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

A polynomial set ${\left\{{P}_{n}\right\}}_{n\ge 0}$ is called quasi-monomial if and only if it is possible to define two operators $\stackrel{^}{𝒫}$ and $\stackrel{^}{ℳ}$, independent of $n$, such that

$\stackrel{^}{𝒫}\left({P}_{n}\right)\left(x\right)=n{P}_{n-1}\left(x\right)\phantom{\rule{1.em}{0ex}}\text{and}\phantom{\rule{1.em}{0ex}}\stackrel{^}{ℳ}\left({P}_{n}\right)\left(x\right)={P}_{n+1}\left(x\right)·$

In this paper, the author shows that every polynomial set is quasi-monomial and presents some useful tools to explicitly express the operators $\stackrel{^}{𝒫}$ and $\stackrel{^}{ℳ}$ 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)].

##### MSC:
 33C45 Orthogonal polynomials and functions of hypergeometric type 39A70 Difference operators