Geodesic
On linear independence of differentiated with respect to parameter functions and their values
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 9 (2019), pp. 27-36

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We consider hypergeometric functions and their derivatives (including with respect to parameter). For such functions we prove theorems concerning their linear independence over the field of rational fractions. In this connection we apply method that has been specially developed for this purpose. The proven independence of the functions under consideration is subsequently used for the investigation of arithmetic properties of their values by means of the modified Siegel's method. Some papers have been previously published containing in case of absence of differentiation with respect to parameter necessary and sufficient conditions for the aforementioned linear independence. For differentiated with respect to parameter functions there have been published many theorems concerning their algebraic independence. But by means of all these theorems one is unable to establish linear independence of the functions which are considered in this paper.
Keywords: hypergeometric function, linear independence, differentiation with respect to parameter. 
@article{IVM_2019_9_a2,
     author = {P. L. Ivankov},
     title = {On linear independence of differentiated with respect to parameter functions and their values},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {27--36},
     publisher = {mathdoc},
     number = {9},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2019_9_a2/}
}
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P. L. Ivankov. On linear independence of differentiated with respect to parameter functions and their values. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 9 (2019), pp. 27-36. http://geodesic.mathdoc.fr/item/IVM_2019_9_a2/