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Sobolev orthogonality for the Gegenbauer polynomials \(\{ C_n^{-N+1/2}\}_{n\geq 0}\). (English) Zbl 0931.33008

In this paper the authors show that the Gegenbauer polynomials \(\{C_n^{N+1/2)}\}_{n\geq 0}\) for \(N=1,2,3, \dots\), are orthogonal with respect to the Sobolev inner product defined by the expression: \[ (f,g)_S= \bigl(F(1) \mid F(-1)\bigr)A \bigl(G(1) \mid G(-1)\bigr)^T+ \int^1_{-1}f^{(2N)} (x)g^{(2N)}(x) (1-x^2)^Ndx \tag{*} \] where \((F(1)\mid F(-1))= (f(1)\), \(f'(1), \dots, f^{(N-1)} (1)\), \(f(-1)\), \(f'(-1), \dots, f^{(N-1)}(-1))\), \(A\) stands for a symmetric and positive definite matrix of order \(2N\), and \(f\) and \(g\) are arbitrary polynomials. Using a certain linear operator \({\mathcal F}\) defined on the space of real polynomials, which is symmetric with respect to (*), i.e. \(({\mathcal F}f,g)_S= (f,{\mathcal F}g)_S\), they also establish several relationships between the sequences of Gegenbauer polynomials \(\{C_n^{(-N +1/2)}\}_{n\geq 0}\) and the classical ones \(\{C_n^{(N+ 1/2)}\}_{n\geq 0}\).

MSC:

33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
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