Multiplicity and vanishing lemmas for differential and $q$-difference equations in the Siegel–Shidlovsky theory

We present a general multiplicity estimate for linear forms in solutions of various types of functional equations, which extends the zero estimates used in some recent works on the Siegel–Shidlovsky theorem and its $q$-analogues. We also present a dual version of this estimate, as well as a new interpretation of Siegel's theorem itself in terms of periods of Deligne's irregular Hodge theory.