Various Series Related to the Polylogarithmic Function

Round 1

Reviewer 1 Report

The paper presents various results about series that involve (classical) polylogarithms, harmonic numbers and similar. In particular, non-trivial results between series representation and (iterated) integral representations are provided. As a practitioner of polylogarithms in various context, the first thing that surprises me is that the paper does not make any mention to the developments that happened in the last 2 decades, with the definition and study of a larger class of functions called hyperlogarithms or multiple polylogarithms (there are slightly different but equivalent definitions in the literature, see below). These functions are particularly important in theoretical physics and they received a lot of attention by physicists and mathematicians, see for example:

• E. Panzer, Comput.Phys.Commun. 188 (2015) 148-166
• A. Goncharov, Math.Res.Lett. 5 (1998) 497-516
• E. Remiddi, J. Vermasere, Int.J.Mod.Phys.A 15 (2000) 725-754
• C. Duhr, H. Gangl, J, Rhodes, JHEP 10 (2012) 075
• J. Vollinga, S. Weinzierl, Comput.Phys.Commun. 167 (2005) 177

and many others. In particular, there exist also a mathematical package to treat these functions in full generality:

- C. Duhr, F. Dulat, JHEP 08 (2019) 135

The integral representations present in this paper are special cases of hyperlogarithms, which in turn admit well understood series representations. In particular, in the reference by Remiddi and Vermaseren, the relation between a special type of hyperlogarithms (called harmonic polylogarithms) and harmonic sums is discussed in detail. Harmonic sums also appear in the results derived in the present paper.

The methods developed in the papers above (and in many others), allow one to find non-trivial functional relations among these functions and to perform series expansions in any relevant region. I believe all  integral representations in this paper can be expressed in terms of these functions and therefore I think it would be appropriate a reference to these functions and a discussion of whether these methods can help to prove these relations. 

The simplest example is indeed the classical polylogarithms Li_s(x), which are defined as

Li_{s+1}(x) = \int_0^x \frac{Li_s(y)}{y} dy,  with Li_1(x) = -\log(1-x), |x| < 1

And which can therefore be written as hyperlogarithms as Li_{s+1}(x) = - G(0,0,...,0,1,x)  with s occurrences of the index '0', following the notation of J. Vollinga and S. Weinzierl.

In conclusions, I think the series presented in this paper are non trivial and might be worth publishing, but that using the modern techniques to handle polylogarithmic functions and the definition of hyperlogarithms should help the authors to derive, or at least verify, their results much more easily.  I suggest the authors to keep this new results into consideration and use them to verify all their theorems. Moreover, it would be interesting to write down explicitly all the integral representations that they use in terms of hyperlogarithms. To make an explicit example, the integral in their Theorem 3, for |x|<1, |z|<1, can be written as

(x/(1-x) + Log[1-x]) Li_s(z) - x/(1-x)* Li_s(x z) + G(1,0,..,0,1/z,x),

with s-2 zeros in total in the function G (I follow here again the conventions in J Vollinga and S Weinzierl above.)

The authors should present all their results in terms of hyperlogarithms.

 

Author Response

Dear Referee,
thank you for the comments that helped to improve our paper.
We have followed your suggestions, and put the correct reference
to hyperlogarithms.
We did also present our results in terms of hyperlogarithms.

 

Reviewer 2 Report

The main results of this paper are a few equalities involving polylogarithms. The method is mostly routine and the results do not seem to be appealing to the general audience of this journal. So I recommend the author to submit the paper to another more specialized number theory journal. 

Author Response

Dear Referee,
thank you for the comments that helped to improve our paper.

Reviewer 3 Report

In the manuscript at hand the authors deal with a couple of series including and related to the polylogarithmic function. As the authors stress themselves, most of their results are extensions of formulas found in a series of the handbooks [6,15,14]. Despite the fact that the motivation for such kind of extensions is missing (i.e. the paper is just an addendum to the handbooks), I have a lot of concerns related to the structure of the manuscript. Usually, in writing a referee report I start with the main topics and end up with minor changes. In this case I will proceed the other way round in order to allow the authors to correct the manuscript in a straightforward way. My concerns (in chronological order, later on in larger units) are the following:

A. title: concerning -> related to

B. abstract and introduction: sum -> series

C. line 9: with evolution -> with the evolution

D. line 10 and later: found here [ ],[ ],[ ] -> found in Refs. [ , , ] (in LaTeX this is programmed with a common \cite command with entries separated by comma).

E. line 20: We will use the following notation throughout the paper. -> To start with, we explain the notation we will use throughout the paper.

F. Definition 1: polylogarithm see [11] is defined ... by -> polylogarithm is defined ... by [11] (similar corrections in the following will not be mentioned).

G. Definition 1: Also the special case we will use frequently is -> The iterative definition via integrals starts with

H. In all the complicated expressions the denominator factor should be pulled in front of the numerator in order to make the latter visible. Of course, for Theorem this will make necessary a line break. Particularly urgent is this solution for Theorems 5 and 6, but also the others will gain from this change (similar corrections also in the following)

I. Corollary 3(a), first part: what is the variable t in the result?

J. Abel's summation formula should be written separately and maybe even in front of Lemma 1, and it should always be cited like this, not "Abel's summation method" or similar.

K. Lemma 4: It is more convenient to call Li_s(z) the polylogarithmic function than Li_s. In order make possible the comments on the steps in the proof and not to start with "=" without reference, I suggest to call the expression to be calculated (i.e. the starting expression in the proof) like L(y,z) and use this notation throughout the proof (but not outside of it).

L. Lemma 4 (cont.): The limit goes to zero due to Lemma 3. -> The limit goes to zero due to Lemma 3. One obtains

M. Lemma 4 (cont.): Which are both from the Lemma 1. -> Using Lemma 1 one obtains

N. Proof of Theorem 1: rewriting harmonic -> rewriting the harmonic

O. Proof of Theorem 1 (cont.): Both sums come from Lemma 4, first one is direct and second one is when we take y to be yu. Plugging in the results from the Lemma 4 we get -> For both sums one can use Lemma 4, directly for the first one and taking yu for y for the second one. The result reads

The proof of Corollary 1(a) should be put behind the Corollary itself. The same for Corollary 2 and Corollary 3.

P. Proof of Theorem 3: "The proof is done." is redundant (also in the following. The square symbol is enough).

Q. What is "the most interesting result in our opinion"? Probably it is meant Theorem 6. This should be named here: "The most interesting result in our opinion is Theorem 6 containing a squared bracket, and Corollary 6(b) with the brackets of the third degree."

R. From the proof of Corollary 6(c) I see that this is not purely a numerical result but a formula, maybe the central formula of the paper. Therefore, it should be formulated as Theorem 7, not with the upper limits being 1/2 and the numerical value given, but with arbitrary upper limits x and sum over the brackets to the third degree as the result. The numerical result for x=1/2 can then be the corollary from this theorem.

S. At this point I can start with my main suggestions. Actually, It does not make any sense to formulate the theorems and Corollaries in the Introduction if the (closely related) proofs are found later. As the theorems are named also in the Main part of the manuscript, they can just be skipped in the introduction. On the other hand, the Corollaries have to be placed in front of their corresponding proofs. This means that the whole part of the Introduction starting with "The main results of the paper are the following." will vanish. In addition:

U. Corollary 3(a), second part: shouldn't this be formulated as a separate corollary? It is obviously 

V. Corollary 3(b): here for sure it should be formulated separately, as it is not a consequence of the previous one.

W. Asymptotic behaviour: where does this come and what is the benefit of having this expression in the paper, i.e. where are the applications of this?

X. Harmonic sum: the same question. In addition: why do the authors change from q to n? They should stay with one of these - even if q is originally not integer, setting q=5 would suffice to make it integral. The lines (6), (7), (8), (9) and (10) should be put separately, the curly brackets removed and the lines named as f(x,1) to f(x,5). I guess that the results are obtained (and the hypothesis up to q=60 checked) with MATHEMATICA Alpha which of course should be cited correctly in the paper, not just by naming it.

Y. Point (2) in the Conclusions is more like an outlook. Therefore, it should be moved to the end and the title should be changed to "Conclusions and Outlook".

Z. Finally: the motivation to be found should be described also in the Conclusions.

Certainly, the manuscript has to take a second round where a profound motivation has to be added. But if this is done, I think that this manuscript can be published in Axioms.

Author Response

Dear Referee,
thank you for the comments that helped to improve our paper.

We have covered all of your points from A to Z, and overhauled the structure
of the paper. Now the corollaries are presented right after the
correspondig theorem, not in a separate section.

 

 

Round 2

Reviewer 1 Report

Thank you for considering my comments and adding references to Hyperlogarithms. Note that the definition given in eq (4) is not appropriate when z_k = 0, in fact one needs to add for n-zeros

G(0,0,...,0,x) = 1/n! Log^n(x)

 

 

Author Response

Dear Referee,

we have added the hyperlogarithms case when z_i=0 in eq. (4).

Reviewer 2 Report

None.

Author Response

Dear Referee,
thank you for the comments that helped to improve our paper.

Reviewer 3 Report

I am happy that the authors accepted (most of) the suggestions and improved thereby the logics and readability of the manuscript. Just a few remnants from my long list:

D. The replacements like [ ],[ ],[ ] -> [ , , ] have not been performed in all instances.

Y. Point (2) in the Conclusions is more like an outlook. Therefore, it should be moved to the end.

To continue with this last issue on the Conclusions, Point (3) is very well suited to be a conclusion and also a motivation for the publication. At the same time, (1) does not work out as first point. Therefore, I suggest the order (3)(1)(2), the latter being the outlook.

I think that the manuscript is approaching the publication. Having a fresh view at the manuscript, though, I would like to ask to improve the following new items:

1. Does it make sense that only the proofs of Theorems are started with a boldface "Proof." but not for Corollaries. I would unify this notation.

2. The sentence "In the following we give a proof of the Theorem ..." (several times) is more than redundant. Actually, at this point the Theorem is stated while the proof is following after that. You can skip these sentences.

3. Starting from Corollary 5, the authors write the insertion of particular values as part of the Corollary - and "prove" this part by doing just the same as in the corollary, i.e. inserting the value. This part can (and should) be skipped from the proof (or, alternatively, from the Corollary, if it is more appropriate).

4. Definition 5: polylogarith -> polylogarithmic (or you skip "function" and replace by "polylogarithm")

5. In all instances, you can replace the addition ", see [...]" with and without full stop by just the reference [...] without full stop. In Definition 4 this "see [8]" should be moved to the end of this line and treated the same way. A good example is Definition 5.

6. A hypothesis on harmonic sum -> A hypothesis of the harmonic sum

Author Response

Dear Referee,
we have corrected points D and Y, overhauling Conclusions.
The new points 1-6 have all been addressed.