For Siegel E-function values at algebraic points there exists an E-function realizing the value at 1 without 1 being a singularity of its minimal DE; the property fails at 0 for Bessel values.
Lepetit, Le th´ eor` eme d’Andr´ e-Chudnovsky-Katz au sens large, North-West
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On the singularities of differential equations satisfied by $E$-functions
For Siegel E-function values at algebraic points there exists an E-function realizing the value at 1 without 1 being a singularity of its minimal DE; the property fails at 0 for Bessel values.