Posted: Jan. 29, 2020
The authors sketch a proof of an interesting identity between two integrals. Although I like their theorem very much, their paper is not publishable as it now stands.
The authors begin their paper by stating their Proposition, providing no background what- soever. (For some unknown reason, the authors choose to describe the paper's only result as a proposition.) It is only at the end of their paper, that some motivation for their article appears. Thus, only after giving the origin of the identity, background, and any previous appearances, should the authors state their identity and give a proof.
The paper hinges on definitions in lines 3,4 of two rational functions R1 (x) and R2 (x), for which no motivation is given. Since these functions are at the heart of their paper, the authors should establish the reasoning behind making these definitions, which are critically used for the identities in lines 6,7. Leading up to these formulas, in the third line of the paper's proof, the authors write, "We cleverly construct the rational functions . . . ." It is nice to know that the authors are clever, but such an assertion does not help the reader. From whence do the formulas involving R1 (x) and R2 (x) come? Without explaining their origins to readers, they will remain in a quandary. The only readers who would be satisfied are those wanting to see a magician "pull a rabbit out of his hat."
The paper is written in a blackboard style with two interjections of "check!" As I indicated above, the paper is written in reverse order from what it should be written. Maybe this is ok for a blackboard lecture, but for a lasting journal, this is not ok.
In the authors' first comment, they ask readers to consult Bailey's book, Generalized Hy- pergeometric Series, for a theorem equivalent to their primary theorem. Do the authors really expect readers to turn to Bailey's book to justify the authors' remark on equivalence? More- over, they tell us that this equivalent formula is "buried" in Bailey's book. The use of the word "buried" would seem to indicate that their source would be an old book or journal article that would be hard to find. Since Bailey's book is a classic and does not fit the above- mentioned criteria, a comment on "burying" does not seem appropriate. If the authors do indeed think that their result is "buried" in Bailey's book, by all means, they should resurrect it for readers and prove the equivalence. It also seems unnecessary to tell readers that one of the authors is an "expert" in finding such formulas. It is nice to know that one of the authors is an "expert", but this reader would prefer to read proofs instead! It is also strange to tell us what each of the authors did in the preparation of their paper. I have never read a multiple authored paper in which readers are told "who did what."
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