2022 Fiscal Year Annual Research Report
スクリャーニン・ホイン演算子とホイン・バンルベ対応に関する研究
| Project/Area Number |
22F21320
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| Allocation Type | Single-year Grants |
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| Research Institution | Kyoto University |
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| Host Researcher |
辻本 諭 京都大学, 情報学研究科, 教授 (60287977)
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| Foreign Research Fellow |
GABORIAUD JULIEN 京都大学, 情報学研究科, 外国人特別研究員
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| Project Period (FY) |
2022-04-22 – 2024-03-31
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| Keywords | Heun operator / Integrable system / orthogonal polynomials |
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| Outline of Annual Research Achievements |
One key question at the core of the research program is: how can we effectively tackle the task of finding and studying novel and increasingly general special functions?
Inspired by previous work on q→-1 limits of various families of orthogonal polynomials, we tried extending it to encompass more general special functions, specifically hypergeometric rational functions. We studied the Wilson biorthogonal rational functions, which can be thought of as a rational function generalization of the Askey-Wilson polynomials. We obtained their q→-1 limits and characterized the properties of the resulting functions. These developments have led to multiple conference presentations and an article on these results is in preparation.
Following the presentation of these results in conferences, discussions with Alexander Stokes from Tokyo University were initiated, on the topic of q→-1 limits of Painleve equations. Painleve equations in the -1 world were not known but one hope was to make some progress through the use of Sklyanin-Heun operators. Alexander Stokes visited Kyoto for a few days of work but so far there has not been any significant breakthrough.
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| Current Status of Research Progress |
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
One remarkable aspect of special functions is their connection with abstract algebra in mathematics. Through the study of representations of algebras, one often encounters special functions. This includes representations of quantum algebras, which possess a parameter q. It has been seen that the representation theory of these algebras when q→-1 is related to q→-1 limits of various families of orthogonal polynomials. We are now trying to extend this to other cases when q tends to other roots of unity.
Another promising avenue of research is the exploration of multivariate families of orthogonal polynomials. This connects to various other fields, such as association schemes, tridiagonal pairs, etc. A number of projects related to this that are currently being developed.
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| Strategy for Future Research Activity |
We are planning to discuss the above questions related to the limits when q tends to other roots of unity with Alexis Langlois-Remillard during his visit in Kyoto in June 2023. We are also hoping to study extensions of Leonard pairs, called tridiagonal pairs, with Nicolas Crampe during his visit in Kyoto in May 2023. On the topic of q→-1 limits of Wilson biorthogonal rational functions, we are planning to complete the first draft of the paper. Note that these functions are very general. As such, it would be desirable to also develop other more simple examples of q→-1 limits of rational functions. A first idea that comes to mind would be to study q-Pastro polynomials and their q→-1 limits. Such an example would provide an easier entry into the world of q→-1 rational functions.
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