Abstract :
Let the polynomialsPn(x),n⩾1, be defned byP0(x)=0,P1(x)=1,anPn+1(x)+an−1Pn−1(x)+bnPn(x)=xPn(x),n⩾1. Ifan>0 andbnare real then there exists at least one measure of orthogonality for the polynomialsPn(x),n=1, 2, ... . The problem of finding conditions on the sequencesanandbnunder which this measure is unique or nonunique still remains open for large classes of sequencesanandbn. Here a new criterion for the nonuniqueness of the measure of orthogonality is proved. This was achieived by proving that the infinite-dimensional Jacobi matrix associated with the sequencesanandbnis not self-adjoint.
