Generalized Hahn's theorem
type: Item
instance of: scholarly article
title: Generalized Hahn's theorem
zbMATH Open document ID: 0947.33009
DOI: 10.1016/S0377-0427(99)00319-2
author: N. Delaunay
published in: Journal of Computational and Applied Mathematics
publication date: 2000-11-01T00:00:00Z
review text: One of the famous methods of classification of orthogonal polynomial systems goes back to \textit{W. Hahn} [Math. Z. 39, 634-638 (1935; Zbl 0011.06202)]: If \(\{P_n(x)\}_{n=0}^{\infty}\) and \(\{P_n'(x)\}_{n=0}^{\infty}\) are positive-definite orthogonal polynomial systems, then \(\{P_n(x)\}_{n=0}^{\infty}\) must be one of the classical systems (Jacobi, Laguerre or Hermite). This classification was later extended by several authors (to include the Bessel case too) and the conditions were weakened to: if \(\{P_n(x)\}_{n=0}^{\infty}\) is an orthogonal polynomial system and \(\{P_n^{(r)}(x)\}_{n=0}^{\infty}\) is a weakly orthogonal polynomial system, then \(\{P_n(x)\}_{n=0}^{\infty}\) must be classical. The authors now solve the following problems: (1) given an orthogonal polynomial system \(\{P_n(x)\}_{n=0}^{\infty}\) with respect to a moment functional \(\sigma\), (2) given a linear differential operator \(L[\cdot]=\sum_{i=0}^k a_i(x)D^i\) with polynomial coefficients, when is the polynomial sequence \(\{Q_n(x)\}_{n=0}^{\infty}\) defined by \(Q_n(x)=L[P_{n+r}^{(r)}(x)]\) also (weakly) orthogonal (\(r\) any non-negative integer)? The main result is: A. The \(Q\)'s are weakly orthogonal if and only if there exists a moment functional \(\tau\) and \(k+r+1\) polynomials \(\{b_i(x)\}_{i=r}^{k+2r}\) with deg\((b_i)\leq i\) and \[ \sum_{i=j}^{k+r} (-1)^j{i\choose j} (a_{i-r}(x)\tau)^{(i-j)}=b_{j+r}(x)\sigma,\quad 0\leq j\leq k+r, \] where \(\text{deg}(b_r)=r\), \(\langle\sigma,b_2r\rangle\not= 0\), \(b_{2r}(x)\not\equiv 0\). B. The \(Q\)'s are orthogonal if the \(b\)'s also satisfy \[ \sum_{i=0}^{k+r} b_{i+r,i+r}n_{(i)}\not= 0,\quad n\geq 0, \] where \(b_i(x)=\sum_{j=0}^i b_{ij}x^j\) (now deg\((b_r)=r\) and \(b_{k+2r}(x)\not= 0\)) and \(n_{(i)}=n(n-1)\cdots (n-i+1)\) for \(i\geq 1,\;n_{(0)}=1\). Furthermore, the authors give the complete analysis for the cases \(k=0,1,2\); \(r=1\) and \(k=r=1\). A nicely written paper.
Mathematics Subject Classification ID: 33C45
Mathematics Subject Classification ID: 34L05
zbMATH DE Number: 1455710
zbMATH Keywords: differential equations
zbMATH Keywords: orthogonal polynomials
zbMATH Keywords: Hahn's theorem
reviewed by: Marcel G. de Bruin
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1455710