## An algebraic description of the bispectrality of the biorthogonal rational functions of Hahn type

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- by Satoshi Tsujimoto, Luc Vinet and Alexei Zhedanov PDF
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## Abstract:

The biorthogonal rational functions of ${_3}F_2$ type on the uniform grid provide the simplest example of rational functions with bispectrality properties that are similar to those of classical orthogonal polynomials. These properties are described by three difference operators $X,Y,Z$ which are tridiagonal with respect to three distinct bases of the relevant finite-dimensional space. The pairwise commutators of the operators $X,Y,Z$ generate a quadratic algebra which is akin to the algebras of Askey–Wilson type attached to hypergeometric polynomials.## References

- James A. Wilson,
*Orthogonal functions from Gram determinants*, SIAM J. Math. Anal.**22**(1991), no. 4, 1147–1155. MR**1112071**, DOI 10.1137/0522074 - Dharma P. Gupta and David R. Masson,
*Contiguous relations, continued fractions and orthogonality*, Trans. Amer. Math. Soc.**350**(1998), no. 2, 769–808. MR**1407490**, DOI 10.1090/S0002-9947-98-01879-0 - Mourad E. H. Ismail and David R. Masson,
*Generalized orthogonality and continued fractions*, J. Approx. Theory**83**(1995), no. 1, 1–40. MR**1354960**, DOI 10.1006/jath.1995.1106 - A. Zhedanov,
*Padé interpolation table and biorthogonal rational functions*, Rokko. Lect. in Math.**18**(2005), 323-363. - Alexei Zhedanov,
*Biorthogonal rational functions and the generalized eigenvalue problem*, J. Approx. Theory**101**(1999), no. 2, 303–329. MR**1726460**, DOI 10.1006/jath.1999.3339 - Satoshi Tsujimoto, Luc Vinet, and Alexei Zhedanov,
*Jordan algebras and orthogonal polynomials*, J. Math. Phys.**52**(2011), no. 10, 103512, 8. MR**2894610**, DOI 10.1063/1.3653482 - E. G. Kalnins and Willard Miller Jr.,
*$q$-series and orthogonal polynomials associated with Barnes’ first lemma*, SIAM J. Math. Anal.**19**(1988), no. 5, 1216–1231. MR**957681**, DOI 10.1137/0519086 - Eiichi Bannai and Tatsuro Ito,
*Algebraic combinatorics. I*, The Benjamin/Cummings Publishing Co., Inc., Menlo Park, CA, 1984. Association schemes. MR**882540** - Ya. I. Granovskiĭ, I. M. Lutzenko, and A. S. Zhedanov,
*Mutual integrability, quadratic algebras, and dynamical symmetry*, Ann. Physics**217**(1992), no. 1, 1–20. MR**1173277**, DOI 10.1016/0003-4916(92)90336-K - Paul Terwilliger,
*Two linear transformations each tridiagonal with respect to an eigenbasis of the other*, Linear Algebra Appl.**330**(2001), no. 1-3, 149–203. MR**1826654**, DOI 10.1016/S0024-3795(01)00242-7 - Douglas A. Leonard,
*Orthogonal polynomials, duality and association schemes*, SIAM J. Math. Anal.**13**(1982), no. 4, 656–663. MR**661597**, DOI 10.1137/0513044 - S. Tsujimoto, L. Vinet, and A. Zhedanov,
*The rational Heun operator and Wilson biorthogonal functions*, arXiv:1912.11571, (2019). - Luc Vinet and Alexei Zhedanov,
*The Heun operator of Hahn-type*, Proc. Amer. Math. Soc.**147**(2019), no. 7, 2987–2998. MR**3973900**, DOI 10.1090/proc/14425 - V. Ginzburg,
*Calabi-Yau algebras*, arXiv: math/0612139, (2006). - Gwyn Bellamy, Daniel Rogalski, Travis Schedler, J. Toby Stafford, and Michael Wemyss,
*Noncommutative algebraic geometry*, Mathematical Sciences Research Institute Publications, vol. 64, Cambridge University Press, New York, 2016. Lecture notes based on courses given at the Summer Graduate School at the Mathematical Sciences Research Institute (MSRI) held in Berkeley, CA, June 2012. MR**3560285**

## Additional Information

**Satoshi Tsujimoto**- Affiliation: Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University, Yoshida-Honmachi, Kyoto, Japan 606-8501
- MR Author ID: 339527
**Luc Vinet**- Affiliation: Centre de recherches mathématiques, Université de Montréal, P. O. Box 6128, Centre-ville Station, Montréal (Québec), Canada H3C 3J7
- MR Author ID: 178665
- ORCID: 0000-0001-6211-7907
**Alexei Zhedanov**- Affiliation: School of Mathematics, Renmin University of China, Beijing 100872, People’s Republic of China
- MR Author ID: 234560
- Received by editor(s): May 7, 2020
- Received by editor(s) in revised form: May 30, 2020
- Published electronically: November 30, 2020
- Additional Notes: The first author’s work was partially supported by JSPS KAKENHI (Grant Numbers 19H01792, 17K18725).

The research of the second author was funded in part by a Discovery Grant from the Natural Sciences and Engineering Council (NSERC) of Canada.

The third author gratefully acknowledges the award of a CRM-Simons Professorship and was supported by the National Science Foundation of China (Grant No.11771015). - Communicated by: Mourad Ismail
- © Copyright 2020 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**149**(2021), 715-728 - MSC (2010): Primary 33C45, 33C80
- DOI: https://doi.org/10.1090/proc/15225
- MathSciNet review: 4198077