Orthogonality Preserving Maps and the Laguerre Functional

Abstract

Let $\Re [x]$ be the usual algebra of all polynomials in the indeterminate $x$ over the field of real numbers $\Re$, and let $\varphi$ be a linear operator mapping $\Re [x]$ into $\Re [x]$. In this paper we show that if $\varphi$ maps every orthogonal polynomial sequence into an orthogonal polynomial sequence, then $\varphi$ is defined by $\varphi ({x^n}) = s{(ax + b)^n},n = 0,1,2, \ldots$, where $s,a,$, and $b$ belong to $\Re$, $s \ne 0$, and $a \ne 0$.