Zeros of Sobolev Orthogonal Polynomials of Gegenbauer Type

AbstractLet {Sn}n denote the monic orthogonal polynomial sequence with respect to the Sobolev inner product〈f, g〉S=∫f(x)g(x)dψ0 (x)+λ∫f(x)g(x)dψ1 (x), where λ>0 and {dψ0, dψ1} is a so-called symmetrically coherent pair, with dψ0 or dψ1 the classical Gegenbauer measure (x2−1)αdx, α>−1. If dψ1 is the Gegenbauer measure, then Sn has n different, real zeros. If dψ0 is the Gegenbauer measure, then Sn has at least n−2 different, real zeros. Under certain conditions Sn has complex zeros. Also the location of the zeros of Sn with respect to Gegenbauer polynomials, is studied