Structure relations for orthogonal polynomials

Abstract

We show how to derive structure relations for general orthogonal polynomials, that is, we find operators whose action on p n is a combination of p n and p n+1 with variable coefficients. We also provide an analogue of the string equation for general orthogonal polynomials. We explore the connection with the Toda lattice and polynomials orthogonal with respect to general exponential weights.

Dunkl shift operators and Bannai–Ito polynomials

“…The main tool will be the Bannai-Ito algebra (BI-algebra for brevity) which is a special case of the AW (3) algebra introduced in [35]. For the Askey-Wilson polynomials the structure relations could be derived in a similar manner using representations of the AW(3) algebra [18], [15]. For a study of the structure relations of the -1 Jacobi polynomials see [28].…”

Phase transitions in the one-dimensional coulomb medium

“…Already J. J. Thomson (discovering electron in 1897) suggested the problem of finding such configurations on the sphere, and the answer has been known for N = 2, 3, 4 for more than 100 years, but for N = 5 the solution was obtained only quite recently [4]. In one-dimensional case T, J, Stieltjss studied the problem with logarithmic interaction and found its connection with zeros of orthogonal polynomials on the corresponding interval, see [2], [3]. However, the problem of finding minimal energy configurations on two-dimensional sphere for any N and power interaction (sometimes it is called the seventh problem of S. Smale, it is also connected with the names of F. Risz and M. Fekete) was completely solved only for quadratic interaction (see [5], [7], [8] and review [6]).…”

One-Dimensional Coulomb Multiparticle Systems

“…More interesting is the case of large N, where the asymptotics N → ∞ is of main interest. In one-dimensional case T. J. Stieltjes studied the problem with logarithmic interaction and found its connection with zeros of orthogonal polynomials on the corresponding interval, see [9], [10]. However, the problem of finding minimal energy configurations on two-dimensional sphere for any N and power interaction (sometimes it is called the seventh problem of S. Smale, it is also connected with the names of F. Risz and M. Fekete) was completely solved only for quadratic interaction (see [12], [14], [15] and review [13]).…”