Some Modifications of the Chebyshev Measures and the Corresponding Orthogonal Polynomials

Gradimir V. Milovanović ; Nevena M. Vasović

Bulletin de l'Académie serbe des sciences. Classe des sciences mathématiques et naturelles, Tome 47 (2022) no. 1.

Résumé

A few rational modifications of the Chebyshev measures of the first and second kind and the corresponding orthogonal polynomials on the finite interval $[-1,1]$ are studied, included also a convex combination of these two Chebyshev measures. Also, an non-rational modification of the Chebyshev measure of the second kind, i.e., ${\D}\sigma ^{n,s}(x)=|\widehat U_{n}(x)|^{2s}(1-x^2)^{s+1/2}\D x$ on $[-1,1]$, for $n\in\NN$ and a real number $s>-1/2$, is studied, as well as certain properties of the corresponding orthogonal polynomials, including explicit expressions for the coefficients in their three-term recurrence relation.