Session III.7 - Special Functions and Orthogonal Polynomials
Poster
Orthogonality of Polynomials Involved in a Linear Combination with Chebyshev Polynomials of the Second Kind
Mirela Vanina de Mello
Universidade Estadual de Santa Cruz - UESC, Brazil - This email address is being protected from spambots. You need JavaScript enabled to view it.
Let $\{U_n\}_{n\geq0}$ and $\{S_n\}_{n\geq0}$ be sequences of polynomials such that $$U_n(x) = S_n(x) + a_{n-1}S_{n-1}(x),$$ $n \geq 1,$ where $\{a_n\}_{n\geq0} \in \mathbb{R}$ and $U_n$ are the orthogonal Chebyshev polynomials of the second kind. Our interest is to find out when $\{S_n\}_{n\geq0}$ is a sequence of orthogonal polynomials.
Marcellán and Petronilho [1] solved this problem by imposing conditions on the coefficients $a_n$. They also obtained a relationship between the linear functionals related to the orthogonal polynomials cited. Using results for recovery the orthogonality measure via Turán determinants [2], we determined both: the sequence of coefficients $a_n$ for which $\{S_n\}_{n\geq0}$ is orthogonal, and not only the linear functional, but also the weight function with respect to which the corresponding polynomials $S_n$ are orthogonal. In other words, the answer to the question posed above was obtained in a completely different and independent way from the approach of Marcellán and Petronilho, with our approach being analytical while the other is entirely algebraic.
Bibliography
[1] F. Marcellán; J. Petronilho, Orthogonal polynomials and coherent pairs: the classical case, Indag. Mathem. 6 (1995), 287-307.
[2] A. Máté; P. Nevai; V. Totik, Asymptotics for orthogonal polynomials defined by a recurrence relation, Constr. Approx. 1 (1985) 231-248.