Orthogonality of some polynomial sets via quasi-monomiality

摘要

The lowering operator σ and the raising operator τ, associated with a polynomial set {Pn}n=0∞, are two operators which are independent of n and satisfy the relationshipsσ(Pn)(x)=nPn−1(x)andτ(Pn)(x)=Pn+1(x)(n∈N0).In this paper, we use these operators to study the orthogonality of some polynomial sets. More precisely, we state sufficient conditions, in terms of the σ and τ operators, to ensure the orthogonality. We also express explicitly, by means of the σ operator, the linear functional for which the orthogonality holds true. We obtain some well-known results as particular cases of the results presented in this paper.