Integral and discrete representations of certain Brenke type orthogonal \(q\)-polynomials
Let \(\mathcal{P}\) be the vector space of polynomials with coefficients in \(\mathbb{C}\), and let \(\mathcal{P}'\) be its topological dual formed by linear forms \(u:\mathcal{P}\to \mathbb{C},\;f\mapsto \langle u,f\rangle \). A linear form \(u\) is said to be regular if there is a sequence of monic polynomials \(\{ P_n\}_{n\geq 0}\) such that \(\langle u,P_n\, P_m\rangle =r_n\, \delta _{n,m}\) with \(\deg P_n=n\) and \(r_n\neq 0\), for any \(n\). In this case the monic orthogonal polynomial sequence (MOPS) \(\{ P_n\}_{n\geq 0}\) satisfies a three-term recurrence relation. A linear form \(u\) is called \(H_q\)-semiclassical if it is regular and there exist a monic polynomial \(\Phi \) and a polynomial \(\Psi \) with \(\deg \Psi \geq 1\) such that \(H_q(\Phi u)+\Psi u=0\), where \(\langle H_qu,f\rangle :=-\langle u,H_qf\rangle \) and \((H_qf)(x)=\frac{f(qx)-f(x)}{(q-1)x}\). The authors investigate the \(H_q\)-semiclassical character, duality and the standard perturbation to characterize a well-known non-symmetric MOPS of Brenke-type related to the Wall's one.
scientific article; zbMATH DE number 6143174