On Algebraic Independence of Solutions of Generalized Hypergeometric Equations^†

Round 1

Reviewer 1 Report

The peer-reviewed paper refers to Siegel's method in number theory. The main task of the theory of transcendental numbers is to prove the transcendence and algebraic independence of various sets of numbers. Usually, the properties of the values of various analytical functions are investigated. Siegel's method allows us to establish the transcendence and algebraic independence of the values of the so-called E-functions, which include exp(x), sin(x), cos(x), Bessel, Kummer functions and other special mathematical functions. In the second half of the 20th century, Siegel's method was further developed in the works of A.B. Shidlovsky, E. Kolchin, D. Bertrand, N. Katz, F. Beukers, Y. Andre, Yu.V. Nesterenko, V.Kh. Salikhov and other mathematicians.

In the peer-reviewed paper, necessary and sufficient conditions are found for the algebraic independence of the values of generalized hypergeometric functions. If these conditions are violated, examples of algebraic dependence are indicated explicitly. Besides, some incorrect statements from the works of F. Beukers and other mathematicians are corrected.

I think the paper deserves to be published in Axioms.

 

Author Response

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Reviewer 2 Report

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Reviewer 3 Report

The article is a continuation of the author's previous works on the algebraic independence of solutions of certain differential equations. The article is written interestingly. Most proofs are presented with sufficiently complete strict and detailed considerations. However, in the motivational part of the introduction, it is desirable to pay attention to the application of the theory of generalized hypergeometric equation and point to unresolved problems in the conclusions. Last but not least, it is desirable to give an example of algebraic independence of solutions of a generalized hypergeometric equation of the third-order or higher.

Author Response

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Round 2

Reviewer 3 Report

The article is well motivated and well written. Most proofs are presented with sufficiently complete strict and detailed considerations. This is an interesting article, so I recommend it for publication in the Axioms.