Overview of Six Number/Polynomial Sequences Defined by Quadratic Recurrence Relations
Round 1
Reviewer 1 Report
Comments and Suggestions for AuthorsYou can find my report in the attachment file.
Comments for author File: 
Comments.pdf
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Author Response
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Author Response File: Author Response.pdf
Reviewer 2 Report
Comments and Suggestions for AuthorsThis paper provides an overview of six number/polynomial sequences defined by quadratic recurrence relations: Fibonacci and Lucas numbers, Pell and Pell/Lucas polynomials, and Chebyshev polynomials of the first and second kinds. Here there are some comments:
- At first, the authors must revise the language of the paper carefully.
- The novelty of the work is not high, in other words, the novelty of the work is very weak. Many of the works that have been presented were shown in the previously published works (eg. Fibonacci and Lucas numbers).
- The details of many presented proofs are neglected and there are no details about them.
- The paper is full of math formulas, but it doesn’t explain how these results could be useful in other areas like computer science, physics, or engineering. It is recommended to add some application of this work in the real world.
- The literature review is not satisfactory. It is recommended to consider these works:
- The binomial sums for four types of polynomials involving floor and ceiling functions
- Convolution identities for Pell-Lucas polynomials.
- An innovative approach to predict the diffusion rate of reactant’s effects on the performance of the polymer electrolyte membrane fuel cell
- The paper jumps from one topic to another without smooth transitions. Please add a road map from start to end, it would be useful.
- The conclusion mentions several topics not covered in this paper that could be explored further.
Without a proper revision, the paper is not suggested to be published in this journal.
Author Response
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Reviewer 3 Report
Comments and Suggestions for AuthorsThe authors deal with some special functions, especially Fibonacci and Lucas numbers, Pell and
Pell/Lucas polynomials as well as Chebyshev polynomials. These sequences are characterized
by quadratic linear recurrence relations and reviewed in a unified manner. The authors presented several properties (such as Binet form expressions, Cassini identities and
Catalan formulae) and remarkable results concerning power sums, ordinary and binomial convolutions, by employing the symmetric feature, series rearrangements and the generating function approach.
The results that were obtained seem to be interesting and well presented. But, in Section 4, the authors did not provide the proofs of the given theorems, and stated that the proofs are made by the standard generating function approach (cf. Comtet [21] and wilf [52]). I encourage the authors to provide the proof of some of these theorems.
Author Response
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Author Response File: Author Response.pdf
Reviewer 4 Report
Comments and Suggestions for AuthorsThe authors derive the following sequences of numbers by using quadratic linear recurrence relations in a unified manner: Fibonacci and Lucas numbers, Pell,
Pell/Lucas and Chebyshev polynomials. Other properties, like Binet form expressions, Cassini identities and Catalan formulae, and results concerning power sums, ordinary
and binomial convolutions are also obtained by employing the symmetric feature,
series rearrangements and the generating function approach.
Actually, in my opinion, although the purpose of unification seems to be reached, the proofs are short and the techniques are very standard, so that the article does not deserve publication in the Symmetry journal.
Author Response
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Author Response File: Author Response.pdf
Round 2
Reviewer 1 Report
Comments and Suggestions for AuthorsThere is no extra comment.
Reviewer 2 Report
Comments and Suggestions for AuthorsThe paper can be accepted.
