Satoru Odake, Ryu Sasaki
The exceptional Racah and q-Racah polynomials are constructed. Together with the exceptional Laguerre, Jacobi, Wilson and Askey-Wilson polynomials discovered by the present authors in 2009, they exhaust the generic exceptional orthogonal polynomials of a single variable.
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Upon solving a finite discrete reduction of the difference Heun equation, we arrive at an elliptic generalization of the Racah polynomials. We exhibit the three-term recurrence relation and the orthogonality relations for these elliptic Racah polynomials. The well-known $q$-Racah polynomials of Askey and Wilson are recovered as a trigonometric limit.
We construct the orthogonal eigenbasis for a discrete elliptic Ruijsenaars type quantum particle Hamiltonian with hyperoctahedral symmetry. In the trigonometric limit the eigenfunctions in question recover a previously studied $q$-Racah type reduction of the Koornwinder-Macdonald polynomials. When the inter-particle interaction degenerates to that of impenetrable bosons, the orthogonal eigenbasis simplifies in terms of generalized Schur polynomials on the spectrum associated with recently found elliptic Racah polynomials.
We present various results on the properties of the four infinite sets of the exceptional $X_{\ell}$ polynomials discovered recently by Odake and Sasaki [{\it Phys. Lett. B} {\bf 679} (2009), 414-417; {\it Phys. Lett. B} {\bf 684} (2010), 173-176]. These $X_{\ell}$ polynomials are global solutions of second order Fuchsian differential equations with $\ell+3$ regular singularities and their confluent limits. We derive equivalent but much simpler looking forms of the $X_{\ell}$ polynomials. The other subjects discussed in detail are: factorisation of the Fuchsian differential operators, shape in…
We present a new set of infinitely many shape invariant potentials and the corresponding exceptional (X_{\ell}) Laguerre polynomials. They are to supplement the recently derived two sets of infinitely many shape invariant thus exactly solvable potentials in one dimensional quantum mechanics and the corresponding X_{\ell} Laguerre and Jacobi polynomials (Odake and Sasaki, Phys. Lett. B679 (2009) 414-417). The new X_{\ell} Laguerre polynomials and the potentials are obtained by a simple limiting procedure from the known X_{\ell} Jacobi polynomials and the potentials, whereas the known X_{\ell} L…
Representations of the $s\ell_q(2)$ algebra are constructed in the space of polynomials of real (complex) variable for $q^N=1$. The spin addition rule based on eigenvalues of Casimir operator is illustrated on few simplest cases and conjecture for general case is formulated.
Two sets of infinitely many exceptional orthogonal polynomials related to the Wilson and Askey-Wilson polynomials are presented. They are derived as the eigenfunctions of shape invariant and thus exactly solvable quantum mechanical Hamiltonians, which are deformations of those for the Wilson and Askey-Wilson polynomials in terms of a degree \ell (\ell=1,2,...) eigenpolynomial. These polynomials are exceptional in the sense that they start from degree \ell\ge1 and thus not constrained by any generalisation of Bochner's theorem.
An interesting discovery in the last two years in the field of mathematical physics has been the exceptional $X_\ell$ Laguerre and Jacobi polynomials. Unlike the well-known classical orthogonal polynomials which start with constant terms, these new polynomials have the lowest degree $\ell=1,2,...$, and yet they form complete sets with respect to some positive-definite measure. In this paper, we study one important aspect of these new polynomials, namely, the behaviors of their zeros as some parameters of the Hamiltonians change.
Recently one of the authors obtained a classification of simple infinite-dimensional Lie superalgebras of vector fields which extends the well-known classification of E. Cartan in the Lie algebra case. The list consists of many series defined by simple equations, and of several exceptional superalgebras, among them E(3,6). In the article we study irreducible representations of the exceptional Lie superalgebra E(3,6). This superalgebra has s\ell(3)\times s\ell(2)\times g\ell(1) as the zero degree component of its consistent \ZZ-grading which leads us to believe that its representation theory ha…
We present a new family of shape invariant potentials which could be called a ``continuous \ell version" of the potentials corresponding to the exceptional (X_{\ell}) J1 Jacobi polynomials constructed recently by the present authors. In a certain limit, it reduces to a continuous \ell family of shape invariant potentials related to the exceptional (X_{\ell}) L1 Laguerre polynomials. The latter was known as one example of the `conditionally exactly solvable potentials' on a half line.
An interesting discovery in the last two years in the field of mathematical physics has been the exceptional $X_\ell$ Laguerre and Jacobi polynomials. Unlike the well-known classical orthogonal polynomials which start with constant terms, these new polynomials have lowest degree $\ell=1,2,...$, and yet they form complete set with respect to some positive-definite measure. While the mathematical properties of these new $X_\ell$ polynomials deserve further analysis, it is also of interest to see if they play any role in physical systems. In this paper we indicate some physical models in which th…
It has been recently discovered that exceptional families of Sturm-Liouville orthogonal polynomials exist, that generalize in some sense the classical polynomials of Hermite, Laguerre and Jacobi. In this paper we show how new families of exceptional orthogonal polynomials can be constructed by means of multiple-step algebraic Darboux transformations. The construction is illustrated with an example of a 2-step Darboux transformation of the classical Laguerre polynomials, which gives rise to a new orthogonal polynomial system indexed by two integer parameters. For particular values of these para…
The s ell_q(2) representations are realized in the space of polynomials for general and exceptional values of deformation parameter q and on finite set of theta-functions for cyclic representation corresponding to q^N = +/- 1, which are a natural extension of the polynomials. The complete set of eigenstates of the Universal R-matrix are constructed and corresponding eigenvalues are calculated.
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