Théorie analytique locale des équations aux $q$-différences de pentes arbitraires

Abstract

La classification analytique locale et la description du groupe de Galois pour les équations aux q-différences linéaires analytiques complexes ont été obtenues par Ramis, Sauloy et Zhang [15,14] sous l'hypothèse que les pentes du polygone de Newton sont entières. Nous relâchons ici cette hypothèse.

On the vanishing of coefficients of the powers of a theta function

“…Note that, since θ q 4 (q 3 ) = θ q 4 (q), we have a 4,1 = a 4,3 as was to be expected by (9). Also note that various transformations are possible; for instance, using Entry 18 one can check that a 4,1 = a 4,3 = 1 2 θ q (1) 3 .…”

“…Let q ∈ C such that |q| > 1. In [3], one uses the function 1 θ q (z) := ∑ m∈Z q −m(m+1)/2 z m , z ∈ C * . This is a holomorphic function over C * .…”

“…For all q such that no t (k) n (q) vanishes, the first author has described in [3] the "wild fundamental group" of analytic complex linear q-difference equations which are irregular at 0 (and with arbitrary slopes). This led to call "good values" of q such values.…”

“…The relation to the formulas in [3] is clear: if |q| > 1 and if p := q −1 , then θ q (z) = θ p (x) with x := pz, and moreover t (k) n (q) = γ k,n (q −1 ); and the "good values" of q, |q| > 1, are those such that none of the γ k,n (q −1 ) vanishes.…”