论文标题
Hecke-Mahler系列值的超越和持续扩展的分数扩展
Transcendence and continued fraction expansion of values of Hecke-Mahler series
论文作者
论文摘要
令$θ$和$ρ$为实数，$ 0 \leθ，ρ<1 $和$θ$不合理。我们表明，Hecke-Mahler系列$$ f_ {θ，ρ}（z_1，z_2）= \ sum_ {k_1 \ ge 1} \，\ sum_ {k_2 = 1}^{\ lfloork_1θ+ρ\ρ\ rfloor}其中$ \ lfloor \ cdot \ rfloor $表示整数函数，在任何代数点$（β，α）$（$ 0 <|β|，|βα^θ|）上采用先验值（β，α）$（β，α）$ <1 $。这扩展了Mahler（1929）和Loxton和van der Poorten（1977）的早期结果，他们解决了案件$ρ= 0 $。此外，对于正整数$ b $和$ a $，带有$ b \ ge 2 $和$ a $，一致，$ 1 $ modulo $ b-1 $，我们给出了数字的持续分数扩展 $$ {（b-1）^2 \ voce b} f_ {θ，ρ} \ left（{1 \ over b}，{1 \ a} \ right）+{\ lfloorθ+ρ\ rfloor（b-rfloor（b-1） $$ 从中，我们得出一个公式，给出了$ f_ {θ，ρ}（1/b，1/a）$的非理性指数。
Let $θ$ and $ρ$ be real numbers with $0 \le θ, ρ< 1$ and $θ$ irrational. We show that the Hecke-Mahler series $$ F_{θ, ρ} (z_1, z_2) = \sum_{k_1 \ge 1} \, \sum_{k_2 = 1}^{\lfloor k_1 θ+ ρ\rfloor} \, z_1^{k_1} z_2^{k_2}, $$ where $\lfloor \cdot \rfloor$ denotes the integer part function, takes transcendental values at any algebraic point $(β, α)$ with $0 < |β|, |βα^θ| < 1$. This extends earlier results of Mahler (1929) and Loxton and van der Poorten (1977), who settled the case $ρ=0$. Furthermore, for positive integers $b$ and $a$, with $b \ge 2$ and $a$ congruent to $1$ modulo $b-1$, we give the continued fraction expansion of the number $$ {(b-1)^2\over b} F_{θ, ρ} \left({1\over b}, {1\over a}\right)+{\lfloor θ+ρ\rfloor(b-1)\over b^2a}, $$ from which we derive a formula giving the irrationality exponent of $F_{θ, ρ} (1/b, 1/a)$.
