Coherent pairs of measures of the second kind on the real line and Sobolev orthogonal polynomials. An application to a Jacobi case.

The aim here is to consider the orthogonal polynomials Sn(ν0,ν1,s;x)$\mathcal {S}_{n}(\nu _0, \nu _1,s; x)$ with respect to an inner product of the type ⟨f,g⟩s=∫f(x)g(x)dν0(x)+s∫f′(x)g′(x)dν1(x)$ \langle f, g \rangle _{\mathfrak{s}} = \int f(x)g(x) d \nu _0(x) + s \int f^{\prime }(x) g^{\prime }(x) d \nu _1(x)$, where s>0$s > 0$ and {ν0,ν1}$\lbrace \nu _0, \nu _1\rbrace$ is a coherent pair of positive measures of the second kind on the real line (CPPM2K on the real line). Properties of Sn(ν0,ν1,s;x)$\mathcal {S}_{n}(\nu _0, \nu _1, s; x)$ and the connection formulas they satisfy with the orthogonal polynomials associated with the measure ν0 are analyzed. It is also shown that the zeros of Sn(ν0,ν1,s;x)$\mathcal {S}_{n}(\nu _0, \nu _1, s; x)$ are the eigenvalues of a matrix, which is a single line modification of the n×n$n \times n$ Jacobi matrix associated with the measure ν0. The paper also looks at a special example of a CPPM2K on the real line, where one of the measures is the Jacobi measure, and provides a much more detailed study of the properties of the orthogonal polynomials and the corresponding connection coefficients. In particular, the relation that these connection coefficients have with the Wilson polynomials is exposed. [ABSTRACT FROM AUTHOR]