On the zeros of m-orthogonal polynomials for Freud weights

AbstractThis paper gives the estimates of the distance between two consecutive zeros of the nth m-orthogonal polynomial Pn for a Freud weight W=e−Q as follows. Let {xkn} be the zeros of Pn in decreasing order, an=an(Q) the nth Mhaskar–Rahmanov–Saff number, and ϕn(x)=max{n−2/3,1−|x|/an}. Assume that Q∈C(R) is even, Q(0)=0,Q′∈C[0,∞),Q′(x)>0,x∈(0,∞),Q″∈C(0,∞), and for some A,B>1, A≤(xQ′(x))′Q′(x)≤B,x∈(0,∞). Then, for 1≤k≤n−1, xkn−xk+1,n≤cannϕn(xkn)−1/2 and xkn−xk+1,n≥{cannϕn(xkn)−1/2,m=2,cannϕn(xkn)(m−2)/2,m≥3.Moreover, we have −an<xnn<⋯<x1n<an and for even m, −an[1−c(W,m)n−2/3]≤xnn<⋯<x1n≤an[1−c(W,m)n−2/3].This paper also gives the estimates of Christoffel type functions with odd order and discusses the convergence of Gaussian quadrature formulas for such a weight

Some notes on the zeros of power orthogonal polynomials

AbstractIn this note some estimates of the largest zeros of power orthogonal polynomials are given and some relations involving the largest zeros of orthogonal polynomials, including the Turán inequality, are also established