Abstract:
A brief biography of A.B. Shidlovskii is given at the beginning of the article. Then it tells about the origins of the method - the Hermite and Lindemann theorems. The main definitions and results of the works of K. Siegel in 1929 and 1949 are formulated. The definition of the $E$-function and the conditions for the normality of a set of functions are given, examples are given. The story about the works of A.B. Shidlovskii began with the condition of irreducibility of a system of functions. Then the three main theorems of A.B. Shidlovskii and their main consequences are formulated. A theorem on the linear independence of the values of a set of $E$-functions with coefficients from an imaginary quadratic field is presented. A similar theorem is formulated in the case of arbitrary algebraic coefficients. The hypothesis of K. Siegel on the structure of the set of $E$-functions satisfying differential equations is formulated and its solution is described. The formulations of theorems are given under which generalized hypergeometric functions are algebraically independent over the field of rational functions, and their values at algebraic points are algebraically independent. Quantitative problems are described - estimates of measures of linear and algebraic independence of function values. Unimproved estimates are given. Another class of functions is considered, to which the Siegel-Shidlovskii method can be applied, the class of $G$-functions. The factorial cancelling condition is formulated, which holds for all considered $G$-functions. The concept of a global relation is given and the possibility of its application to series diverging in the field of complex numbers is described. It describes the arithmetic nature of results of summation of divergent series.
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