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2-orthogonal polynomials and Darboux transformations. Applications to the discrete Hahn-classical case. (English) Zbl 1473.33005

In [P. Maroni, Ann. Fac. Sci. Toulouse, V Ser., Math. 10, No. 1, 105–139 (1989; Zbl 0707.42019)] \(d\)-orthogonal polynomials were introduced which satisfy recurrence relations of order \(d+1\) and which obey orthogonality w.r.t. \(d\) linear functionals. In this paper the authors solve the problem of existence of the sequence of the monic 2-orthogonal polynomials \(Q_n\) satisfying the relation \(P_n(x)=Q_n(x)+a_nQ_{n-1}(x),\,n\geq 0,\) where \(a_0=0,\,a_n\neq 0\) for \(n>0.\) The related condition can be written as a linear difference equation with respect to \(\{b_n\}\) such that \(a_n=\frac{b_n}{b_{n+1}}, b_1=1.\) The solution of this difference equation is given in a recurrence form.
The following particular cases are studied in more detail: 1) \(\{P_n\}\) is discrete Hahn-classical; 2) \(\{P_n\}\) is threefold symmetric; 3) the sequence \(\{a_n\}_{n\geq 1}\) is constant. Also some examples of discrete Hahn-classical 2-orthogonal families are given as an illustration.

MSC:

33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
39A70 Difference operators

Citations:

Zbl 0707.42019
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References:

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