Related papers: An embedding of the Bannai-Ito algebra in $\mathca…
A two-variable generalization of the Big $-1$ Jacobi polynomials is introduced and characterized. These bivariate polynomials are constructed as a coupled product of two univariate Big $-1$ Jacobi polynomials. Their orthogonality measure is…
The Racah problem for the quantum superalgebra $\mathfrak{osp}_{q}(1|2)$ is considered. The intermediate Casimir operators are shown to realize a $q$-deformation of the Bannai-Ito algebra. The Racah coefficients of $\mathfrak{osp}_q(1|2)$…
A quantum superintegrable model with reflections on the three-sphere is presented. Its symmetry algebra is identified with the rank-two Bannai-Ito algebra. It is shown that the Hamiltonian of the system can be constructed from the tensor…
The equitable presentation of the quantum superalgebra $\mathfrak{osp}_q(1|2)$, in which all generators appear on an equal footing, is exhibited. It is observed that in their equitable presentations, the quantum algebras…
We obtain some properties of a class $\mathcal{A}$ of $q$-hypergeometric orthogonal polynomials with $q=-1$, described by a uniform parametrization of the recurrence coefficients. We construct a class $\mathcal{C}$ of complementary $-1$…
The $(-1)$-Jacobi, Bannai-Ito, and $(-1)$-Meixner-Pollaczek polynomials are studied in [Trans. Amer. Math. Soc. 364 (2012), 5491-5507], [Adv. Math. 229 (2012), 2123-2158], and [Stud. Appl. Math. 153 (2024), e12728], respectively, through…
The multi-indexed Jacobi polynomials are the main part of the eigenfunctions of exactly solvable quantum mechanical systems obtained by certain deformations of the P\"oschl-Teller potential (Odake-Sasaki). By fine-tuning the parameter(s) of…
Assume that $\mathbb F$ is an algebraically closed with characteristic $0$. The Bannai--Ito algebra $\mathfrak{BI}$ is a unital associative $\mathbb F$-algebra generated by $X,Y,Z$ and the relations assert that each of \begin{gather*}…
The generating function of the Bannai-Ito polynomials is derived using the fact that these polynomials are known to be essentially the Racah or $6j$ coefficients of the $\mathfrak{osp}(1|2)$ Lie superalgebra. The derivation is carried in a…
We study universal enveloping Hopf algebras of Lie algebras in the category of weakly complete vector spaces over the real and complex field.
The little and big q-Jacobi polynomials are shown to arise as basis vectors for representations of the Askey-Wilson algebra. The operators that these polynomials respectively diagonalize are identified within the Askey-Wilson algebra…
The nonabelian two-dimensional Lie algebra over a field $\mathbb{F}$ has a presentation by generators $A$, $B$ and relation $\left[ A,B\right]=A$, with the universal enveloping algebra having a presentation by generators $A$, $B$ and…
We look for spectral type differential equations satisfied by the generalized Jacobi polynomials which are orthogonal on the interval [-1,1] with respect to a weight function consisting of the classical Jacobi weight function together with…
Sequences of orthogonal polynomials that are alternative to the Jacobi polynomials on the interval $[0,1]$ are defined and their properties are established. An $(\alpha,\beta)$-parameterized system of orthogonal polynomials of the…
We find all spectral type differential equations satisfied by the symmetric generalized ultraspherical polynomials which are orthogonal on the interval [-1,1] with respect to the classical symmetric weight function for the Jacobi…
We classify the subalgebras of the general Lie conformal algebra $\gc_N$ that act irreducibly on $\C[\partial]^N$ and that are normalized by the $\operatorname{sl}_2$--part of a Virasoro element. The problem turns out to be closely related…
Given a finite-dimensional, complex simple Lie algebra we exhibit an integral form for the universal enveloping algebra of its map algebra, and an explicit integral basis for this integral form. We also produce explicit commutation formulas…
Let $n\geq 0$ denote an integer. Let $\mathscr M_n$ denote the space of Dunkl monogenics of degree $n$ associated with the reflection group $\mathbb Z_2^3$. The universal Bannai--Ito algebra $\mathfrak{BI}$ is a unital associative algebra…
We introduce two explicit examples of polynomials orthogonal on the unit circle. Moments and the reflection coefficients are expressed in terms of Jacobi elliptic functions. We find explicit expression for these polynomials in terms of a…
Schur-Delsarte-Genin (SDG) maps and Bannai-Ito polynomials are studied. SDG maps are related to dressing chains determined by quadratic algebras. The Bannai-Ito polynomials and their kernel polynomials -- the complementary Bannai-Ito…