超越函数导数密度的下界
Lower Bounds for Densities of Transcendental Gamma-Function Derivatives
AI总结：
研究超越函数导数在正整点和有理移位整点上的超越性，推导其密度下界。
摘要：
在近期工作中，我们证明对于所有q∈1/2Z\Z≤0，序列{Γ^(n)(q)}_{n≥1}中包含超越元素无限次，其中超越Γ^(n)(q)在n∈{1,2,…,N}中的密度下界为β(N)=max{0,√N−5/2}/N。对于固定和可变的n，我们研究Γ^(n)(q)在正整点q=m∈{1,2,…}和有理移位整点q=˜m∈{κ,±1+κ,±2+κ,…}（其中κ∈(0,1)∩Q且Γ(κ)为超越数）的超越性。对于n∈Z≥2，我们发现最多有n−1个代数Γ^(n)(m)，对于n∈Z≥1，每个单侧移位整点最多有n个代数Γ^(n)(˜m)。这些结果为构造超越Γ^(n)(m)在m∈{1,2,…,M}中的密度下界以及超越Γ^(n)(˜m)在˜m∈{κ,1+κ,2+κ,…,M+κ}或˜m∈{κ,−1+κ,−2+κ,…,−M+κ}中的密度下界提供了基础。允许n变化，我们推导了双变量密度下界。
英文摘要：
In recent work, we showed that for all $q\in\tfrac{1}{2}\mathbb{Z}\setminus\mathbb{Z}_{\leq0}$ the sequence $\left\{Γ^{\left(n\right)}\left(q\right)\right\} _{n\geq1}$ contains transcendental elements infinitely often, with the density of transcendental $Γ^{\left(n\right)}\left(q\right)$ among $n\in\left\{1,2,\ldots,N\right\}$ bounded below by $β\left(N\right)=\max\left\{0,\sqrt{N}-5/2\right\}/N$. For both fixed and variable $n$, we now study the transcendence of $Γ^{\left(n\right)}\left(q\right)$ at both positive lattice points $q=m\in\left\{1,2,\ldots\right\}$ and rationally shifted lattice points $q=\widetilde{m}\in\left\{κ,\pm1+κ,\pm2+κ,\ldots\right\}$ (for $κ\in\left(0,1\right)\cap\mathbb{Q}$ such that $Γ\left(κ\right)$ is transcendental). For $n\in\mathbb{Z}_{\geq2}$, we find there are at most $n-1$ algebraic $Γ^{\left(n\right)}\left(m\right)$, and for $n\in\mathbb{Z}_{\geq1}$, there are at most $n$ algebraic $Γ^{\left(n\right)}\left(\widetilde{m}\right)$ for each one-sided shifted lattice (i.e., $\widetilde{m}\geqκ$ or $\widetilde{m}\leqκ$). These results form the basis for constructing lower bounds for the densities of transcendental $Γ^{\left(n\right)}\left(m\right)$ among $m\in\left\{1,2,\ldots,M\right\}$ and transcendental $Γ^{\left(n\right)}\left(\widetilde{m}\right)$ among either $\widetilde{m}\in\left\{κ,1+κ,2+κ,\ldots,M+κ\right\}$ or $\widetilde{m}\in\left\{κ,-1+κ,-2+κ,\ldots,-M+κ\right\}$. Allowing $n$ to vary, we derive lower bounds for the bivariate densities of both transcendental $Γ^{\left(n\right)}\left(m\right)$ among $\left(n,m\right)\in\left\{2,3,\ldots,N\right\}\times\left\{1,2,\ldots,M\right\}$ and transcendental $Γ^{\left(n\right)}\left(\widetilde{m}\right)$ among one-sided shifted-lattice analogues.
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