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 = ∫ a b p ( x ) q ( x ) d μ ( x ) + λ p ( 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.