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Integrable Systems and Special Functions

Lecturer:  Masatoshi NOUMI

KTH Royal Institute of Technology ( / Kobe University (

May 07, 2021:  Today's lecture was the last of 12 lectures.  I am happy so many people participated in the lectures.  Thank you all! 

Mar 26, 2021:  The next lecture (Lecture 9) will be on Friday April 16, 15:00 - 17:00.   It will be the first lecture of the second part of the course.  Topics will be taken mainly from (B)  Symmetries of Painlevé equations.  

Mar 12, 2021:  We will not have lectures on April 2 and April 9, 2021.  Lectures will be given on March 19 (Lecture 7), March 26 (Lecture 8) and resume from April 16 (Lecture 9) on.  

Feb 08, 2021:  The next lecture will be on Friday 12, 2021, 15:00-17:00.  We are going to use the same zoom link as the one we used in the first lecture.  

Jan 26, 2021:   The lectures will be delivered on zoom.  The first lecture will be on Friday, February 5, 2021, 15:00-17:00.   If you are interested to participate in this course, you are welcome to send a request to  (or ) to get the zoom link for the course.


Summaries and Notes

Lecture A01:   SummaryA01_2021_02_05.pdf / NotesA01_2021_02_05.pdf

Lecture A02:  SummaryA02_2021_02_12.pdf / NotesA02_2021_02_12.pdf

Lecture A03:  SummaryA03_2021_02_19.pdf / NotesA03_2021_02_19.pdf

Lecture A04:  SummaryA04_2021_02_26.pdf / NotesA04_2021_02_26.pdf

Lecture A05:  SummaryA05_2021_03_05.pdf / NotesA05_2021_03_05.pdf

Lecture A06:  SummaryA06_2021_03_12.pdf / NotesA06_2021_03_12.pdf

Lecture A07:  SummaryA07_2021_03_19.pdf / NotesA07_2021_03_19.pdf

Lecture A08:  SummaryA08_2021_03_26.pdf / NotesA08_2021_03_26.pdf

Lecture B01:  SummaryB01_2021_04_16.pdf / NotesB01_2021_04_16.pdf

Lecture B02:  SummaryB02_2021_04_23.pdf / NotesB02_2021_04_23.pdf

Lecture B03:  SummaryB03_2021_04_30.pdf / NotesB03_2021_04_30.pdf

Lecture B04:  SummaryB04_2021_05_07.pdf / NotesB04_2021_05_07.pdf


Course Description

This introductory course will cover several topics around which the theories of integrable systems, representation theory and special functions interact.  It will consist of 12 double-hour lectures in Periods 3 and 4.   

The course is intended for PhD students, postdocs and interested faculty.   We plan to give out homework problems and/or mini-projects to allow PhD students to earn course points.  

As to the contents of the course, we have three main subjects in mind as indicated below, and a substantial time will be used for the introductory parts.   The allotment of time on individual topics will be flexible, depending on backgrounds and interests of the audience.  


(A)  Introduction to Macdonald polynomials

1.  Symmetric functions and Schur functions

2.  Definition of Macdonald polynomials and some examples

3.  Commuting family of q-difference operators

4.  Fundamental properties of Macdonald polynomials

5.  Asymptotically free eigenfunctions

6.  Affine Hecke algebras and Dunkl operators


I.G. Macdonald: Symmetric Functions and Hall Polynomials, Second Edition. Oxford University Press, 1995, x+475pp.

I.G. Macdonald: Affine Hecke Algebras and Orthogonal Polynomials. Cambridge University Press, 2003, x+175 pp.

(B)  Symmetries of Painlevé equations

1.  Some examples of Painlevé equations

2.  Bäcklund transformations and affine Weyl group symmetry 

3.  Relation to nonlinear integrable hierarchies

4.  Birational representations of affine Weyl groups and discrete Painlevé equations


M. Noumi: Painlevé Equations through Symmetry.  American Mathematical Society, 2004. x+156 pp.

(C)  Elliptic hypergeometric functions

1.  Summation and transformation formulas for hypergeometric series

2.  q-Hypergeometric series and elliptic hypergeometric series

3.  Spiridonov’s elliptic beta integral and elliptic hypergeometric integrals

4.   q-Selberg integrals and elliptic Selberg integrals


G. Gasper and M. Rahman:  Basic Hypergeometric Series, Second Edition. Cambridge University Press, 2004, xxvi+428 pp. 

H. Rosengren: Elliptic Hypergeometric Functions.  Lectures at OPSF-S6 (arXiv:1608.06161)



The schedule for subsequent lectures will be decided on the first day, taking requests by the audience into consideration.  


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  • Integrable Systems and Special Functions