\(d\)-Orthogonality of Hermite type polynomials
Consider polynomials of the form \(P_n(x)=\sum_{k=0}^{[n/ m]}\gamma_{n,m}(k)\mathcal B_{n-mk}(x)\), where \(m\geq2\), \(\mathcal B_{0} (x)=1\), \(\mathcal B_{k}(x)=\prod_{r=0}^{k-1}(x+\pi(r))\), \(k=1,2, \dotsc\), \(\pi\) an entire function, \(\{\gamma_{n,m}(k)\}\) is the sequence defined by the recurrence relation \(\gamma_{n,m}(k+1)/ \gamma_{n,m}(k)=(-n+mk)_m/(k+1)\cdot N(k)/(D(k)c^m)\), \(k=0,\dots, [n/m]\), \(\gamma_{n,m}(0)=1\), where \(D\) and \(N\) are two monic coprime polynomials with non positive integer roots and \(c\) is a complex number. A problem \(P_1\): Find necessary and sufficient conditions on \(D\), \(N\), \(\pi\) and \(c\) such that the polynomial system \(\{P_n(x)\}\) of Hermite type is a \(d\)-orthogonal polynomial system (\(d\)-OPS) [see \textit{J. van Iseghem}, J. Comput. Appl. Math. 19, 141--150 (1987; Zbl 0626.65013)] is considered. Problem \(P_1\) is solved in Theorem 1.1: The only \(d\)-OPS \(P_n(x)\) arise for \(D=N=1\), \(\pi\) a polynomial and \(d=m\deg\pi\) if \(\deg\pi\geq1\), \(d=m-1\) if \(\pi \equiv0\). Two characterization problems related to Gould-Hopper polynomials and Charlier type polynomials are also solved. Some properties of the obtained polynomials: limit relations, generating function, inversion formula and connection coefficients are obtained. The \(d\)-dimensional functional vectors associated with Gould-Hopper, Charlier type and Laguerre type polynomials are given.
scientific article; zbMATH DE number 5318692
scientific article