Monotonicity results on the zeros of generalized Laguerre polynomials

AbstractIt is shown that, for x > −1 and n = 1, 2,…, the sequence (n + (α + 1)2) xnk − 14xnk2 increases with n, where xnk = xnk(α) denotes the kth zero of the generalized Laguerre polynomial, in increasing order. As a consequence of this result, the inequality xnkxn + m,k + 4 < xn,k + 1xn + m,k, m = 1, 2,…, 1 ⩽ k < k + 1 ⩽ n, is established. Similar results are proved for the zeros of Hermite polynomials. The principal tool used is Sturm's comparison theorem in a variation due to Szegö