Factors of 𝐸-operators with an 𝜂-apparent singularity at zero

Abstract

In 1929, Siegel defined 𝐸-functions as power series in \overline{ℚ}[[𝑧]], with Taylor coefficients satisfying certain growth conditions, and solutions of linear differential equations with coefficients in \overline{ℚ}(𝑧). The Siegel–Shidlovskii theorem (1956) generalized to 𝐸-functions the Diophantine properties of the exponential function. In 2000, André proved that the finite singularities of a differential operator in \overline{ℚ}(𝑧)[𝑑/𝑑𝑧] ∖ {0} of minimal order for some non-zero 𝐸-function are apparent, except possibly 0 which is always regular singular. We pursue the classification of such operators and consider those for which 0 is 𝜂-apparent, in the sense that there exists 𝜂 ∈ ℂ such that 𝐿 has a local basis of solutions at 0 in 𝑧^𝜂 ℂ[[𝑧]]. We prove that they have a ℂ-basis of solutions of the form 𝑄_𝑗(𝑧)𝑧^𝜂 𝑒^{𝛽_𝑗 𝑧}, where 𝜂 ∈ ℚ, the 𝛽_𝑗 ∈ \overline{ℚ} are pairwise distinct and the 𝑄_𝑗(𝑧) ∈ \overline{ℚ}[𝑧] ∖ {0}. This generalizes a previous result by Roques and the author concerning 𝐸-operators with an apparent singularity or no singularity at the origin, of which certain consequences are also given here.