Event Details
Title: On polynomials satisfying a special $R_{II}$ type recurrence formula
Speaker: Alagacone Sri Ranga, Departamento de Matemática Aplicada, IBILCE, UNESP - Universidade Estadual Paulista, SP, Brazil
Venue: July, 14 (12:00) Seminario 1ª planta (IEMath-Gr)
Abstract: We consider the sequence of polynomials $\{P_n\}_{n\ge0}$ satisfying the recurrence formula
$$P_{n+1}(x) = (x − c_{n+1})P_{n}(x) − d_{n+1}(x^2 + 1)P_{n−1}(x), n ≥ 1,$$
with $P_0(x) = 1$, $P_1(x) = x − c_1$, where $\{c_n\}_{n≥1}$ is a real sequence and $\{d_{n+1}\}_{n≥1}$ is a positive chain sequence. The above recurrence formula can be classified as belongs to the class of recurrence formulas known as $R_{II}$ type recurrence formulas. It turns out that the polynomials $P_n$ are characteristic polynomials associated with certain generalized eigenvalue problems involving two tri-diagonal matrices. Even though the zeros of $P_n$ are simple and lie on the real line, with our $R_{II}$ type recurrence formula one can always associate a positive measure on the unit circle. The orthogonality properties satisfied by the polynomials $P_n$ with respect to this measure is also studied. Examples are given to justify the results.
