A study of a sequence of classical orthogonal polynomials of dimension 2
The notion of `\(d\)-orthogonality' and the connection with the length of the recurrence relation for the sequence of `orthogonal polynomials of dimension \(d\)' has been introduced by \textit{J. van Iseghem} [Approximants de Padé vectoriels, Thèse d'état, Univ. des Sciences et Techniques de Lille-Flandres-Artois (1987)] in the context of vector Padé approximation to \(d\) formal power series. The result given there ``a sequence of polynomials is orthogonal of dimension \(d\) if and only if it satisfies a recurrence relation of order \(d+1\)'' is used by the author to generalize even further. Starting with a sequence of monic orthogonal polynomials satisfying the recurrence relation \[ B_{n+d+1}(x)=xB_{n+d}(x)-\sum_{k=1}^{d+1} \gamma_kB_{n+d+1-k}(x),\;n\geq 0, \] with initial values \[ B_0(x)=1;\;B_j(x)=xB_{j-1}(x)-\sum_{k=1}^j \gamma_kB_{j-k}(x),\;1\leq j\leq d, \] the generating function \[ G(x,t)=\sum_{n=0}^{\infty} B_n(x)t^n=\left(1-xt+\sum_{k=1}^{d+1} \gamma_k t^k\right)^{-1} \] is used to introduce a new sequence of polynomials for \(\alpha\not= -1,-2,\ldots\) by \[ G_{\alpha}(x,t)=\left(1-xt+\sum_{k=1}^{d+1} \gamma_k t^k\right)^{-\alpha} = \sum_{n=0}^{\infty} B_n^{\alpha}(x)t^n. \] With the aid of relations between \(\partial G_{\alpha}/\partial t,\;\partial G_{\alpha}/\partial x\) and \(G_{\alpha}\) a number of interesting results is derived -- \(\{B_n^{\alpha}\}\) satisfies a recurrence relation of order \(d+1\) and -- it is a sequence of orthogonal polynomials of dimension \(d\), -- the same holds for \(D^{m+1}\{B_n^{\alpha}\}\) for \(n\geq m+1\), \(D=d/dx\), -- an integral representation for \(B_n^{\alpha}(x)\), -- interpretation of the moments of the underlying linear forms \({\mathcal L}_n\), -- recurrence relations and an integral representations for the \({\mathcal L}_n\). Finally the author gives a treatment of the case \(d=2\) including the derivation of a third order differential equation. In a special case (generalisation of the Gegenbauer polynomials) this reduces to the \({}_3F_2\)-equation. A nicely written paper (the misprints are easily spotted).
scientific article; zbMATH DE number 1076162