On the Linear Independence of Some Functions over the Field of Rational Fractions

In 1955 A.B. Shidlovski's general theorems were published. They allow us to reduce the problem of algebraic independence of the analytic function values, belonging to the specific class, to a simpler problem of algebraic independence of these functions. Since the abovementioned general theorems can be applied to the generalized hyper-geometric functions with rational parameters, there appeared many works in which the algebraic independence of such functions (and their derivatives) had been established. The A.B. Shidlovski's results generalize and develop a Siegel's method well known in the theory of transcendental numbers. Besides the Siegel's method to solve the problems concerning the arithmetic nature of the values of analytic functions one also applies methods based on the effective construction of linear approximating forms. Such methods enabled finding the most accurate estimates of linear forms and obtaining the numerous results concerning the arithmetic properties of the values of hyper-geometric functions with irrational parameters. This shows that effective methods are of some value for the development of the theory of transcendental numbers.Recently, in the context of studied arithmetic nature of the values of differentiated hypergeometric functions with respect to parameter, there was a need in results concerning the linear independence of such functions over the field of rational fractions. Similar investigations were also conducted earlier because of applications of A.B. Shidlovski's general theorems, but in that case a more difficult problem of algebraic independence had to be solved, and therefore only the simplest functions were considered. The paper studies the issue of linear independence of hypergeometric functions, differentiated with respect to parameter, and this parameter is included both in the numerator and in the denominator of the common member of the appropriate power series. The paper defines a condition (in some cases, it is necessary and sufficient) of linear independence of such functions, which is very convenient for checking in concrete cases. The paper results are obtained by calculating some determinants, which, naturally, arise from the problems under consideration. In the future, the theorems proved in this paper can be used to have the diverse statements concerning the arithmetic nature of the values of the appropriate functions.DOI: 10.7463/mathm.0415.081732