Abstract : Orthogonal polynomials play crucial roles in a variety of fields including integrable systems, combinatorics, quantum information and so on. Generalization of orthogonal polynomials has thus been considered from many points of view and has led to successful application to such areas. This minisymposium aims to bring together latest research results on theory and applications of generalized orthogonal polynomials and aims to promote interdisciplinary discussions.
[04542] Meta algebras, biorthogonal rational functions and the Askey scheme
Format : Talk at Waseda University
Author(s) :
Satoshi Tsujimoto (Kyoto University)
Luc Vinet (IVADO & CRM, Université de Montréal)
Alexei Zhedanov (School of Mathematics, Renmin University)
Abstract : Algebras that subsume those of the Askey-Wilson type and are designated by the suffix meta are introduced to explain in a unified way the bispectral properties of the orthogonal polynomials of the Askey scheme and of biorthogonal rational functions that can be associated to the entries of that scheme. The Hahn and Racah families will be used to illustrate the framework.
[04588] Introducing q→-1 limits of biorthogonal rational functions: two instructive examples
Format : Talk at Waseda University
Author(s) :
Julien Gaboriaud (Kyoto University)
Satoshi Tsujimoto (Kyoto University)
Abstract : We recall how $q\to-1$ limits of orthogonal polynomials have been introduced and we introduce analogous limits for biorthogonal rational functions (BRF). In order to illustrate the main properties of these $q\to-1$ BRF, we look at two "extremal" cases: the most general one (Wilson) and one of the simplest ones (Pastro).
[03379] CMV bispectrality of polynomials orthogonal on the unit circle
Format : Online Talk on Zoom
Author(s) :
Alexei Zhedanov (School of Mathematics, Renmin University)
Abstract : We present new explicit results and examples concerning CMV bispectrality of the polynomials orthogonal on the unit circle.
[04976] The Element Distinctness Problem Revisited
Format : Online Talk on Zoom
Author(s) :
Hajime Tanaka (Tohoku University)
Abstract : The element distinctness problem is the problem of deciding whether or not a list contains identical elements. In this talk, I will revisit Ambainis' famous quantum algorithm for the problem (2007) and its refinement by Portugal (2018) in terms of the Grover quantum walk on the Johnson graphs. I will explain how a result about orthogonal polynomials (i.e., Leonard pairs) plays a role here.
[02693] Christoffel Transformations for (Partial-)Skew-Orthogonal Polynomials and Applications
Format : Talk at Waseda University
Author(s) :
Guofu Yu (Shanghai Jiao Tong University)
Abstract : In this talk, we consider the Christoffel transformations for skew-orthogonal polynomials and partial-skew-orthogonal polynomials. We demonstrate that the Christoffel transformations can act as spectral problems for discrete integrable hierarchies, and therefore we derive certain integrable hierarchies from these transformations. Some reductional cases are also considered. This is a joint work with Shi-Hao Li.
[02730] Multiple skew orthogonal polynomials and two-component Pfaff lattice
Format : Talk at Waseda University
Author(s) :
Shi-Hao Li (Sichuan University)
Abstract : The relation between orthogonal polynomials and integrable system is a long-standing focus in mathematical physics, and sheds light in many related fields like random matrices, random walks, and so on. Skew orthogonal polynomials, which were proposed in the study of random matrices with orthogonal/symplectic symmetry, were found to be wave functions for integrable systems of DKP type. In this talk, we will generalize this frame by considering multiple skew orthogonal polynomials which involve several different weights. In particular, connections with integrable systems are also considered. Pfaffian expressions, recurrence relations and Cauchy transforms for multiple skew orthogonal polynomials will be performed to give rise to multiple-component Pfaff lattice hierarchy.
[04996] Another Type of Forward and Backward Shift Relations for Orthogonal Polynomials in the Askey Scheme
Format : Talk at Waseda University
Author(s) :
Satoru Odake (Shinshu University)
Abstract : The forward and backward shift relations are basic properties of the (basic) hypergeometric orthogonal polynomials in the Askey scheme (Jacobi, Askey-Wilson, $q$-Racah, big $q$-Jacobi etc.) and they are related to the factorization of the differential or difference operators. Based on other factorizations, we obtain another type of forward and backward shift relations.
