Monotonicity and asymptotics of zeros of Sobolev type orthogonal polynomials: A general case

We investigate the location, monotonicity, and asymptotics of the zeros of the polynomials orthogonal with respect to the Sobolev type inner product〈p,q〉λ,c,j=∫abp(x)q(x)dμ(x)+λp(j)(c)q(j)(c), where μ is a positive Borel measure, λ⩾0, j∈Z+, and c∉(a,b). We prove that these zeros are monotonic function of the parameter λ and establish their asymptotics when either λ converges to zero or to infinity. The precise location of the extreme zeros is also analyzed.