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Orthogonality with respect to a Jacobi differential operator and applications. (English) Zbl 1304.42062

Summary: Let \(\mu\) be a finite positive Borel measure on \([-1,1]\), \(m\) a fixed natural number and \(\mathcal L^{(\alpha,\beta)}[f]=(1-x^2)f'''+(\beta-\alpha-(\alpha+\beta+2)x)f'\), with \(\alpha,\beta>-1\). We study algebraic and analytic properties of the sequence of monic polynomials \((Q_n)_{n>m}\) that satisfy the orthogonality relations \[ \int\mathcal L^{(\alpha,\beta)}[Q_n](x)x^kd\mu(x)=0\quad\text{for all }0\leq k\leq n-1. \] A fluid dynamics model for source points location of a flow of an incompressible fluid with preassigned stagnation points is also considered.

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.)
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References:

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