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Monotonicity and asymptotics of zeros of Sobolev type orthogonal polynomials: A general case. (English) Zbl 1256.33005

Summary: We investigate the location, monotonicity, and asymptotics of the zeros of the polynomials orthogonal with respect to the Sobolev-type inner product \[ \langle p, q\rangle _{\lambda ,c,j} = \int^b_a p(x)q(x)d \mu(x)+ \lambda p^{(j)} (c)q^{(j)}(c), \] where \(\mu\) is a positive Borel measure, \(\lambda \geqslant 0\), \(j \in \mathbb {Z}_+\) and \(c \notin (a,b)\). We prove that these zeros are monotonic function of the parameter \(\lambda\) and establish their asymptotics when either \(\lambda\) converges to zero or to infinity. The precise location of the extreme zeros is also analyzed.

MSC:

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
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