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On zeros of Sobolev-type orthogonal polynomials. (English) Zbl 0768.33008
Some properties of the zeros of orthogonal polynomials \(Q_ n\) corresponding with the inner product \[ \langle f,g\rangle=\int_ I f(x)g(x)d\mu(x)+\lambda f'(c)g'(c) \] are analyzed. In particular various interlacing properties with the zeros of the orthogonal polynomials associated with the measure \(\mu\) and corresponding kernel polynomials are given and for \(c\) to the right of the interval \(I\) it is shown that the largest zero of \(Q_ n\) is to the right of \(c\) for \(n\) large enough. Furthermore it is shown that the zeros of \(Q_ n\) are an increasing function of \(\lambda\).

33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis