Journal of the Mathematical Society of Japan
Online ISSN : 1881-1167
Print ISSN : 0025-5645
ISSN-L : 0025-5645
Factors of 𝐸-operators with an 𝜂-apparent singularity at zero
Tanguy Rivoal
ジャーナル 認証あり

2022 年 74 巻 3 号 p. 719-733


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 𝑧𝜂 ℂ[[𝑧]]. We prove that they have a ℂ-basis of solutions of the form 𝑄𝑗(𝑧)𝑧𝜂 𝑒𝛽_𝑗 𝑧, where 𝜂 ∈ ℚ, the 𝛽𝑗 ∈ \overline{ℚ} are pairwise distinct and the 𝑄𝑗(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.



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