Three General Double-Series Identities and Associated Reduction Formulas and Fractional Calculus

Round 1

Reviewer 1 Report

The authors of the article cited many well-known formulas. On the one hand, the reader can independently find it in reference books and scientific articles. But on the other hand, it makes it easier for the reader to read. Starting from Theorem 1, the main question is what is the condition for the convergence of the variable z. What are the requirements for the Psi function? It would probably be advisable to add information that the conditions for the convergence of the series need to be considered additionally.

Author Response

Many thanks for the constructive and encouraging comments.

Author Response File: Author Response.pdf

Reviewer 2 Report

Paper:  Three general double-series identities and associated reduction formulas, and fractional calculus

Section 1- Introduction and preliminaries

This section is very well organized, and has the required structure and background supported by excellent references. The big abnormal size of this section is justified due to the intended and clarified relations between the contents presented in this work.

Section 2 - Three general double-series identities

Contains three nice theorems and the proofs are ok. Structurally and scientifically the proofs hold on their main ideas.

Section 3 - Three main transformation formulas between Srivastava-Daoust and Kampé de Fériet functions

Can be the title shorter? Since this is a very short section, with only one theorem and its proof, the title perhaps can anticipate to the reader the existence of extensive developments, so my suggestion is: Transforming Srivastava-Daoust to Kampé de Fériet functions.

Section 4 - Application of fractional calculus

Since this section only contains the statements of the theorems, can the authors add some words referring that their proof come exactly from the structure described in each presentation text before each Theorem statement, just to be clearer. The ideas kneading is very well done by the way. Congrats.

Section 5 – Certain particular instances of transformations (50), (51) and (52)

Can the “particular” word in the title be removed? An instance is already a particular case of something.

The presented examples are great, but are they really all needed?

Section 6 - Summation formulas for Kampé de Fériet and p+1Fp

Just a collection of applications. Can the sections 5 and 6 be merged? It is ok like this, but after a careful reading we noticed that they can be just one, just because they contain just “instances”.

Author Response

Many thanks for the constructive and encouraging comments.

Author Response File: Author Response.pdf

Reviewer 3 Report

The authors present a rather exhaustive treatment of three particular double series identities for generalized hypergeometric functions. They present numerous examples and give several applications which I think will broaden the reach of the paper to a larger audience and it has good pedagogical value for the reader. The thoroughness of the presentation and the quality of the presentation lead me to suggest publication in Fractal and Fractional. The novelty and significance are average in my opinion.  I have just a couple minor suggestions for the authors neither of which I'm insistent upon.

1) The bottom of page 5 through page 7 is listing of relevant facts. It might help the reader to break this off as a subsection. Or maybe even enumerate the listed facts. 

2) If possible I think it would add to the presentation to very briefly motivate the 22 examples. Why were those particular special case chosen? Is there a parameter set that appears in common physical or mathematical applications? etc. 

3) I'm not sure I understand the difference between and example and an instance. It seems to me they are subexamples of examples 1, 7, and 13?? 

Author Response

Many thanks for the constructive and encouraging comments.

Author Response File: Author Response.pdf