## 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.)
Full Text:

### References:

 [1] M. Alfaro, M.L. Rezola, T.E. Pérez, M.A. Piñar, Sobolev orthogonal polynomials: the discrete-continuous case, submitted.; M. Alfaro, M.L. Rezola, T.E. Pérez, M.A. Piñar, Sobolev orthogonal polynomials: the discrete-continuous case, submitted. · Zbl 0980.42017 [2] Chihara, T. S., An Introduction to Orthogonal Polynomials (1978), Gordon and Breach: Gordon and Breach New York · Zbl 0389.33008 [3] Kwon, K. H.; Littlejohn, L. L., The Orthogonality of the Laguerre polynomials {$$L_n^{(−k)$$}(x)\) · Zbl 0831.33003 [4] K.H. Kwon, L.L. Littlejohn, Sobolev orthogonal polynomials and second-order differential equations, Rocky Mt. J. Math., to appear.; K.H. Kwon, L.L. Littlejohn, Sobolev orthogonal polynomials and second-order differential equations, Rocky Mt. J. Math., to appear. · Zbl 0930.33004 [5] Pérez, T. E.; Piñar, M. A., On Sobolev orthogonality for the generalized Laguerre polynomials, J. Approx. Theory, 86, 278-285 (1996) · Zbl 0864.33009 [6] Pérez, T. E.; Piñar, M. A., Sobolev orthogonality and properties of the generalized Laguerre polynomials, (Jones, W. B.; Sri Ranga, A., Orthogonal Functions. Orthogonal Functions, Moment Theory and Continued Fractions: Theory and Applications, 18 (1998), Marcel Dekker: Marcel Dekker New York), 375-385 · Zbl 0930.33006 [7] Szegö, G., Orthogonal polynomials, (Amer. Math. Soc. Colloq. Publ., 23 (1975), Amer. Math. Soc: Amer. Math. Soc Providence, RI) · JFM 65.0278.03
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.