论文标题
具有不同变化的广义性聚类的线性独立标准
Linear independence criteria for generalized polylogarithms with distinct shifts
论文作者
论文摘要
对于给定的有理数$ x $和一个整数$ s \ geq 1 $，让我们考虑一个广义的polygarithmic函数，通常称为lerch函数，由$$φ__{s}（x，z）=定义\ sum_ {k = 0}^{\ infty} \ frac {z^{k+1}}} {（k+x+1） $ x_1，\ ldots，x_d $带$ 0 \ le x_1 <\ ldots <x_d <1 $，以及任何选择深度$ 1 \ leq s_1 \ leq s_1 \ leq r_1，\ ldots，1 \ leq s_d s_d s_d \ leq s_d \ leq r_d $ abgebra $ a $ a $ k y rd rd ld ld ld ld ld ld ld ld ld ld ld ld ld ld公制条件。与该理论中的平常一样，相对于给定的固定位置$ v_0 $ $ k $，acragimedean或有限的点需要足够接近零。 这是第一个线性独立性结果，具有不同的偏移$ x_1，\ ldots，x_d $，允许在不同点的广义polyrogarithmic函数下值。先前的标准仅用于一个固定偏移或一个点的功能。此外，我们建立了另一个线性独立性标准，用于具有循环系数的广义性聚类函数的值。令$ q \ geq 1 $为整数，$ \ boldsymbol {a} =（a_1，\ ldots，a_q）\ in k^q $是$ q $ - tuple，其坐标应该与周期$ q $一起循环。考虑具有系数的广义pologarithmc函数$$φ_ {\ boldsymbol {a}，s}（x，x，z）= \ sum_ {k = 0}^{\ infty} \ frac { z^{k+1}}} {（k+x+1）^s} \ enspace。$$在适当的条件下，我们表明这些功能的值在$ k $上是线性独立的。我们的关键工具是一种与我们在这个广义性聚类函数家族的Padé近似值相关的广义的Hermite类型的新型非逐渐属性。
For a given rational number $x$ and an integer $s\geq 1$, let us consider a generalized polylogarithmic function, often called the Lerch function, defined by $$Φ_{s}(x,z)= \sum_{k=0}^{\infty}\frac{z^{k+1}}{(k+x+1)^s}\enspace.$$ We prove the linear independence over any number field $K$ of the numbers $1$ and $Φ_{s_j}(x_j,α_i)$ with any choice of distinct shifts $x_1,\ldots, x_d$ with $0\le x_1<\ldots<x_d<1$, as well as any choice of depths $1\leq s_1\leq r_1,\ldots, 1\leq s_d\leq r_d$, at distinct algebraic numbers $α_1,\ldots,α_m\in K$ subject to a metric condition. As is usual in the theory, the points $α_i$ need to be chosen sufficiently close to zero with respect to a given fixed place $v_0$ of $K$, Archimedean or finite. This is the first linear independence result with distinct shifts $x_1, \ldots, x_d$ that allows values at different points for generalized polylogarithmic functions. Previous criteria were only for the functions with one fixed shift or at one point. Further, we establish another linear independence criterion for values of the generalized polylogarithmic function with cyclic coefficients. Let $q\geq 1$ be an integer and $\boldsymbol{a}=(a_1,\ldots, a_q)\in K^q$ be a $q$-tuple whose coordinates supposed to be cyclic with the period $q$. Consider the generalized polylogarithmc function with coefficients $$Φ_{\boldsymbol{a},s}(x,z)= \sum_{k=0}^{\infty}\frac{a_{k+1\bmod(q)}\cdot z^{k+1}}{(k+x+1)^s}\enspace.$$ Under suitable condition, we show that the values of these functions are linearly independent over $K$. Our key tool is a new non-vanishing property for a generalized Wronskian of Hermite type associated to our explicit constructions of Padé approximants for this family of generalized polylogarithmic function.
