Riemann–Hilbert Problem for the Matrix Laguerre Biorthogonal Polynomials: The Matrix Discrete Painlevé IV

Round 1

Reviewer 1 Report

In this paper, the authors obtained Sylvester systems of differential equations for the orthogonal polynomials and its second kind functions, directly from a Riemann–Hilbert problem, with jumps supported on appropriate curves on the complex plane, which are constructed in terms of a given matrix Pearson equation. They also extended to the non-commutative case the discrete Painlevé I and the alternate discrete Painlevé I quations. Finally, they gave an explicit and general example to illustrate the theoretical results of the work. Thus, this paper generalizes and improves the theories of the classical Riemann-Hilbert problems and integral equations, which is valuable and meaningful.

Author Response

1. The introduction was left without definitions. 
2. We pass on that information to section 2 (where we introduce the definitions and key results).
3. We end section 1 with an outline of the work.
4. We added Zhou's bibliographical reference (the other one was already in the text).
5. We also write a summary of the strengths of the paper and announce the paper on the Jacobi case.
6. We improved the English.

 

Reviewer 2 Report

In this paper, the authors applied the Riemann-Hilbert problem with jump supported on an appropriate curve on the complex plane with a finite endpoint at the origin to the study of the corresponding matrix biorthogonal polynomials associated with Laguerre type matrices of weights, which are constructed in terms of a given matrix Pearson equation. The authors obtained Sylvester systems of differential equations for the orthogonal polynomials and its second kind functions, directly from a Riemann-Hilbert problem. They considered a Sylvester type differential Pearson equation for the matrix of weights. They also studied whenever the orthogonal polynomials and its second kind functions are solutions of a second order linear differential operators with matrix eigenvalues and found nonlinear equations for the matrix coefficients of the corresponding three term relations, which extends to the non-commutative case the discrete Painlevé I and the alternate discrete Painlevé I quations. Thus, this paper generalizes and improves the theories of the classical Riemann-Hilbert problems and integral equations, and also provides theoretical support to the Matrix Laguerre Biorthogonal Polynomials which is valuable and meaningful.

Author Response

1. The introduction was left without definitions. 
2. We pass on that information to section 2 (where we introduce the definitions and key results).
3. We end section 1 with an outline of the work.
4. We added Zhou's bibliographical reference (the other one was already in the text).
5. We also write a summary of the strengths of the paper and announce the paper on the Jacobi case.
6. We improved the English.

Reviewer 3 Report

In this paper, the authors study the Riemann–Hilbert problem with jump supported on an appropriate curve on the complex plane with a finite endpoint at the origin, is used for the study of the corresponding matrix biorthogonal polynomials associated with Laguerre type matrices of weights —which are constructed in terms of a given matrix Pearson equation. At the same time, The authors also derive the non-Abelian extensions of a family of discrete Painlevé IV equations.

  From my opinion, these results are reasonable in terms of mathematical methods and the manuscript is thus interesting.

The Riemann–Hilbert problem mentioned by the authors has been studied extensively (See, Giovanni A. Cassatella-Contra Manuel Manas, Stud Appl Math,2019; X. Zhou, SIAM J. MATH. ANAL.,1989, 966-986). So, I suggest the authors make a comment for the two related articles.

Moreover, there are some typing and grammar errors in the paper:

In addition, the references need to be broader than just those mentioned in this paper.

After revision, I would like to recommend the acceptance of the paper for publication.

Author Response

1. The introduction was left without definitions. 
2. We pass on that information to section 2 (where we introduce the definitions and key results).
3. We end section 1 with an outline of the work.
4. We added Zhou's bibliographical reference (the other one was already in the text).
5. We also write a summary of the strengths of the paper and announce the paper on the Jacobi case.
6. We improved the English.

Reviewer 4 Report

In this paper, the Riemann–Hilbert problem with a jump on a corresponding curve in the complex plane with a finite end at the origin is used to study the corresponding matrix biorthogonal polynomials associated with Laguerre-type weight matrices that are constructed in terms of a given Pearson matrix equation. From a scientific point of view, the article is framed correctly, reference material is given, the evidence is presented correctly. However, there are methodological comments that can improve the article.
Remarks
1. It is necessary to divide the introduction, remove definitions and reference material from it into a separate section.
2. At the end of the introduction, it would be nice to give a brief outline of the study.
3. After line 95, different spellings of the inverse matrix U are striking. It is necessary to bring to uniformity.
4. The article does not have a Conclusion section. Here we can summarize the results and outline the further development of the study.
5. The article has abbreviations that can be displayed in a separate table at the end of the article.
 After elimination of these minor remarks, the article can be recommended for publication in the journal.

Author Response

1. The introduction was left without definitions. 
2. We pass on that information to section 2 (where we introduce the definitions and key results).
3. We end section 1 with an outline of the work.
4. We added Zhou's bibliographical reference (the other one was already in the text).
5. We also write a summary of the strengths of the paper and announce the paper on the Jacobi case.
6. We improved the English.