Geodesic
Multiplicity and vanishing lemmas for differential and $q$-difference equations in the Siegel--Shidlovsky theory
Fundamentalʹnaâ i prikladnaâ matematika, Tome 16 (2010) no. 5, pp. 19-30

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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. 
@article{FPM_2010_16_5_a1,
     author = {D. Bertrand},
     title = {Multiplicity and vanishing lemmas for differential and $q$-difference equations in the {Siegel--Shidlovsky} theory},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
     pages = {19--30},
     publisher = {mathdoc},
     volume = {16},
     number = {5},
     year = {2010},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_2010_16_5_a1/}
}
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D. Bertrand. Multiplicity and vanishing lemmas for differential and $q$-difference equations in the Siegel--Shidlovsky theory. Fundamentalʹnaâ i prikladnaâ matematika, Tome 16 (2010) no. 5, pp. 19-30. http://geodesic.mathdoc.fr/item/FPM_2010_16_5_a1/