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This paper is devoted to the study of the location of the zeros of the Sobolev polynomials \(S_n\), where \(\{S_n\}_n\) are the monic orthogonal polynomial sequences with respect to the Sobolev inner product. First the author recalls some well known properties of Gegenbauer polynomials, which are used in the following. He divides the symmetrically coherent pairs of Gegenbauer type in five classes (type \(A,B,C,D\) and \(E)\), investigating their properties in detail. Then the author introduces so-called moments, determining their signs. Establishing the position of zeros, he proves that \(S_n\) has \(n\) different real zeros for type \(A,B,C\) and \(D\), and at least \(n-2\) different real zeros for the type \(E\). It is proved also, that under certain conditions \(S_n\) has complex zeros. The major results of the paper are contained in the ten theorems, the proofs of which are straightforward.
Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.), Sobolev orthogonal polynomials, Gegenbauer polynomials, Mathematics(all), Numerical Analysis, zeros of polynomials, Applied Mathematics, zeros, symmetrically coherent pairs, Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis, Analysis
Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.), Sobolev orthogonal polynomials, Gegenbauer polynomials, Mathematics(all), Numerical Analysis, zeros of polynomials, Applied Mathematics, zeros, symmetrically coherent pairs, Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis, Analysis
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