The Siegel-Shidlovskii method remains one of the basic methods in the theory of transcendental numbers. By using this method, it is possible to prove the transcendence and algebraic independence of the values of entire functions of a certain class (so-called E-functions). A necessary condition for applying this method is that all of the considered functions must constitute a solution of a system of linear differential equations and were algebraically independent over . The question about algebraic independence of the solutions of linear differential equations and systems of such equations is of great importance in differential algebra, analytical theory of differential equations, theory of special functions, and calculus (in the broad sense of the word). As is shown in papers by E. Kolchin, F. Beukers, W.D. Brownawell, and G. Heckman, this question boils down in many instances to verification of the cogredience and contragredience condition. Two systems of 1st order linear homogeneous differential equations with coefficients from are said to be cogredient (or, respectively, contragredient), if for arbitrary fundamental matrices and Ψ of these systems one of the equations Φ=gBΨC, Φ (ΨC)^*=gB, is fulfilled, where C∈GL(C), B∈GL(C(z)), g=g(z) is a function with the condition g^'/g∈C(z), and A^* is the matrix transposed to . The notions of cogredience and contragredience for linear homogeneous differential equations of arbitrary order are defined similarly. Another, more restricted definitions of cogredience and contragredience were in fact used in some papers of the author, devoted to generalized hypergeometric functions. According to these definitions, the function in the presented equalities is the product of a power function and an exponential function of some kind. The conditions for equivalence of these definitions are found.
On the estimates of the algebraic independence measures of the values of E-functions
