Linear independence criteria for generalized polylogarithms with distinct shifts (original) (raw)

For a given rational number x and an integer s ≥ 1, let us consider a generalized polylogarithmic function, often called the Lerch function, defined by Φs(x, z) = ∞ k=0 z k+1 (k + x + 1) s. We prove the linear independence over any number field K of the numbers 1 and Φs j (xj, αi) with any choice of distinct shifts x1,. .. , x d with 0 ≤ x1 <. .. < x d < 1, as well as any choice of depths 1 ≤ s1 ≤ r1,. .. , 1 ≤ s d ≤ r d , at distinct algebraic numbers α1,. .. , αm ∈ K subject to a metric condition. As is usual in the theory, the points αi need to be chosen sufficiently close to zero with respect to a given fixed place v0 of K, Archimedean or finite. This is the first linear independence result with distinct shifts x1,. .. , 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 ≥ 1 be an integer and a = (a1,. .. , aq) ∈ K q be a q-tuple whose coordinates supposed to be cyclic with the period q. Consider the generalized polylogarithmc function with coefficients Φa,s(x, z) = ∞ k=0 a k+1 mod (q) • z k+1 (k + x + 1) s. 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.