Linear independence criteria for generalized polylogarithms with distinct shifts

Abstract

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 $$\Phi_{s}(x,z)= \sum_{k=0}^{\infty}\frac{z^{k+1}}{(k+x+1)^s}\enspace.$$ We prove the linear independence over any number field $K$ of the numbers $1$ and $\Phi_{s_j}(x_j,\alpha_i)$ 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 $\alpha_1,\ldots,\alpha_m\in K$ subject to a metric condition. As is usual in the theory, the points $\alpha_i$ 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 $$\Phi_{\boldsymbol{a},s}(x,z)= \sum_{k=0}^{\infty}\frac{a_{k+1\bmod(q)}\cdot z^{k+1}}{(k+x+1)^s}\enspace.$$ 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 Pad\'e approximants for this family of generalized polylogarithmic function.