New Applications of a Result of Galochkin on Linear Independence
We propose a slightly improved version of a criterion of Galochkin for the linear independence of the values of functions which are analytic in some neighborhood of the origin. Whereas Galochkin himself applied his criterion only to functions satisfying very special Mahler functional equations, we present new applications to solutions of certain Poincaré type functional equations. In particular, we propose a generalization of a recent irrationality result due to Duverney, and we give a new proof of the Tschakaloff-Skolem theorem on the linear independence of values, at appropriate points, of the Tschakaloff function T q which is intimately connected with the Jacobian theta function ϑ 3, in the usual notation.
Keywordslinear independence Poincaré functional equations q-functions
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