Deformed Laguerre-Hahn orthogonal polynomials on the real line

Orador: Maria das Neves Rebocho (Departamento de Matemática, UBI). Data, hora e local: 19 de Junho de 2014, início às 15h na sala de reuniões do Departamento de Matemática.

Resumo: A sequence of orthogonal polynomials on the real line (OPRL), say $\{ Pn \}$ , is said to be Laguerre-Hahn if the corresponding Stieltjes function, $S$ , satisfies a Riccati type differential equation with polynomial coeficients

$\displaystyle A(x)S'(x) = B(x)S^2(x) + C(x)S(x) + D(x)$ . (1)

As particular cases, some well-known families of orthogonal polynomials are obtained: the semi-classical OPRL, when $B = 0$ ; the classical OPRL (Hermite, Laguerre, Jacobi), when $B = 0$ and $deg(A) \leq 2$ , $deg(C) = 1$ , $deg(D) = 0$ .
In this talk we focus on the following problem: given a time dependence $t$ on the polynomials $A, B, C, D$ of (1), to describe the deformations of the three-term recurrence relation coefficients of $\{ Pn \}$ . Such deformations are described by nonlinear (difference in $n$ and differential in $t$ ) equations. We deduce discrete Lax equations which lead to difference equations for the corresponding three term recurrence relation coefficients, and we analyze the continuous $t$ -differential equations.

Seminário realizado com o apoio do Centro de Matemática – 212 (Pest-OE/MAT/UI0212/2014).