A Survey on Orthogonal Polynomials from Monomiality Principle Point of View

A Survey on Orthogonal Polynomials from Monomiality Principle Point of View: History

Please note this is an old version of this entry, which may differ significantly from the current revision.

Contributor: Clemente Cesarano , Yamilet Quintana , William Ramírez

This survey highlights the significant role of exponential operators and the monomiality principle in the theory of special polynomials. Using operational calculus formalism, we revisited classical and current results corresponding to a broad class of special polynomials. For instance, we explore the 2D Hermite polynomials and their generalizations. We also present an integral representation of Gegenbauer polynomials in terms of Gould–Hopper polynomials, establishing connections with a simple case of Gegenbauer–Sobolev orthogonality. The monomiality principle is examined, emphasizing its utility in simplifying the algebraic and differential properties of several special polynomial families. This principle provides a powerful tool for deriving properties and applications of such polynomials. Additionally, we review advancements over the past 25 years, showcasing the evolution and extensive applicability of this operational formalism in understanding and manipulating special polynomial families.

operational calculus
exponential operators
Hermite polynomials
Gegenbauer polynomials
monomiality principle

This monograph was brought about by the current operational methods involving exponential operators and the corresponding identities commonly used for the study of special functions and diverse classes of orthogonal polynomials, both in one and several variables.

We only present a limited sampling of the many results related with methods involving exponential operators and their connection with special functions and orthogonal polynomials, placing emphasis on some of the contributions of the last 25 years (see, for instance ^[1]^[2]^[3]^[4]^[5]^[6]^[7]^[8]^[9]^[10]^[11]^[12] and the references therein). We would like to include all results but the length of this paper would not suffice. In addition, we do not prove most of the results we quote. We hope, nonetheless, that the readers will find something here of interest.

This entry is adapted from the peer-reviewed paper 10.3390/encyclopedia4030088

References

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