Triple Symmetric Sums of Circular Binomial Products
Round 1
Reviewer 1 Report
Comments and Suggestions for AuthorsDear Author,
Please find the attached file
Comments for author File: 
Comments.pdf
Author Response
see uploaded pdf-file
Author Response File: Author Response.pdf
Reviewer 2 Report
Comments and Suggestions for AuthorsThis paper considers sixteen triple sums for circular binomial products of binomial coefficients. The main results are recurrence relations and generating functions for these sums. However, the paper does not conclude applications motivated by concrete real-life problems, where the results apply, and the hypotheses are verified. Providing relevant examples would have made the paper much more interesting with not only the theoretic value.
Authors should take into account the following remarks:
1. The object of research should be clearly defined: is it "triple symmetric sums of circular binomial products" or "quasi symmetric triple sums of circular binomial products"? In the second case, it should be explained what "quasi symmetric" means.
2. The practical application of binomial sums given in works [1-4] should be described in detail.
3. The idea of ​​a single proof of the statements of lemmas, propositions and theorems in Sections 2-4 is not good one. It would be much better to write proofs for each statement separately. At the same time, differences in the proof of the remaining three expressions, recurrence relations, and generating functions should be noted.
Minor editing of English language required.
Author Response
see uploaded pdf-file
Author Response File: Author Response.pdf
Reviewer 3 Report
Comments and Suggestions for Authors
Reviewer Report On Manuscript Id: Mathematics-3081031
Title: Triple Symmetric Sums of Circular Binomial Products
Authors: Marta Na Chen and Wenchang Chu
Introduction: The authors deployed a generating function method to construct recursively recurrence relations and generating functions for sixteen (16) triple sums of binomial circular products. This same process was used to establish the counterpart results corresponding to double sums U_n, V_n and W_n and the corresponding sums U_ny^n, V_ny^n and W_ny^n. The limitation of the rational generating functions was provided by the authors and suggestions offered.
The title of the work seems appropriate considering the focus of the paper and the abstract is fairly presented but can be improved. For instance, the authors failed to identify the knowledge gap, mention the motivation either in the abstract or the introduction which is very necessary. and the introduction is well illustrated to include adequate review of relevant literature.
-
The authors are advised to avoid the use of “we shall“ since the research is conducted already.
Results: The results are well presented and adjudged to be correct. However, the following observations are raised for necessary corrections:
-
No statement of the Theorem for Theorem 2.3
-
No statement of the Lemma for Lemma 2.1 and no reference.
-
No statement of the Lemma for Lemma 3.1 and no reference
-
No statement of the Lemma for Lemma 4.1 and no reference.
-
No statement of the Lemma for Lemma 5.1 and no reference.
-
No statement of the Theorem for Theorem 3.3
-
No statement of the Theorem for Theorem 4.3
-
No statement of the Theorem for Theorem 5.3
The authors should kindly revise each of the Lemmas, including the statements of the lemmas and the references. A Lemma is expected to have been proven before as a Theorem. Hence, each of the Lemmas requires a reference.
In the same manner, each of the Theorems needs to have a statement of the Theorem. Hence, Theorem 2.3, Theorem 3.3, Theorem 4.3 and Theorem 5.3 should be adequately revisited and reviewed to have statements.
The phrase “Concluding Comments” should be changed to “Conclusion”.
References: No case of self-citation was observed. Three (3) out of twenty (20) references (representing 15%) of the references were observed to be self cited.
I suggest that the authors should add one (1) or two (2) current reference(s) more.
All cited papers were adequately referenced and no referenced paper was noticed not to be cited in the manuscript. The authors also followed the MDPI citation and reference styles of references appearing in the ways and manners in which they are cited. For instance, the first cited work is first referenced in the order until the last cited work.
In general, the manuscript is considered publishable in the journal, Mathematics but some modifications are required. I therefore recommend its publication in the journal after some corrections. Moderate English editing is also recommended.
Decision: Accepted for publication in the journal after correction.
Comments for author File: Comments.pdf
Author Response
see uploaded pdf-file
Author Response File: Author Response.pdf
Round 2
Reviewer 1 Report
Comments and Suggestions for Authorscan be accepted.
Reviewer 2 Report
Comments and Suggestions for AuthorsAn updated version of the paper may be published in the journal Mathematics.
Comments on the Quality of English LanguageMinor editing of English language required, for instance on page 1 instead of "... the Fibonacci numbers as usual. we record ..." should be "... the Fibonacci numbers as usual. We record ...".
