Orthogonal functions related to Lax pairs in Lie algebras

Abstract

We study a Lax pair in a 2-parameter Lie algebra in various representations. The overlap coefficients of the eigenfunctions of L and the standard basis are given in terms of orthogonal polynomials and orthogonal functions. Eigenfunctions for the operator L for a Lax pair for $$\mathfrak {sl}(d+1,\mathbb {C})$$ sl ( d + 1 , C ) is studied in certain representations.

Gaudin model for the multinomial distribution

“…This approach provided yet another proof of the orthogonality and established their bispectral properties by showing that the polynomials are eigenfunctions of commuting partial difference operators parametrized by Cartan subalgebras of sl d+1 . Connections to the Racah algebra, numerous probabilistic applications, Lax pairs and the analysis of the reproducing kernel of the multinomial distribution can be found in the recent works [6,8,18,32]. We review below the construction of these polynomials together with the commutative algebras of partial difference operators diagonalized by them which will be needed later following the approach in [19].…”