On the Algebraic Independence of the Values of Functions Associated with Hypergeometric Functions^†

Round 1

Reviewer 1 Report

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Comments for author File: Comments.pdf

Author Response

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Reviewer 2 Report

The manuscript is a piece of well done work, and therefore should be published.

Author Response

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Reviewer 3 Report

Dear Editor,

 

The new referee report you requested is attached containing the questions and answers you want,

 

Respects.

 

New Review Report(mdpi-Axioms–A-Manuscript ID Number-axioms-2095264)

 

Article title:         On algebraic independence of the values of functions associated with hypergeometric functions

 

Possible questions and answers about the article:

 

1. What is the main question addressed by the research?

 

                In the article, the illustrates, necessary properties, definitions and theorems,formulas in finding variationality features of a special “hypergeometric function equation” structure modelling are characterized different type equations(based Compound optimal design algebraic independence rules).

 

2. Do you consider the topic original or relevant to the field? Does it address a specific gap in the field?

 

                This topic fills the specific gap in characterizing the original “algebraic,topological, geometric and Application maps , applied mathematics, representation approach, comparative analysis, operator theory, fixed point theory, optimization theory” model issue in applied science fields (with long-term (asymptotic)- hypergeometric solution property).

 

3. What does it add to the subject area compared with other published material?

 

                The compared with other published material,there are adding “in determining special differential equation approach field, sufficient conditions for a special complex hypergeometric equations and its contents to be the modelling with special structures on the special Multi- hypergeometric objective-approach theory” the subject area(under special hypergeometric cases).

 

4. What specific improvements should the authors consider regarding the methodology? What further controls should be considered?

 

                Authors can improve by developing new aspects of a special hypergeometric function-approach structure on connecting with the structure hypergeometric function problem equation to the algebraic applied representation theory and equations of common applied problem transforms regarding the Methodology, The authors can provide a control mechanism by making new different aspects from the phenomenon of developing special approach features(in Siegel's methodial and criterional operators).

 

5. Are the conclusions consistent with the evidence and arguments presented and do they address the main question posed?

 

                The results appear consistent with the evidence and arguments.

 

The arguments presented adequately address the main question.

 

6. Are the references appropriate?

 

                In the article, references appear to be relevant to the study subject.

 

7. Please (if any) include any additional comments on the tables and figures.

 

                model mechanisms formed from special Axiomity features on applied competing models-approach theory and special counting applied-mapping forms are developed and shown in terms of article images (by minimizing the maximum variance of the E-function values).

 

report:In the article, it is seen that  the special " in algebraic and hypergeometric applied topology,and applied mathematical field method(the theory of applied mathematics and science)( applied theoretic method)" featured definitions and calculations are made (the special hypergeometric mapping forms and enumerating the numbers of algebraic on a minimizity maximum hypergeometric variance- structure) for some special applied approach form equations (by strengthening with algebraic independence).

 

findings: In the article, the special “algebraic and applied math and structural properties of the common complex hypergeometric -topology” process approach form equations are seen (the special hypergeometric calculations) with the help of some special forms (under the hypergeometric solution-(asymptotic) property On Model independence Equation).

 

strengths: In the article, there are proficiency-enhancing conditions for the special pre hypergeometric applied and algebraic approach forms.

 

weaknesses: In the article, it may be necessary to develop some extra new algebraic hypergeometric conditions for the special pre- hypergeometric form algorithms and applied approach forms of this type(with axiomatics).

 

any minor issue: Although there are some minor weaknesses in terms of mathematical language, the article can be considered sufficient.

 

result: The article is suitable for publication for the journal.

Comments for author File: Comments.pdf

Author Response

Please see the attachment

Author Response File: Author Response.pdf