Some monotonicity properties of the zeros of ultraspherical polynomials
Using the Sturm comparison theorem, the authors prove results concerning the signs of the determinants \[ x_{ni}(\lambda)x_{m,j+\ell}(\lambda)- x_{mj}(\lambda)x_{n,i+\ell}(\lambda) \] and \[ \theta_{ni}(\lambda)\theta_{nk}(\lambda ')- \theta_{nk}(\lambda)\theta_{ni}(\lambda '), \] on the monotonicity in n of \(x_{nk}(\lambda)/x_{ni}(\lambda)\) and on the monotonicity in \(\ell\) of \(x_{n,i+\ell}(\lambda)/x_{n,j+\ell}(\lambda)\), where \(x_{n,k}(\lambda)\) is the k-th positive zero of the ultraspherical (or Gegenbauer) polynomial \(P_ n^{(\lambda)}(x)\), and \(\theta_{nk}(\lambda)=\arccos x_{nk}(\lambda)\).
scientific article; zbMATH DE number 3987687