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Zbl 1041.33008
Ben Cheikh, Youssèf
Some results on quasi-monomiality. (English)
[J] Appl. Math. Comput. 141, No. 1, 63-76 (2003). ISSN 0096-3003

A polynomial set $\{P_n\}_{n\ge 0}$ is called quasi-monomial if and only if it is possible to define two operators $\widehat{{\Cal P}}$ and $\widehat{{\Cal M}}$, independent of $n$, such that $$\widehat{{\Cal P}}(P_n)(x)= nP_{n- 1}(x)\quad\text{and}\quad \widehat{{\Cal M}}(P_n)(x)= P_{n+1}(x).$$ In this paper, the author shows that every polynomial set is quasi-monomial and presents some useful tools to explicitly express the operators $\widehat{{\Cal P}}$ and $\widehat{{\Cal M}}$ 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 {\it G. Dattoli}, the reviewer and {\it C. Cesarano} [Appl. Math. Comput. 124, No. 1, 117--127 (2001; Zbl 1036.33008)].
[Hari M. Srivastava (Victoria)]

MSC 2000:: *33C45 Orthogonal polynomials and functions of hypergeometric type
39A70 Difference operators

Keywords: quasi-monomiality; Boas-Buck polynomial set; $d$-symmetry; binomial identity

Citations: Zbl 1036.33008

Cited in: Zbl 1050.33006

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