Une caractérisation des polynômes \(d\)-orthogonaux ''classiques''
The authors study \(d\)-orthogonal sequences of polynomials \((B_n)_n\) using the definition of \textit{J. van Iseghem} [`Approximants de Padé vectoriels', Thèse d'État, Univ. des Sciences et Techniques de Lille-Flandres-Artois (1987)]: \[ \langle {\mathcal L}_k, x^m B_n (x) \rangle=0, \quad n\geq md+k, \;m\geq 0,\qquad \langle {\mathcal L}_k, x^m B_{md+ k+1} (x) \rangle \neq 0, \quad m\geq 0, \] for each integer \(k\), \(1\leq k\leq d\). The \({\mathcal L}_k\) are linear scalar forms on the set of all polynomials, with actions defined from duality: \[ \langle D{\mathcal L}, p(x) \rangle:=- \langle {\mathcal L}, p'(x) \rangle, \qquad \langle q(x) {\mathcal L}, p(x) \rangle:= \langle {\mathcal L}, q(x) p(x) \rangle, \] for all polynomials \(p\), \(q\). This is equivalent to the existence of a \(d+2\)-term recurrence relation of the form \[ B_{m+ d+1} (x)= \biggl( x- \beta_{m+d} B_{m+d} (x)= \sum^{d- 1}_{k= 0} \gamma^{d- 1-k}_{m+ d-k} B_{m+ d-1 -k} (x) \biggr), \qquad m\geq 0, \] with appropriate starting values. Generalizing the situation in the classical case of orthogonal polynomials \((d=1)\), the authors call \((B_n)_n)\) classical \(d\)-orthogonal if the sequence \(Q_n (x):= B'_{n+1} (x)/ (n+1)\) is again \(d\)-orthogonal with respect to the same moment functionals \(\Lambda:= {}^t({\mathcal L}_1, \dots, {\mathcal L}_d)\). The main result is the equivalence between classical \(d\)-orthogonality and the existence of a vectorial functional equation for \(\Lambda\) of the form \(\Psi \Lambda+ D(\Phi \Lambda) =0\), where \(\Psi\), \(\Phi\) are \(d\times d\) matrices with polynomial elements of certain (bounded) degrees. For \(d=1\), the classical results are recovered.
\(d\)-orthogonal sequences of polynomials
scientific article; zbMATH DE number 803689