On Degenerate Truncated Special Polynomials
Round 1
Reviewer 1 Report
Authors introduce degenerate truncated forms of special polynomials and numbers, in particular, degenerate truncated exponential, Stirling, Fubini, Bernoulli, Euler, Bell, Bernstein numbers and polynomials. They derive the summation formula, recurrence relations, connection formulas and set up the connection with the original form (non-degenerate) of truncated polynomials.
The article contains all the necessary proofs and all declared results.
There is, however, a number of corrections concerning the style of exposition, which has to be done:
(1) In the paper, including the abstract, authors use the word "some" incorrectly. For instance in expressions "the some", "some special", "their some" and so on. Authors have to replace such incorrect expressions.
(2) In the introduction, it has to be clearly explained what the difference between the ordinary truncated polynomials and degenerate truncated polynomials. It is important.
(3) In the abstract and in the conclusion section, authors write "series manipulation methods and some special proof techniques". The methods and techniques have to be explained in words explicitly, what kind of techniques and methods are used.
(4) In the conclusion section: "interesting formulas". The word "interesting" is not appropriate in this context.
The article can be published after revision.
Author Response
Thank you for your comments and suggestions.
Please see the attached corrected version.
Author Response File: 
Author Response.pdf
Reviewer 2 Report
My opinion on the article being evaluated is basically, but not extremely, positive. The papare is founded on a well-known extension of the Pochhammer symbol that is used to generalize the notion of Truncated Special Polynomials and from here in sequence are derived properties similar to the non-degenerate case when $ \ lamda = 1 $. I have not found relevant errors. I would like to point out a small inaccuracy on page 3 which can be found in the annex. Notwithstanding this, for me, this is not an extremely impactful work for novelties, but a long exercise of adaptations induced by the definition of $ \ lambda $ - falling factorial, I believe that, since it is a repertoire owned by a certain interest, the work deserve to be published
Comments for author File: Comments.pdf
Author Response
Thank you for your comments and suggestions.
Please see the attached corrected version.
Author Response File: Author Response.pdf
Reviewer 3 Report
The paper, after minor revision conteined in the report, can be accepted.
Comments for author File: Comments.pdf
Author Response
Thank you for your comments and suggestions.
Please see the attached corrected version.
Author Response File: Author Response.pdf
Reviewer 4 Report
This paper considered the degenerate truncated forms of the various special polynomials and numbers and have investigated their several properties and interesting relationships by using the series manipulation method and generating function techniques. The authors introduced the degenerate truncated exponential polynomials and have given their several properties. The degenerate truncated Stirling polynomials of the second kind are designed and they have proved elementary properties and relations. Some other types of polynomials have also been considered. In general, the results are new. I welcome any advance in this promising field. However, the presentation can be improved and there are some technical issues that need to be clarified. I will give my final recommendation based on the revised version. The detailed comments are as follows.
(1) The abstract can be improved. For example, "Moreover, ..... Furthermore,...." These sentences can be better written.
(2) In the first sentence, it is mentioned that special functions have "a lot of importances" in a number of fields. It would be beneficial to mention one or two concrete examples.
(3) What is the mathematical motivation for considering degenerate forms for special polynomials? It is not mentioned in Introduction.
(4) Last line on page 2, I would suggest delete "some".
(5) Generating function is a key tool used here. It is worth mentioning some applications of them in other fields. e.g. 'False positive and false negative effects on network attacks'.
(6) In (2.6), can the definition for n=0 be somehow justified?
(7) The several equalities at the bottom of page 4 are not very straightforward. Some explanations are appreciated.
(8) A proof of Proposition 3 should be given. Or at least some key steps should be added.
(9) A combinatorial explanation for Stirling numbers would be useful.
(10) Page 9, the first line, "settting" should be "setting". There are ohter errors and typos. Please carefully walk through them.
(11) I suggest many remarks can be combined in a nicer way. For example, Remarks 7-10. At present, they are just one sentence each.
(12) In the proof of Theorem 3, how |y|<1 is used? The similar problem occurs in other results.
(13) Last line in page 11, "is" should be "are".
(14) Theorem 22 is not obvious. The full proof should be given.
(15) In the conclusion section, some open problems and future directions should be mentioned.
Author Response
Thank you for your comments and suggestions.
Reply to "(6) In (2.6), can the definition for n=0 be somehow justified?" : It is just definition of the Pochhammer symbol.
Reply to "In the proof of Theorem 3, how |y|<1 is used? The similar problem occurs in other results.": To get geometric series, we take |y|<1 for convergence of the series.
The other valuable comment are corrected and the updated sentences are indicated in the attached corrected pdf version.
Author Response File: Author Response.pdf
Round 2
Reviewer 4 Report
The authors have made good effort to patch the paper. All my previous concerns have been addressed. I am glad to recommend this paper for publication in Mathematics.
Author Response
Thank you for your comments.
