On the zeros of mm-orthogonal polynomials for Freud weights

This paper gives the estimates of the distance between two consecutive zeros of the nnth mm-orthogonal polynomial PnPn for a Freud weight W=e−Q as follows. Let {xkn}{xkn} be the zeros of PnPn in decreasing order, an=an(Q)an=an(Q) the nnth Mhaskar–Rahmanov–Saff number, and ϕn(x)=max{n−2/3,1−|x|/an}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,∞)Q(0)=0,Q′∈C[0,∞),Q′(x)>0,x∈(0,∞),Q″∈C(0,∞), and for some A,B>1A,B>1, A≤(xQ′(x))′Q′(x)≤B,x∈(0,∞). Then, for 1≤k≤n−11≤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