We consider hypergeometric functions and their derivatives (including derivatives with respect to a parameter). For such functions, we prove theorems on their linear independence over the field of rational fractions. For this purpose we apply a specially developed method. The proven independence of the functions under consideration is subsequently used for the investigation of arithmetic properties of their values with the help of modified Siegel’s method.
In order to obtain some results on arithmetic properties of the values of hypergeometric functions, one usually has to justify their linear independence over the field of rational fractions. In the case of absence of derivatives with respect to a parameter, there are theorems containing necessary and sufficient conditions for the aforementioned linear independence. For the functions differentiated with respect to a parameter, there are many theorems on their algebraic independence. Nevertheless, these theorems do not allow justification of linear independence for the functions considered in this paper.
hypergeometric function linear independence differentiation with respect to a parameter
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Ivankov, P.L. “On linear independence of the values of entire hypergeometric functions with irrational parameters”, Siberian Mathematical Journal 34 (5), 839–847 (1993).MathSciNetCrossRefGoogle Scholar