Recently, $(\beta,\gamma)$-Chebyshev functions, as well as the corresponding zeros, have been introduced as a generalization of classical Chebyshev polynomials of the first kind and related roots. They consist of a family of orthogonal functions on a subset of $[-1,1]$, which indeed satisfies a three-term recurrence formula. In this paper we present further properties, which are proven to comply with various results about classical orthogonal polynomials. In addition, we prove a conjecture concerning the Lebesgue constant's behavior related to the roots of $(\beta,\gamma)$-Chebyshev functions in the corresponding orthogonality interval.

翻译：近期，$(\beta,\gamma)$-Chebyshev函数及其相应的零点被引入，作为经典第一类Chebyshev多项式及相关根的一种推广。它们构成$[-1,1]$子集上一族正交函数，并满足三项递推关系。本文进一步探讨了其性质，证明这些性质与经典正交多项式的若干结论一致。此外，我们证明了关于$(\beta,\gamma)$-Chebyshev函数在对应正交区间内根的勒贝格常数行为的一个猜想。