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Some results on quasi-monomiality. (English) Zbl 1041.33008

A polynomial set {P n } n0 is called quasi-monomial if and only if it is possible to define two operators 𝒫 ^ and ^, independent of n, such that

𝒫 ^(P n )(x)=nP n-1 (x)and ^(P n )(x)=P n+1 (x)·

In this paper, the author shows that every polynomial set is quasi-monomial and presents some useful tools to explicitly express the operators 𝒫 ^ and ^ for some polynomial families given by their generating functions. The obtained results are then applied to the Boas-Buck polynomial sets. Some closely-related earlier works include (among others cited by the author) a recent paper by G. Dattoli, the reviewer and C. Cesarano [Appl. Math. Comput. 124, No. 1, 117–127 (2001; Zbl 1036.33008)].

33C45Orthogonal polynomials and functions of hypergeometric type
39A70Difference operators