On the Relation between Lambert W-Function and Generalized Hypergeometric Functions
Round 1
Reviewer 1 Report
Present article is devoted to the study of some types of special functions, namely the Lambert W-function, H-function and R-function. The article presents well-known formulas obtained in articles from the list of references, which express the solution of the trinomial equation through H-functions, as well as through the branches of the W-function, as well as formulas (25)-(11) determine the R-function and its integral representation. Formula (26) elementarily follows from these formulas, which determines the relationship between R-function and W-function. Also, the relationship between the branches of W-function and H-functions has been obtained.
All obtained formulas are direct consequences from the definition of special functions and formulas obtained earlier in the works indicated in the list of references. It does not require proof and, in my opinion, does not have independent scientific interest.
Author Response
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Reviewer 2 Report
The manuscript is devoted to the finding new relationships between special functions used for statistical applications.
The method used in the study is based on the well-known results published in journals and books. The start point was Lambert's trinomial equation and its limitation. Authors solve the equation in term of the H-function and then build new relation between H- and R-functions from one side and different branches of W-function from other side.
The study and the result seem to me quite useful for statistics and probability theory research purposes.
Author Response
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Reviewer 3 Report
In this paper, the authors studied the relationship between Lambert W-function and generalized hypergeometric functions is discussed. The relation between these functions is shown by the following two different strategies: by means of the direct and inverse Mellin transform of Lambert W-function and by solving the trinomial equation originally studied by Lambert and Euler. The paper's findings are interesting and potentially applicable in the special function field. Besides, the following points must be considered at the time of revision.
1. Describe all the parameters before its use. For example, see equation 4
2. The introduction needs to improve by providing the definitions/properties of the special functions used in this paper. For example, H-function (Refer Mathai et al., MA Pathan et al.), generalized hypergeometric function (Refer, HM Srivastava et al., KS Nisar et al. )
3. The paper's contents are too short. The authors can provide some examples, corollaries / some real-world applications/ possible applications in statistics other than the relationship of the function.
4. The authors must proofread the paper carefully by checking the convergence conditions provided with many equations.
Author Response
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Round 2
Reviewer 1 Report
see the attached file
Comments for author File: Comments.pdf
Author Response
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Reviewer 3 Report
The authors revised the paper based on my comments
Author Response
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