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Sobolev-Gegenbauer-type orthogonality and a hydrodynamical interpretation. (English) Zbl 1231.42024

The paper is devoted to study orthogonality with respect to a discrete-continuous Sobolev inner product defined in the space of algebraic polynomials as follows: \[ \langle f, g \rangle_{\lambda,\zeta}= \eta f(\zeta) g(\zeta)+ \int_{-1}^1 f'(x)g'(x) d \mu_{\lambda}(x), \] where \(\eta >0, \zeta\) is a fixed complex point and \(d \mu_{\lambda}(x)=(1-x^2)^{\lambda-\frac{1}{2}}dx\), with \(\lambda>- \frac{1}{2}\), is the classical Gegenbauer or ultraspherical measure.
Several properties of the corresponding monic orthogonal polynomials are obtained such as the recurrence relation that they satisfy and the relative asymptotic with respect to the Gegenbauer polynomials and their derivatives.
Besides, the location of the zeros of the orthogonal discrete-continuous Sobolev polynomials is obtained and a hydrodynamical interpretation like the location of source points and its corresponding strength is also given.

MSC:

42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
33C47 Other special orthogonal polynomials and functions
76B99 Incompressible inviscid fluids
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