Published papers

We considered a sequence of non-negative integers consisting of the exponents of terms with a coefficient of 1 in a power series whose coefficients are 0 and 1. For cases where such a sequence is expressed as the sum of an arithmetic progression and a geometric progression, we described a necessary and sufficient condition for the infinite set—comprising all the values and derivatives of any order of the power series at any non-zero algebraic numbers within the circle of convergence—to be algebraically independent.

In this paper, we proved the algebraic independence of the values at orbits represented by algebraic points of a certain three-variable function, which is generated by a linear recurrence sequence of positive integers satisfying specific congruence conditions and is invariant under the action of the dihedral group. In a previous study, a similar result was demonstrated for a two-variable function, which is a special case of the aforementioned function and is invariant under the action of the Klein four-group. Furthermore, as an application, we derived a necessary and sufficient condition for the set of values of the three-variable function to be algebraically dependent, expressed as an explicit linear relation.

Using a descent method, we construct certain power series generated by linear recurrences, each of which possesses the following property: The infinite set consisting of all its values and all the values of its derivatives of any order, at any nonzero algebraic numbers within its domain of existence, is algebraically independent. The main theorems of this paper assert that the power series generated by a linear recurrence simpler than ever before, have this property.

We construct a complex entire function with arbitrary number of variables which has the following property: The infinite set consisting of all the values of all its partial derivatives of any orders at all algebraic points, including zero components, is algebraically independent. In Section 2 of this paper, we develop a technique involving linear isomorphisms and infinite products to replace the algebraic independence of the values of functions in question with that of functions easier to deal with. In Sections 2 and 3, using the technique together with Mahler's method, we can reduce the algebraic independence of the infinite set mentioned above to the linear independence of certain rational functions modulo the rational function field of many variables. The latter one is solved by the discussions involving a certain valuation and a generic point in Sections 3 and 4.

Hecke-Mahler series is the generating function of the sequence {[nω]} of the integral parts [nω] of the positive integer multiple nω for ω real. In this paper, we prove that the Hecke-Mahler series has the following property: Its values and its derivatives of any order, at any nonzero distinct algebraic numbers inside the unit circle, are algebraically independent if ω is a quadratic irrational number satisfying a suitable condition, including the case where ω is in addition an algebraic integer. Using Mahler's method, we reduce the algebraic independence of the values of the Hecke-Mahler series to the linear independence of the Hecke-Mahler series themselves modulo the rational function field. For proving this linear independence, we observe the distribution of the sequence of points whose components consist of the fractional parts of the components of scalar multiple of a real vector.

We give positive characteristic analogues of complex entire functions having remarkable property that their values as well as their derivatives of any order at any nonzero algebraic numbers are algebraically independent. These results are obtained by establishing a criterion for the algebraic independence of the values of Mahler functions as well as that of the algebraic independence of the Mahler functions themselves over any function fields of positive characteristic.

In the previous work the author constructed a function of two variables taking algebraically independent values at any distinct points belonging to the cartesian product of the set of nonzero algebraic numbers and that of nonzero algebraic numbers inside the unit circle. In this paper the author constructs a balanced analogue of the two-variable function stated above, which is defined on a wider domain, the cartesian product of the set of complex numbers and that of complex numbers except the unit circle. This balanced function can also be regarded as a map defined on the set of orbits of the action of Klein four-group on the domain of definition of the balanced function and the latter map takes algebraically independent values at any distinct orbits represented by algebraic points.

Mahler関数において変数の変換を表す行列は非負行列に限る、という従来の制約の外にある多変数Mahler関数の新しいクラスについてその値の代数的独立性を示し、2項回帰数列で生成されるLaurent級数の値の代数的独立性を証明した。

無限積表示される整関数であって零点以外の相異なる代数的数における値及びそれら代数的数における逐次微分をすべて併せた無限集合が代数的独立となるものを、2項回帰数列の等比数列に対応する部分列を用いて構成した。これを達成するため多変数Mahler関数の特殊値の新たな代数的独立性判定定理を証明した。

フィボナッチ数列およびルカ数列の等比級数に対応する部分列を零点としてもつ無限積として表示される整関数について、実代数点における値が代数的従属となる極小関係式の新しい実例を与えた。

等比級数のフィボナッチ数列およびルカ数列に対応する部分列を零点としてもつ無限積として表示される整関数について、零点を除く代数的数における逐次微分をすべて含めた値が代数的独立となることを証明した。

フィボナッチ数列およびルカ数列の等比級数に対応する部分列を零点としてもつ無限積として表示される整関数について、整数点における値が代数的独立となる必要十分条件を決定した。

q超幾何級数において q の指数を線形回帰数列で置き換えた級数について、相異なる代数的パラメータの任意の組に対する値が代数的従属となる必要十分条件を２個の代数点間の同値関係として求めた。更に、q超幾何級数のフィボナッチ類似については、すべての相異なる代数的パラメータの組に対する値が代数的独立となることを証明した。

q超幾何級数において q の指数を線形回帰数列で置き換えた級数の、相異なる代数的パラメータの任意の組に対する値が代数的従属となる必要十分条件は２個の代数点間の同値関係となること、および q超幾何級数のフィボナッチ類似については、すべての相異なる代数的パラメータの組に対する値が代数的独立となることについて解説した。

Mahlerの方法による Mahler-Manin 予想の証明を解説した。また、独自に考案した最も簡単な場合のMahlerの方法による代数的独立性のself-contained な証明を詳述した。

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

望遠鏡級数に関する一般的な恒等式を証明し、それを用いて収束速度が異なる線形回帰数列の逆数和の間の１次関係式を求めた。更に、それを応用し種々の線形回帰数列の逆数和の代数的独立性を証明した。

線形回帰数列の逆数から成る緩やかに収束する正項級数の和と、その等比数列に対応する部分級数の和との代数的独立性を証明した。