A positive definite linear functional of class $$s=2$$, generalization of Chebyshev polynomials

Abstract

In the present work we deal with the quadratic decomposition of symmetric semiclassical polynomial sequences of class 2 orthogonal with respect to the positive definite weight $$ | x^2-\frac{1}{2} |^p(1-x^2)^{-\frac{1}{2}}$$, $$ p > -1$$, on $$[-1,1]$$. The coefficients of the three-term recurrence relation, the structure relation, the differential equation as well as some information about the zeros of the corresponding orthogonal polynomials are given. These results reduce to the Chebyshev case for $$p=0$$.