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Cesarano, C.; Quintana, Y.; Ramírez, W. A Survey on Orthogonal Polynomials from Monomiality Principle Point of View. Encyclopedia. Available online: https://encyclopedia.pub/entry/57091 (accessed on 05 December 2025).
Cesarano C, Quintana Y, Ramírez W. A Survey on Orthogonal Polynomials from Monomiality Principle Point of View. Encyclopedia. Available at: https://encyclopedia.pub/entry/57091. Accessed December 05, 2025.
Cesarano, Clemente, Yamilet Quintana, William Ramírez. "A Survey on Orthogonal Polynomials from Monomiality Principle Point of View" Encyclopedia, https://encyclopedia.pub/entry/57091 (accessed December 05, 2025).
Cesarano, C., Quintana, Y., & Ramírez, W. (2024, September 20). A Survey on Orthogonal Polynomials from Monomiality Principle Point of View. In Encyclopedia. https://encyclopedia.pub/entry/57091
Cesarano, Clemente, et al. "A Survey on Orthogonal Polynomials from Monomiality Principle Point of View." Encyclopedia. Web. 20 September, 2024.
Peer Reviewed
A Survey on Orthogonal Polynomials from Monomiality Principle Point of View
This is a peer-reviewed entry published in Encyclopedia Journal, full entry can be found at 10.3390/encyclopedia4030088

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.

References

  1. Cesarano, C. Generalized Hermite Polynomials in the Description of Chebyshev-like Polynomials. Ph.D. Thesis, Universidad Complutense de Madrid, Madrid, Spain, 2015.
  2. Dattoli, G. Generalized polynomials, operational identities and their applications. J. Comput. Appl. Math. 2000, 118, 111–123.
  3. Dattoli, G. Hermite-Bessel and Laguerre-Bessel functions: A by-product of the monomiality principle. Adv. Spec. Funct. Appl. 2000, 1, 147–164.
  4. Dattoli, G.; Licciardi, S. Monomiality and a new family of Hermite polynomials. Symmetry 2023, 15, 1254.
  5. Dattoli, G.; Migliorati, M.; Srivastava, H.M. Sheffer polynomials, monomiality principle, algebraic methods and the theory of classical polynomials. Math. Comput. Model. 2007, 45, 1033–1041.
  6. Dattoli, G.; Ottaviani, P.L.; Torre, A.; Vázquez, L. Evolution operator equations: Integration with algebraic and finite-difference methods. Applications to physical problems in classical and quantum mechanics and quantum field theory. Riv. Nuovo C. 1997, 20, 3–133.
  7. Riyasat, M.; Shahid, A.W.; Subuhi, K. Differential and integral equations associated with some hybrid families of Legendre polynomials. Tbilisi Math. J. 2018, 11, 127–139.
  8. Subuhi, K.; Riyasat, M.; Yasmin, G. Certain results for the 2-variable Apostol type and related polynomials. Comput. Math. Appl. 2015, 69, 1367–1382.
  9. Subuhi, K.; Riyasat, M. A determinantal approach to Sheffer–Appell polynomials via monomiality principle. J. Math. Anal. Appl. 2015, 421, 806–829.
  10. Yasmin, G.; Subuhi, K.; Ahmad, N. Operational methods and truncated exponential–based Mittag–Leffler polynomials. Mediterr. J. Math. 2016, 13, 1555–1569.
  11. Zayed, M.; Shahid, A.W. Exploring the versatile properties and applications of multidimensional degenerate Hermite polynomials. AIMS Math. 2023, 8, 30813–30826.
  12. Zayed, M.; Shahid, A.W.; Quintana, Y. Properties of multivariate Hermite polynomials in correlation with Frobenius–Euler polynomials. Mathematics 2023, 11, 3439.
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Subjects: Mathematics
Contributors MDPI registered users' name will be linked to their SciProfiles pages. To register with us, please refer to https://encyclopedia.pub/register : Clemente Cesarano , Yamilet Quintana , William Ramírez
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Online Date: 20 Sep 2024
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