## Determination of all coherent pairs.(English)Zbl 0880.42012

In the study of Sobolev orthogonal polynomials, corresponding to the inner product $\langle f, g \rangle_S=\int_a^b f g d\Psi_0 + \lambda \int_a^b f' g' d\Psi_1 ,$ where $$\Psi_0$$ and $$\Psi_1$$ are distribution functions and $$\lambda \geq 0$$, an essential role has been played by the concept of coherence. If $$\{P_n\}$$ and $$\{T_n\}$$ are monic orthogonal polynomial sequences corresponding to $$\Psi_0$$ and $$\Psi_1$$, respectively, then $$(\Psi_0,\Psi_1)$$ is a coherent pair if $T_n = \frac{P_{n+1}'}{n+1} - \sigma_n \frac{P_{n}'}{n} , \quad \sigma_n=\text{const. } \neq 0 . \tag{*}$ This concept can be extended to quasi-definite linear functionals $$u_0$$ and $$u_1$$. When $$u_0$$ and $$u_1$$ are coherent, the Sobolev polynomial sequence has a structure suitable for further study. Nevertheless, it is important to know all the coherent pairs precisely, and this is the aim of this paper. The author proves that if $$(u_0, u_1)$$ is a coherent pair, then at least one of the functionals is classical (Hermite, Laguerre, Jacobi or Bessel). Moreover, all the coherent pairs are given explicitly. Analogous results are established for symmetric coherent pairs, when the subindices involved in the r.h.s.of (*) are $$n+1$$ and $$n-1$$.

### 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:

 [1] Abramowitz, M.; Stegun, I. A., Handbook of Mathematical Functions (1970), Dover: Dover New York · Zbl 0515.33001 [2] de Bruin, M. G.; Meijer, H. G., Zeros of orthogonal polynomials in a non-discrete Sobolev space, Ann. Numer. Math., 2, 233-246 (1995) · Zbl 0833.33009 [3] Chihara, T. S., An Introduction to Orthogonal Polynomials (1978), Gordon and Breach: Gordon and Breach New York · Zbl 0389.33008 [4] Iserles, A.; Koch, P. E.; Nørsett, S. P.; Sanz-Serna, J. M., On polynomials orthogonal with respect to certain Sobolev inner products, J. Approx. Theory, 65, 151-175 (1991) · Zbl 0734.42016 [5] Marcellán, F.; Alfaro, M.; Rezola, M. L., Orthogonal polynomials on Sobolev spaces: Old and new directions, J. Comput. Appl. Math., 48, 113-131 (1993) · Zbl 0790.42015 [6] Marcellán, F.; Pérez, T. E.; Piñar, M. A., Orthogonal polynomials on weighted Sobolev spaces: The semiclassical case, Ann. Numer. Math., 2, 93-122 (1995) · Zbl 0835.33005 [7] Marcellán, F.; Petronilho, J. C., Orthogonal polynomials and coherent pairs: The classical case, Indag. Math. N.S., 6, 287-307 (1995) · Zbl 0843.42010 [8] Marcellán, F.; Petronilho, J. C.; Pérez, T. E.; Piñar, M. A., What is beyond coherent pairs of orthogonal polynomials?, J. Comput. Appl. Math., 65, 267-277 (1995) · Zbl 0855.42016 [9] Maroni, P., Une théorie algébrique des polynômes orthogonaux. Application aux polynômes orthogonaux semi-classiques, (Brezinski, C.; Gori, L.; Ronveaux, A., Orthogonal Polynomials and Their Applications (1991), BaltzerIMACS Annals on Comp. and Appl. Math: BaltzerIMACS Annals on Comp. and Appl. Math Basel), 95-130 · Zbl 0944.33500 [10] Meijer, H. G., Coherent pairs and zeros of Sobolev-type orthogonal polynomials, Indag. Math. N.S., 4, 163-176 (1993) · Zbl 0784.33004 [11] Meijer, H. G., A short history of orthogonal polynomials in a Sobolev space, I. The non-discrete case, Nieuw Arch. Wisk. (4), 14, 93-112 (1996) · Zbl 0862.33001 [12] Szegö, G., Orthogonal Polynomials. Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publ., 23 (1975), Amer. Math. Soc., Providence: Amer. Math. Soc., Providence RI
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