Introduction Multivariate discrete orthogonal polynomials The higher rank Racah algebra The Racah algebra and multivariate Racah polynomials Hendrik De Bie Ghent University joint work with Vincent Genest, Luc Vinet (CRM, Montreal) Plamen Iliev (GAtech) Wouter van de Vijver (UGent) Dubrovnik, 27 June 2019 Hendrik De Bie The Racah algebra and multivariate Racah polynomials
Introduction Multivariate discrete orthogonal polynomials The higher rank Racah algebra Introduction Discrete orthogonal polynomials The Askey scheme Univariate Racah polynomials and Racah algebra Multivariate discrete orthogonal polynomials The higher rank Racah algebra Construction Connection with multivariate Racah polynomials Algebraic reinterpretation Hendrik De Bie The Racah algebra and multivariate Racah polynomials
Introduction Discrete orthogonal polynomials Multivariate discrete orthogonal polynomials The Askey scheme The higher rank Racah algebra Univariate Racah polynomials and Racah algebra Outline Introduction Discrete orthogonal polynomials The Askey scheme Univariate Racah polynomials and Racah algebra Multivariate discrete orthogonal polynomials The higher rank Racah algebra Construction Connection with multivariate Racah polynomials Algebraic reinterpretation Hendrik De Bie The Racah algebra and multivariate Racah polynomials
Introduction Discrete orthogonal polynomials Multivariate discrete orthogonal polynomials The Askey scheme The higher rank Racah algebra Univariate Racah polynomials and Racah algebra Discrete orthogonal polynomials What? ◮ family of polynomials φ n ( x ), n = 0 , 1 , . . . ◮ deg φ n ( x ) = n ◮ orthogonal w.r.t. discrete measure � w ( x ) φ m ( x ) φ n ( x ) = γ n δ mn x ∈ S Hendrik De Bie The Racah algebra and multivariate Racah polynomials
Introduction Discrete orthogonal polynomials Multivariate discrete orthogonal polynomials The Askey scheme The higher rank Racah algebra Univariate Racah polynomials and Racah algebra Example: Krawtchouk polynomials Put, for n = 0 , . . . , N � − n , − x � ; 1 K n ( x ; p , N ) = 2 F 1 − N p ◮ x variable ◮ p parameter, 0 < p < 1 ◮ N grid length ◮ of hypergeometric type Orthogonality: � N � � N p x (1 − p ) N − x K m ( x ; p , N ) K n ( x ; p , N ) = γ n δ mn x � �� � x =0 weight w ( x ) Hendrik De Bie The Racah algebra and multivariate Racah polynomials
Introduction Discrete orthogonal polynomials Multivariate discrete orthogonal polynomials The Askey scheme The higher rank Racah algebra Univariate Racah polynomials and Racah algebra The Askey scheme Orthogonal polynomials: ◮ univariate ◮ of hypergeometric type ◮ continuous or discrete orthogonality ◮ satisfying some generalization of Bochner’s theorem have been classified in the so-called Askey scheme R. Askey, J. Wilson, Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Memoirs of the American Mathematical Society, 54 (1985): iv+55 R. Koekoek, P. A. Lesky, and R. F. Swarttouw. Hypergeometric Orthogonal Polynomials and Their q -Analogues. Springer , 2010. Hendrik De Bie The Racah algebra and multivariate Racah polynomials
Introduction Discrete orthogonal polynomials Multivariate discrete orthogonal polynomials The Askey scheme The higher rank Racah algebra Univariate Racah polynomials and Racah algebra Hendrik De Bie The Racah algebra and multivariate Racah polynomials
Introduction Discrete orthogonal polynomials Multivariate discrete orthogonal polynomials The Askey scheme The higher rank Racah algebra Univariate Racah polynomials and Racah algebra Racah polynomials Definition The Racah polynomials are defined as r n ( α, β, γ, δ ; x ) := ( α + 1) n ( β + δ + 1) n ( γ + 1) n � � − n , n + α + β +1 , − x , x + γ + δ +1 × 4 F 3 ; 1 α +1 ,β + δ +1 ,γ +1 ◮ most complicated discrete OPs in Askey scheme ◮ appear in many different contexts ◮ highly complicated ◮ rather unpleasant to work with ◮ no need to remember definition! Hendrik De Bie The Racah algebra and multivariate Racah polynomials
Introduction Discrete orthogonal polynomials Multivariate discrete orthogonal polynomials The Askey scheme The higher rank Racah algebra Univariate Racah polynomials and Racah algebra Racah algebra Algebra with 2 generators satisfying: [ K 1 , K 2 ] = K 3 [ K 2 , K 3 ] = K 2 2 + { K 1 , K 2 } + dK 2 + e 1 [ K 3 , K 1 ] = K 2 1 + { K 1 , K 2 } + dK 1 + e 2 d , e 1 and e 2 structure constants Y.A. Granovskii, A.S. Zhedanov, Nature of the symmetry group of the 6 j -symbol. Sov. Phys. JETP 67:1982-1985, 1988. Hendrik De Bie The Racah algebra and multivariate Racah polynomials
Introduction Discrete orthogonal polynomials Multivariate discrete orthogonal polynomials The Askey scheme The higher rank Racah algebra Univariate Racah polynomials and Racah algebra The standard realization K 1 := x ( x + γ + δ + 1) K 2 := B ( x ) E x − ( B ( x ) + D ( x )) I + D ( x ) E − 1 x with the shift operator E x ( x ) = x + 1 and B ( x ) := ( x + α + 1)( x + β + δ + 1)( x + γ + 1)( x + γ + δ + 1) (2 x + γ + δ + 1)(2 x + γ + δ + 2) D ( x ) := x ( x − α + γ + δ )( x − β + γ )( x + δ ) (2 x + γ + δ )(2 x + γ + δ + 1) Hendrik De Bie The Racah algebra and multivariate Racah polynomials
Introduction Discrete orthogonal polynomials Multivariate discrete orthogonal polynomials The Askey scheme The higher rank Racah algebra Univariate Racah polynomials and Racah algebra Racah polynomials Definition The Racah polynomials are defined as r n ( α, β, γ, δ ; x ) := ( α + 1) n ( β + δ + 1) n ( γ + 1) n � � − n , n + α + β +1 , − x , x + γ + δ +1 × 4 F 3 ; 1 α +1 ,β + δ +1 ,γ +1 K 2 has Racah polynomials as eigenvectors: K 2 r n ( α, β, γ, δ ; x ) = n ( n + α + β + 1) r n ( α, β, γ, δ ; x ) Hendrik De Bie The Racah algebra and multivariate Racah polynomials
Introduction Discrete orthogonal polynomials Multivariate discrete orthogonal polynomials The Askey scheme The higher rank Racah algebra Univariate Racah polynomials and Racah algebra Racah polynomials Definition The Racah polynomials are defined as r n ( α, β, γ, δ ; x ) := ( α + 1) n ( β + δ + 1) n ( γ + 1) n � � − n , n + α + β +1 , − x , x + γ + δ +1 × 4 F 3 ; 1 α +1 ,β + δ +1 ,γ +1 K 2 has Racah polynomials as eigenvectors: K 2 r n ( α, β, γ, δ ; x ) = n ( n + α + β + 1) r n ( α, β, γ, δ ; x ) ◮ algebra simpler than polynomials ◮ can be made even simpler, by taking linear combinations of generators K 1 , K 2 and K 3 Hendrik De Bie The Racah algebra and multivariate Racah polynomials
Introduction Discrete orthogonal polynomials Multivariate discrete orthogonal polynomials The Askey scheme The higher rank Racah algebra Univariate Racah polynomials and Racah algebra The centrally extended Racah algebra Rewrite Racah algebra as: C 123 = C 12 + C 23 + C 13 − C 1 − C 2 − C 3 [ C 12 , C 23 ] =: 2 F [ C 23 , C 13 ] = 2 F [ C 13 , C 12 ] = 2 F [ C 12 , F ] = C 23 C 12 − C 12 C 13 + ( C 2 − C 1 ) ( C 3 − C 123 ) [ C 23 , F ] = C 13 C 23 − C 23 C 12 + ( C 3 − C 2 ) ( C 1 − C 123 ) [ C 13 , F ] = C 12 C 13 − C 13 C 23 + ( C 1 − C 3 ) ( C 2 − C 123 ) with C 1 , C 2 , C 3 and C 123 central elements S. Gao, Y. Wang, and B. Hou. The classification of Leonard triples of Racah type. Linear Algebra and Appl. , 439:1834–1861, jan 2013. V. X. Genest, L. Vinet, and A. Zhedanov. The equitable Racah algebra from three su (1 , 1) algebras. J. Phys. A , 47:025203, 2014. Hendrik De Bie The Racah algebra and multivariate Racah polynomials
Introduction Multivariate discrete orthogonal polynomials The higher rank Racah algebra Outline Introduction Discrete orthogonal polynomials The Askey scheme Univariate Racah polynomials and Racah algebra Multivariate discrete orthogonal polynomials The higher rank Racah algebra Construction Connection with multivariate Racah polynomials Algebraic reinterpretation Hendrik De Bie The Racah algebra and multivariate Racah polynomials
Introduction Multivariate discrete orthogonal polynomials The higher rank Racah algebra Multivariate discrete orthogonal polynomials Two possible generalizations: 1. Macdonald-Koornwinder polynomials related to root systems 2. Tratnik-Gasper-Rahman polynomials we are concerned with type 2 Hendrik De Bie The Racah algebra and multivariate Racah polynomials
Introduction Multivariate discrete orthogonal polynomials The higher rank Racah algebra Tratnik-Gasper-Rahman polynomials Multivariate Racah (or XX) polynomials ◮ are a product of univariate Racah (or XX) polynomials ◮ entangled: variable of first polynomial appears as parameter of subsequent one etc. ◮ explicit formulas cumbersome ◮ discrete orthogonality on subset of R n M.V. Tratnik, Some multivariable orthogonal polynomials of the Askey tableau-discrete families. J. Math. Phys. 32 (1991), 2337–2342. G. Gasper and M. Rahman, Some systems of multivariable orthogonal Askey-Wilson polynomials. In: Theory and applications of special functions, p. 209–219, Dev. Math. 13 , Springer, New York, 2005. G. Gasper and M. Rahman, Some systems of multivariable orthogonal q -Racah polynomials. Ramanujan J. 13 (2007), 389–405. Hendrik De Bie The Racah algebra and multivariate Racah polynomials
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