Evaluating Infinite Series Involving Harmonic Numbers by Integration

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

I advise the author to demonstrate the value of the obtained results by application to some similar series. In this form we see only some results and review we do not see the importance and scientific value of the results obtained in the paper.

Also, adding some additional references recommended below will allow the authors to cover all aspects of the considered problems.

Shafiq, M., Srivastava, H. M., Khan, N., Ahmad, Q. Z., Darus, M., & Kiran, S. (2020). An upper bound of the third Hankel determinant for a subclass of q-starlike functions associated with k-Fibonacci numbers. Symmetry, 12(6), 1043.

T. Usman, N. U. Khan, M. Aman, Y. Gasimov. A unified family of multivariable Legendre poly-Genocchi polynomials. Tbilisi Mathematical Journal, 14(2), 2021, 153-170. 

Comments on the Quality of English Language

A minor correction is needed.

Author Response

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Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

Review on the paper “Evaluating Infinite Series Involving Harmonic Numbers by Integration” by C. Li and W. Chu

The paper deals with studies of infinite series involving harmonic-like numbers (called Euler sums). The authors evaluate them in closed form exclusively by integration method together with calculus and complex analysis. In addition, the authors write down special integral identities.

The representation of harmonic-like numbers by Riemann zeta function plays the important role in the paper.

Note that the authors reprove the Euler Theorem (Theorem 1) and it is important for understanding the methods in the paper. Also, it should be noted four interesting identities (3)-(6).

The paper is interesting. I think that after minor revision the paper can be published in the journal.

 

Comments.

1) It is important to define ₄F₃ (page 2)

2) The authors cited [18] for the definition of the trilogarithm function, but for readers would be much more comfortable and convenient to have here in the paper the exact definition of this function.

3) The authors need to clearly explain in the introductory part of the paper what’s new in their work and what’s well-known. Theorems 5, 6 and 7, 8 seem to be new.

Author Response

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Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

Dear Authors,

The report is attached to this submission.

Best wishes

Comments for author File: Comments.pdf

Author Response

see uploaded pdf-file.

Author Response File: Author Response.pdf