Monotonicity results on the zeros of generalized Laguerre polynomials
Let \(L_ n^{(\alpha)}(x)\) be the generalized Laguerre polynomial. The authors prove two theorems on the zeros of this polynomial. We state: Theorem 1. For \(-1<\alpha \leq 1\) and \(k=1,2,...,n\), let \(x_{nk}=x_{nk}(\alpha)\) be the kth positive zero of the generalized Laguerre polynomial \(L_ n^{(\alpha)}(x)\) in increasing order. Then \((n+(\alpha +1)/2)x_{nk}+(1/4)x^ 2_{nk}\) increases with n.
scientific article; zbMATH DE number 4052015