Applying the monomiality principle to the new family of Apostol Hermite Bernoulli-type polynomials

Ramírez, William; Cesarano, Clemente

doi:10.2478/caim-2024-0010

Abstract

In this article, we introduce a new class of polynomials, known as Apostol Hermite Bernoulli-type polynomials, and explore some of their algebraic properties, including summation formulas and their determinant form. The majority of our results are proven using generating function methods. Additionally, we investigate the monomiality principle related to these polynomials and identify the corresponding derivative and multiplicative operators.

References

P. Appell and F. Kampé, Fonctions Hypergéométriques et Hypersphériques Polynomes d ′Hermite. Gautier Villars, 1926.
Search in Google Scholar Back to article
R. Tremblay, S. Gaboury, and B. Fugére, Some new classes of generalized Apostol–Euler and Apostol-Genocchi polynomials, International Journal of Mathematics and Mathematical Sciences, vol. 2012, pp. 1–14, 2012.
Search in Google Scholar Back to article
R. Tremblay, S. Gaboury, and B. Fugére, A further generalization of Apostol-Bernoulli polynomials and related polynomials, Honam Mathematical Journal, vol. 34, no. 3, pp. 311–326, 2012.
Search in Google Scholar Back to article
C. Cesarano, G. Cennamo, and L. Placidi, Humbert polynomials and functions in terms of Hermite polynomials towards applications to wave propagation, Wseas Transactions on Mathematics, vol. 13, pp. 596–602, 2014.
Search in Google Scholar Back to article
C. Cesarano and D. Assante, A note on generalized Bessel functions, International journal of mathematical models and methods in applied sciences, vol. 7, no. 6, pp. 625–629, 2013.
Search in Google Scholar Back to article
C. Cesarano, B. Germano, and P. Ricci, Laguerre-type Bessel functions, Integral Transforms and Special Functions, vol. 16, no. 4, pp. 315–322, 2005.
Search in Google Scholar Back to article
G. Dattoli and C. Cesarano, On a new family of Hermite polynomials associated to parabolic cylinder functions, Applied Mathematics and Computation, vol. 141, no. 1, pp. 143–149, 2003.
Search in Google Scholar Back to article
D. Lim, Some identities of degenerate Genocchi polynomials, Bulletin of the Korean Mathematical Society, vol. 53, no. 2, pp. 569–579, 2016.
Search in Google Scholar Back to article
K. Subuhi, N. Tabinda, and R. Mumtaz, Degenerate Apostol-type polynomials and applications, Boletín de la Sociedad Matemática Mexicana, vol. 25, no. 1, pp. 509–528, 2018.
Search in Google Scholar Back to article
L. Andrews, Special functions for Engineers and Applied Mathematicians. Macmillan, 1985.
Search in Google Scholar Back to article
W. Ramírez, A. Urieles, R. Herrera, and M. Ortega, New family of Bernoulli-type polynomials and some application, Dolomites Research Notes on Approximation, vol. 16, no. 1, pp. 1–11, 2023.
Search in Google Scholar Back to article
D. Bedoya, C. Cesarano, W. Ramírez, and S. Díaz, New classes of degenerate unified polynomials, Axioms, vol. 12, no. 1, pp. 1–10, 2023.
Search in Google Scholar Back to article
D. Bedoya, C. Cesarano, W. Ramírez, and L. Castilla, A new class of degenerate biparametric Apostol-type polynomials, Dolomites Research Notes on Approximation, vol. 16, no. 1, pp. 10–19, 2023.
Search in Google Scholar Back to article
C. Cesarano and W. Ramírez, Some new classes of degenerated generalized Apostol–Bernoulli, Apostol–Euler and Apostol–Genocchi polynomials, Carpathian Mathematical Publications, vol. 14, no. 2, pp. 1–10, 2022.
Search in Google Scholar Back to article
C. Cesarano, W. Ramírez, and S. Khan, A new class of degenerate Apostol–type Hermite polynomials and applications, Dolomites Research Notes on Approximation, vol. 15, no. 1, pp. 1–10, 2022.
Search in Google Scholar Back to article
C. Cesarano, W. Ramírez, S. Díaz, A. Shamaoon, and W. Khan, On Apostol–type Hermite degenerated polynomials, Mathematics, vol. 11, no. 8, pp. 1–13, 2023.
Search in Google Scholar Back to article
F. Costabile and E. Longo, A determinantal approach to Appell polynomials, Journal of Computational and Applied Mathematics, vol. 234, no. 5, pp. 1528–1542, 2010.
Search in Google Scholar Back to article
J. Steffensen, The poweriod, an extension of the mathematical notion of power, Acta Mathematica, vol. 73, pp. 33–366, 941.
Search in Google Scholar Back to article
G. Dattoli, Hermite-bessel and Laguerre-Bessel functions: a by-product of the monomiality principle, Advanced Special functions and applications, vol. 1, no. 2000, pp. 147–164, 1999.
Search in Google Scholar Back to article
G. Dattoli, Generalized polynomials, operational identities and their applications, Journal of Computational and Applied Mathematics, vol. 118, no. 1, pp. 111–123, 2000.
Search in Google Scholar Back to article
M. Zayed, S. Wani, and A. Mahnashi, Certain properties and characterizations of multivariable Hermite-Based Appell polynomials via factorization method, Fractal and Fractional, vol. 7, no. 8, p. 605, 2023.
Search in Google Scholar Back to article
R. Alyusof and S. S.A Wani, Several characterizations of δ h-doped special polynomials associated with Appell sequences, Symmetry, vol. 15, no. 7, p. 1315, 2023.
Search in Google Scholar Back to article
B. Alkahtani, T. Saad, I. Alazman, and S. Wani, Some families of differential equations associated with multivariate Hermite polynomials, Fractal and Fractional, vol. 7, no. 5, p. 390, 2023.
Search in Google Scholar Back to article

Applying the monomiality principle to the new family of Apostol Hermite Bernoulli-type polynomials

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