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.
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][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
.The outline of this paper is as follows. In Section 2, we give some basic features of exponential operators. Section 3 and Section 4 are devoted to showing a brief state of the art about recent and interesting results involving some generalizations of Hermite and Gegenbauer polynomials via exponential operators, as well as relevant implications of monomiality in the theory of special polynomials. Finally, we provide concluding remarks in Section 5.
