Abstract
This paper deals with the general theory of sets of polynomials verifying a (d+1)-order recurrence. The cased=2 is specially carried out. First, we introduce the notion of associated set of a set of monic polynomials. A general formula for successive associated polynomials is given. The co-recursive sets of a two-dimensional orthogonal set are introduced. We calculate the corresponding formal Stieltjes functions.
Finally, we determine the self-associated two-dimensional orthogonal sets and we show they are “classical” two-dimensional orthogonal sets, that is to say, their set of derivatives is also a two-dimensional orthogonal set.
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References
A. Boukhemis, Etude de quelques familles particulières de polynômes demi-orthogonaux, Thèse de l'Univ. P. et M. Curie, Paris (1985)
A. Boukhemis and P. Maroni, Une caractérisation des polynômes strictement 1/p orthogonaux de type Scheffer. Etude du casp=2, J. Approx. Theory 54 (1988) 67–91.
J. Devisme, Sur l'équation de M. Pierre Humbert, Ann. Fac. Sci. Toulouse 25 (1933) 143–238.
J. Dini, Sur les formes linéaires et les polynômes orthogonaux de Laguerre-Hahn, Thèse de l'Univ. P. et M. Curie, Paris (1988).
J. Dini and P. Maroni, La multiplication d'une forme linéaire par une fraction rationnelle. Application aux formes de Laguerre-Hahn, Ann. Polon. Math. 52 (1990) 175–185.
K. Douak, Les polynômes orthogonaux “classiques” de dimensiond, Thèse de troisième cycle, Univ. P. et M. Curie, Paris (1988).
K. Douak and P. Maroni, Les polynômes orthogonaux “classiques” de dimension deux, Analysis 12 (1992) 71–107.
P. Maroni, Une généralisation du théorème de Favart-Shohat sur les polynômes orthogonaux, C.R. Acad. Sci. Paris 293 I (1981) 19–22.
P. Maroni, Le calcul des formes linéaires et les polynômes orthogonaux semi-classiques, in:Orthogonal Polynomials and Their Applications, Lecture Notes in Mathematics, 1329, eds. M. Alfaro et al. (Springer, Berlin, 1988), pp. 279–290.
P. Maroni, L'orthogonalité et les récurrences de polynômes d'ordre supérieur à deux, Ann. Fac. Sci. Toulouse 10 (1989) 105–139.
P. Maroni, L'orthogonalité de dimensiond. Certains polynômes orthogonaux “classiques” de dimension deux, in:Orthogonal Polynomials and Their Applications, Lecture Notes in Pure and Applied Mathematics, 117, ed. J. Vinuesa (M. Dekker, New York, 1989) pp. 73–93.
P. Maroni, Une théorie algébrique des polynômes orthogonaux. Application aux polynômes orthogonaux semi-classiques in:Orthogonal Polynomials and Their Applications, IMACS 9, eds. C. Brezinski et al. (J.C. Baltzer AG, 1991), pp. 95–130.
S.M. Roman and G.C. Rota, The umbral calculus, Adv. Math. 27 (1978) 95–188.
J. Sherman, On the numerators of the convergents of the Stieltjes continued fraction, Trans. Amer. Math. Soc. 35 (1933) 64–87.
W. Van Assche, Orthogonal polynomials, associated polynomials and functions of the second kind, J. Comp. Appl. Math. 37 (1991) 237–249.
J. Van Iseghem, Approximants de Padé vectoriels, Thèse de l'Univ. des Sci. et Tech. Lille-Flandre-Artois (1987).
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Maroni, P. Two-dimensional orthogonal polynomials, their associated sets and the co-recursive sets. Numer Algor 3, 299–311 (1992). https://doi.org/10.1007/BF02141938
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DOI: https://doi.org/10.1007/BF02141938