Monotonicity properties of the zeros of ultraspherical polynomials
Let \(x_{n,k}^{(\lambda)}\), \(k=1,2,\dots,[n,2]\), denote the \(k\)-th positive zero in increasing order of the ultraspherical polynomial \(P_n^{(\lambda)}(x)\). The authors prove that the function \([\lambda+(2n^2+1)/(4n+2)]^{1/2}x_{n,k}^{(\lambda)}\) is increasing for \(\lambda>-1/2\). This proves the Ismail-Letessier-Askey conjecture [\textit{M. E. H. Ismail}, in \(q\)-series and partitions, Proc. Workshop, Minneapolis/MN(USA) 1988, IMA Vol. MAth. Appl. 18, 177-190 (1989; Zbl 0698.33008)], the function \((\lambda+1)^{1/2}x_{n,k}(\lambda)\) increases as \(\lambda\) increases for \(\lambda>-1/2\).
scientific article; zbMATH DE number 1286386
scientific article