On linear independence of the values of some hypergeometric functions over the imaginary quadratic field
The main difficulty one has to deal with while investigating arithmetic nature of the values of the generalized hypergeometric functions with irrational parameters consists in the fact that the least common denominator of several first coefficients of the corresponding power series increases too fast with the growth of their number. The last circumstance makes it impossible to apply known in the theory of transcendental numbers Siegel’s method for carrying out the above mentioned investigation. The application of this method implies usage of pigeon-hole principle for the construction of a functional linear approximating form. This construction is the first step in a long and complicated reasoning that leads ultimately to the required arithmetic result. The attempts to apply pigeon-hole principle in case of functions with irrational parameters encounters insurmountable obstacles because of the aforementioned fast growth of the least common denominator of the coefficients of the corresponding Taylor series. Owing to this difficulty one usually applies effective construction of the linear approximating form (or a system of such forms in case of simultaneous approximations) for the functions with irrational parameters. The effectively constructed form contains polynomials with algebraic coefficients and it is necessary for further reasoning to obtain a satisfactory upper estimate of the modulus of the least common denominator of these coefficients. The known estimates of this type should be in some cases improved. This improvement is carried out by means of the theory of divisibility in quadratic fields. Some facts concerning the distribution of the prime numbers in arithmetic progression are also made use of. In the present paper we consider one of the versions of effective construction of the simultaneous approximations for the hypergeometric function of the general type and its derivatives. The least common denominator of the coefficients of the polynomials included in these approximations is estimated subsequently by means of the improved variant of the corresponding lemma. All this makes it possible to obtain a new result concerning the arithmetic values of the aforesaid function at a nonzero point of small modulus from some imaginary quadratic field.