Orthogonality of some polynomial sets via quasi-monomiality.
The authors present sufficient conditions to ensure the orthogonality of some polynomial sets. Let \(\{P_n\}_{n=0}^\infty\) be a polynomial sequence with \(\deg P_n=n\). It was earlier shown [\textit{Y. Ben Cheikh}, Appl. Math. Comput. 141, 63--76 (2003; Zbl 1041.33008)] that there exist a lowering operator \(\sigma\) and raising operator \(\tau\), independent of \(n\), such that \[ \sigma(P_n(x))=n P_{n-1}(x), \quad r(P_n(x))=P_{n+1}(x). \] The authors claim that if \[ \tau=(a_1+(a_2+a_3 x)\sigma+a_4\sigma^2)^{-1}(a_5+a_6x+a_7\sigma), \] and \[ a_1\neq 0, \quad (a_3, a_6)\neq (0,0)\neq (a_4, a_7), \] where \(a_1,\dots, a_7\) are complex numbers, then \(\{P_n\}_{n=0}^\infty\) is orthogonal with respect to a linear functional \(\mathcal L\) and express \(\mathcal L\) via the lowering operator \(\sigma\). Some well-known results for Hermite polynomials, Laguerre polynomials, Charlier polynomials, Meixner polynomials are obtained as particular cases.
scientific article; zbMATH DE number 1981465
scientific article