one must confirm the algebraic independence of the Mahler functions themselves. This can be reduced, in many cases, to their linear independence modulo the rational function field, that is, the problem of determining whether a nonzero linear combination of them is a rational function or not. In the case of one variable, this can be treated by arguments involving poles of rational functions. However, in the case of several variables, this method is not available. In this paper we shall overcome this difficulty by considering a generic point of an irreducible algebraic variety. Theorems 1 and 2 in this paper assert that certain types of functional equations in several variables have no nontrivial rational function solutions. As applications, we shall prove the algebraic independence of various kinds of reciprocal sums of linear recurrences in Theorems 3 and 4, and that of the values at algebraic numbers of power series, Lambert series, and infinite products generated by linear recurrences in Theorem 5. Let Ω = (ojij) be an n x n matrix with nonnegative integer entries. If z = (z i , . . . , zn) is a point of C n with C the set of complex numbers, we define a transformation Ω : C -> C by

Algebraic independence of the values of power series, Lambert series, and infinite products generated by linear recurrences

Abstract In Theorem 1 of this paper, we establish the necessary and sufficient condition for the values of a power series, a Lambert series, and an infinite product generated by a linear recurrence at the same set of algebraic points to be algebraically dependent. In Theorem 4, from which Theorems 1–3 are deduced, we obtain an easily confirmable condition under which the values more general than those considered in Theorem 1 are algebraically independent, improving the method of [5].

Algebraic independence of the values of the Hecke–Mahler series and its derivatives at algebraic numbers

We show that the Hecke–Mahler series, the generating function of the sequence {[nω]}n=1∞ for ω real, has the following property: Its values and its derivatives of any order, at any nonzero distinct...

Algebraic independence properties related to certain infinite products

In this paper we establish algebraic independence of the values of a certain infinite product as well as its all successive derivatives at algebraic points other than its zeroes, using the fact that the logarithmic derivative of an infinite product gives a partial fraction expansion. Such an infinite product is generated by a linear recurrence. The method used for proving the algebraic independence is based on the theory of Mahler functions of several variables.

Algebraic independence of power series generated by linearly independent positive numbers

In this paper we establish, using Mahler’s method, the algebraic independence of the values at an algebraic number of power series closely related to decimal expansion of linearly independent positive numbers. First we consider a simpler case in Theorem 1 and then generalize it to Theorem 3, which includes Nishioka’s result quoted as Theorem 2 of this paper. Lemma 7 plays an essential role in the proof of Theorems 1 and 3.