Classical orthogonal polynomials: A functional approach
The Rodriguez formula for the classical orthogonal polynomials (here Hermite, Laguerre, Jacobi and Bessel) leads to a distributional differential equation for the corresponding moment functional \(u\) of the form \(D(\Phi u)=\Psi u\) with polynomials \(\Phi,\Psi\) and \(\deg \Phi \leq 2\), \(\deg \Psi=1\). Moreover, the moment functionals of the classical orthogonal polynomials are characterized by this formula. In the present paper, the authors use this characterization and derived further characterizations for the classical orthogonal polynomials. Some of these characterizations are known, but the new proofs given here are quite natural. Moreover, the approach of this paper leads to some new characterizations. One of them is as follows: Let \((P_ n)_{n \geq 0}\) be a system of monic orthogonal polynomials with moment functional \(u\). Then this system is classical if and only if \(P_ n=Q_ n+r_ n Q_{n-1}+s_ n Q_{n-2}\) \((n \geq 2)\) with \(Q_ n=P_{n+1}' /(n+1)\) for suitable constants \(r_ n,s_ n\).
scientific article; zbMATH DE number 572240
scientific article