Four Classes of Symmetric Sums over Cyclically Binomial Products
Round 1
Reviewer 1 Report
Comments and Suggestions for AuthorsAttached is my opinion on the paper
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Reviewer 2 Report
Comments and Suggestions for AuthorsAbstract: At the paper, the authors investigate four classes of multiple symmetric sums over cyclic products of binomial coefficients. By incorporating the generating function approach and recursive construction method, they are expressed analytically as coefficients of some rational functions. Several recurrence relations and generating functions are explicitly determined when the dimension of the multiple sums does non exceed 5.
Overwiev: At the paper, the authors investigated four classes of multiple symmetric sums over cyclic products of binomial coefficients. By incorporating the generating function approach and recursive construction method, they are expressed analytically as coefficients of some rational functions. The topic and results are interesting, paper is well-written and its language is fluent. It does includes a comprehensive literature review.
What is the main question addressed by the research? : By examining the characteristic polynomial of a certain binomial matrix, an excellent formula for multiple sums of circular binomial products has been defined by Carlitz. The authors, at this paper, investigate symmetric sums about cyclic products of binomial coefficients.
Do you consider the topic original or relevant to the field? Does it address a specific gap in the field? Please also explain why this is/ is not the case: I consider that the topic is original and interesting. The authors succesfully obtained analytic expressions of rational functions in figuring out the related generating functions in ordinary sense only for small m from 1 to 5. They can develop the results when m is an integer as a parameter.
What does it add to the subject area compared with other published material? : By combining the generating function approach with the recursive constraction method, the authors obtain analytic expressions in terms of x-coefficients of rational functions, by introducing the four corresponding classes of lambda polynomials.
Are the references appropriate? : I consider that the references are appropriate and enough.
Additional comments on the tables and figures: The authors examine four classes of symmetric sums by incorporating the generating function approach and recursive constraction method. They illustrated all these classes in four different tables.
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Reviewer 3 Report
Comments and Suggestions for AuthorsThe authors investigate the symmetric sums over cyclic products of binomial coefficients are examined. They present some important relationships.
The article is very simple and represent an elementary generalization of the existing theory. Its scientic soundness is not clear.
The whole article is below the standarts of this journal.
In my opinion, the submission is far below the high standards of ‘Symmetry’. I strongly recommend rejecting it.
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Reviewer 4 Report
Comments and Suggestions for AuthorsIn this manuscript, the authors focus on several families of sequences related to the Fibonacci numbers. Many of them appear as coefficients in certain classes of recursively defined polynomials. The introduction provides enough preliminary information. The paper presents algebraic proofs of some identities, and these proofs are not very interesting and do not require nice ideas, but they do require some effort.
I have some remarks. Some of the polynomials, notably the $p_n$ on page 3, are very similar to the Chebyshev polynomials. Is it possible to establish relations to the Chebyshev polynomials for all classes of polynomials discussed? Next, the authors should mention that the sequences such as $A_m(1),A_m(2),\dots$ are linear recurrence sequences. This is obvious from their generating functions. It would be interesting to prove that all these generating functions (for any of $A_m(n),B_m(n),\dots$) are rational functions, maybe in a similar manner as in the proof of Proposition 2, without writing them explicitly. At a separate note, it would be nice to avoid the upper indices, such as $B_m^2$, because they can be confused with powers. Also, there are some typos that should be fixed, e.g. "When $m$ is small integers" (bottom of p.17)
Therefore, in my opinion the paper needs some minor revision before publication.
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