Gauss’ Second Theorem for ${}_{2}F_1(1/2)$-Series and Novel Harmonic Series Identities

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The paper is nicely written and may be accepted for publication provided the following observations are considered:

1.  Conclusion and future observations must be added in Conclusion Section which is missing.

2. The generating function for Harmonic numbers must be written clearly.

The values of H ⟨m⟩ n (λ) in line 26 must be presented, if possible.

Author Response

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Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

This paper contains results in number theory. Several formulae are proved for some combinations of parametric harmonic numbers. I recommend to publish this paper in its present form.

Author Response

Thanks to Reviewer 2 for the recommendation.

Reviewer 3 Report

Comments and Suggestions for Authors

In this manuscript, the authors investigated two theorems related to the ₂F₁(1/2)-series due to Gauss and Bailey by employing the coefficient extraction method. Many theorems are given to evaluate infinite series concerning harmonic numbers and binomial/multinomial coefficients including eight conjectured ones made by Z.-W. Sun. This work is an extension of a recent work of the second author.

Obviously, the authors made good effort to write good although the results appear to be simple .In fact,

- The introduction provides sufficient and summarized preliminaries.

- The language is simple and clear.

-  The results are clearly presented.

I suggest to add a conclusion section to summarize the results ( optional).  

Without any negative comment,  I think the paper is suitable for publication in its current form.

Author Response

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Author Response File: Author Response.pdf

Reviewer 4 Report

Comments and Suggestions for Authors

Dear Editor

 

The paper is interesting but need major revision. I suggest revising paper according to the following comments and suggestion

 

1.       The abstract should be rewritten?

2. The English language needs improvement?
3. The introduction part should be improved by including the novelty, significance, and motivation of the paper?
4. In line 59, why   and , are there other values that achieve this?
5. There is an error in the signal in proposition 5 (b).
6. The proof of theorem 6 is not clear? It must be clarified.

7.      Observe the punctuation at the end of equations.

8.      What is the best value for  in line 89 and line 101?

9.  Researchers found 18 theorems. Is there a relationship between them? What is the benefit of them and where these theorems can be used must be explained in detail

11.  References should be updated from 2023.

12.  Please clarify the novelty of this work respect to published papers?

 

Comments for author File: Comments.pdf

Comments on the Quality of English Language

.

Author Response

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Author Response File: Author Response.pdf

Reviewer 5 Report

Comments and Suggestions for Authors

The authors examined two summation theorems concerning the 2F1(1/2)-series due to Gauss and Bailey by using the "coefficient extraction method". They also calculated some infinite series concerning harmonic numbers and binomial/multinomial coefficients analytically, including the cases made by Z.-W.~Sun. The results seem interesting and the calculations seem correct. I recommend it for publishing with some minor corrections.

1) The confluent hypergeometric functions play an important role in physics or chemistry and other fields, in particular for those soluble quantum systems. Their solutions are connected with the hypergeometric functions, from them it is not difficult to get the energy spectrum. For example, the works were done by Wei, Jia, Dong et al including the study for a new anharmonic oscillator PLA 340, 94 (2005), Phys. Scr. 76 (2007) 442 (2007); PLA 312, 78 (2003); PLA 333, 212 (2004); Physica Scripta 81, 035009 (2010), etc. 

2) There is no Conclusion in this work. It would be helpful if the author could address the potential applications. 

Comments on the Quality of English Language

Moderate improvement.

Author Response

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Author Response File: Author Response.pdf

Round 2

Reviewer 4 Report

Comments and Suggestions for Authors

There are no recommendations 

Author Response

Thanks to the Reviewer for approving the revised version.

Reviewer 5 Report

Comments and Suggestions for Authors

I have studied the revised manuscript and found that the authors did not consinder all. Honestly speaking, the calculations in the present study can be done by mathematica or mathlab  or other tools, the authors should address its potential applications, in particular the connection with physics etc. In quantum soluble systems, the confluent hypergeometric functions play an important role. This is the reason why the authors should address its apolication because the study or calculations could be done easily. Recent study related to this topic has been discussed by Wei, Jia, Dong et al, PLA 312, 78 (2003); Physica Scripta 81, 035009 (2010), etc.  

Comments on the Quality of English Language

Moderate modification is required.

Author Response

The effort made by the Reviewer in assessing this submission is appreciable. After having carefully examined these papers proposed by Reviewer as new references, the authors find that it is true that confluent hypergeometric 
functions are utilized, but irrelevant to harmonic series (the central topic 
of our paper) even though we have employed hypergeometric series, but not 
confluent ones. Therefore, it is not appropriate to include these proposed papers
in the reference list. In addition, Reviewer raised also two minor issues.  One is  
about potential applications of the results presented in this paper, which is already addressed at the end of "Concluding Comments (page 11)". Another  comment by Reviewer 5 reads as "Honestly speaking, the calculations in the present study can be done by mathematica or mathlab  or other tools." This is, in fact, declared at the end of Introduction (page 3)  "numerical tests for all the equations have been made by appropriately devised Mathematica commands." 

In conclusion, there is no justifiable reason to make further modifications.

 

Round 3

Reviewer 5 Report

Comments and Suggestions for Authors

The authors clarified it.

Comments on the Quality of English Language

Moderate revising is required.