Zeros of the Wronskian of Chebyshev and ultraspherical polynomials
If \(P(z)\) is a polynomial then the Wronskian of \(P\) is defined to be \(WP(z)=P(z)P''(z)-(P'(z))^ 2\). In an earlier paper [J. Math. Anal. Appl. 162, No. 2, 430-451 (1991; Zbl 0748.30007)], the authors showed that if all zeros of \(P(z)\) lie on the interval \([-1,1]\) then all the zeros of \(WP(z)\) lie inside or on the unit circle. Here they study the Wronskian of the Chebyshev polynomials and more generally the ultraspherical polynomials of order \(0<\lambda\leq 1\). For \(P(z)=T_ n(z)\), they show that the zeros of \(WT_ n(z)\) lie inside the ellipse \((x/B_ n)^ 2+(y/A_ n)^ 2=1\), where \(A_ n\sim\log n/n\) and \(B_ n=(1+A^ 2_ n)^{1/2}\), and hence are very close to the real axis for large \(n\). They obtain a similar but somewhat less sharp result in the case \(P(z)=C^ \lambda_ n(z)\).
scientific article; zbMATH DE number 218632