On polynomials associated with an Uvarov modification of a quartic potential Freud-like weight

Abstract

In this contribution we consider sequences of monic polynomials orthogonal with respect to the standard Freud-like inner product involving a quartic potentialWe analyze some properties of these polynomials, such as the ladder operators and the holonomic equation that they satisfy and, as an application, we give an electrostatic interpretation of their zero distribution in terms of a logarithmic potential interaction under the action of an external field. It is also shown that the coefficients of their three term recurrence relation satisfy a nonlinear difference string equation. Finally, an equation of motion for their zeros in terms of their dependence on t is given.

A Generalized Freud Weight

“…with c 1 , c 2 and c 3 constants. Note that making the transformation w(z) = ϕ(t), with z = ( 1 3 t) 3 , in (3.4) gives…”

“…Several sequences of monic orthogonal polynomials related to the weight (1.5) have been studied in the literature. For instance, for t = 0, λ = − 1 2 , the asymptotic and analytic properties of the corresponding orthogonal polynomials were studied in [49], while the case when t > 0 and λ = − 1 2 is discussed in [3]. The recurrence coefficients in the three-term recurrence relations associated with semiclassical orthogonal polynomials can often be expressed in terms of solutions of the Painlevé equations and associated discrete Painlevé equations.…”

On Freud–Sobolev type orthogonal polynomials

“…Similar polynomials have been previously studied in [20], when the discrete mass points are located outside the support of the perturbed measure. Here, we find a slightly different situation because the support of the measure is the whole real line and the discrete masses M 0 and M 1 are both located at x = 0 ∈ R. As stated before, M 0 only affects the even degree polynomials, and the dynamics for the zeros of {Q 2n (x)} n≥0 has been already obtained in [3]. Next, we extend those results for the odd sequence {Q 2n+1 (x)} n≥0 .…”

“…In the sequel, we provide the expressions for the odd case (κ [0] n = 0, κ [1] 2n = 0, r 2n+1 = 1, r 2n = 0), together with an electrostatic interpretation of the zeros of {Q n (x)} n≥0 . The even case was analyzed in [3]. We found R(x; 2n + 1) = 2 x − 4x 3 − u ′ (x; 2n + 1) u(x; 2n + 1) , where u(x; 2n + 1) is the biquartic polynomial u(x; 2n + 1) = u 4 (n) x 4 + u 2 (n) x 2 + u 0 (n)…”

“…[σ(x)ω(x)] ′ = τ (x)ω(x), where σ(x) = 1 and τ (x) = −4x 3 . Notice that, in terms of the weight function, the above relation means that…”