On Linear Independence of Some Functions

Abstract

To study an arithmetic nature of the values of hyper-geometric functions (and their derivatives including those with respect to parameter), it is common practice to use one either Siegel's method or the method based on the effective construction of the linear approximating form. The main distinction between these methods consists in the mode of construction of the first approximating form. Applying Siegel's method allows us to construct such a form by means of a pigeonhole principle that makes it possible to establish, in certain cases, the algebraic independence of the values of corresponding functions. The Siegel's method can be usually applied just while considering hyper-geometric functions with rational parameters. The effective method has a certain advantage here, for in some cases this method enables us to carry out corresponding investigation also for the functions with irrational parameters. Another advantage of the effective method is that it provides obtaining of high- accuracy quantitative results. By quantitative results one usually implies the estimates of the moduli of the linear forms in the values of corresponding functions. The effective method has made it possible to obtain, in some cases, estimates being precise regarding the height of such forms with calculation of the corresponding constants. A drawback of the effective method is the narrow domain of its applications. The effective construction of the linear approximating form, which initiates reasoning, is always a challenge. So far, an effective construction of the approximating form for the product of powers of hyper-geometric functions (with the rare exceptions) failed.In both aforementioned methods one proves previously linear independence (in Siegel's method, as a rule, algebraic independence,) of the functions under consideration. Such a proof is often considered as an independent result.In this paper, by means of the method especially elaborated for this purpose we establish linear independence of some hyper-geometric functions and their derivatives (including those with respect to parameter) over the field of rational fractions. Subsequently, it will be possible to apply this result to investigate arithmetic properties of the values of such functions. Herewith we mean the application of the effective method to achieve the sufficiently accurate quantitative result.