Determinant approach of the $ (p, q) $-Hermite-Appell polynomials and some of their components

Mohammed Fadel; Ugur Duran; Clemente Cesarano; William Ramírez; Mohammed Fadel; Ugur Duran; Clemente Cesarano; William Ramírez

doi:10.3934/nhm.2026004

Networks and Heterogeneous Media

2026, Volume 21, Issue 1: 70-91. doi: 10.3934/nhm.2026004

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Determinant approach of the $ (p, q) $-Hermite-Appell polynomials and some of their components

1.
Department of Mathematics, University of Lahej, Lahej 73560, Yemen
2.
Department of Basic Sciences of Engineering, Faculty of Engineering and Natural Sciences, Iskenderun Technical University, Hatay 31200, Turkiye
3.
Section of Mathematics, International Telematic University Uninettuno, Corso Vittorio Emanuele Ⅱ, 39, 00186 Roma
4.
Universidad de la Costa, Department of Natural and Exact Sciences, Calle 58, 55-66, Barranquilla 080002, Colombia

Received: 09 September 2025 Revised: 31 December 2025 Accepted: 08 January 2026 Published: 19 January 2026

In this work, we offer the novel class of $ ({p}, {q}) $-Hermite-Appell polynomials. Some attributes of this class are constructed, along with the generating function, series definition, $ ({p}, {q}) $-derivative properties, $ ({p}, {q}) $-integral representation, summation formulas, and determinate representation. Additionally, we consider a few components for the $ ({p}, {q}) $ -Hermite-Appell polynomials and infer certain elements of their traits. The generating function and series expansions of some classes of two-dimensional $ ({p}, {q}) $-Hermite-Appell polynomials are provided. Moreover, we acquire a $ ({p}, {q}) $-differential operator formula for $ ({p}, {q}) $-Hermite-Appell polynomials. Finally, the Wolfram Mathematica software is used to plot the graphical diagrams of select components of $ ({p}, {q}) $-Hermite-Appell, along with two-dimensional $ ({p}, {q}) $-Hermite-Appell polynomials.
- $ (p, q) $-polynomials,
- $ (p, q) $-Appell polynomials,
- $ (p, q) $-Hermite polynomials,
- determinant approach
Citation: Mohammed Fadel, Ugur Duran, Clemente Cesarano, William Ramírez. Determinant approach of the $ (p, q) $-Hermite-Appell polynomials and some of their components[J]. Networks and Heterogeneous Media, 2026, 21(1): 70-91. doi: 10.3934/nhm.2026004

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Abstract

In this work, we offer the novel class of $ ({p}, {q}) $-Hermite-Appell polynomials. Some attributes of this class are constructed, along with the generating function, series definition, $ ({p}, {q}) $-derivative properties, $ ({p}, {q}) $-integral representation, summation formulas, and determinate representation. Additionally, we consider a few components for the $ ({p}, {q}) $ -Hermite-Appell polynomials and infer certain elements of their traits. The generating function and series expansions of some classes of two-dimensional $ ({p}, {q}) $-Hermite-Appell polynomials are provided. Moreover, we acquire a $ ({p}, {q}) $-differential operator formula for $ ({p}, {q}) $-Hermite-Appell polynomials. Finally, the Wolfram Mathematica software is used to plot the graphical diagrams of select components of $ ({p}, {q}) $-Hermite-Appell, along with two-dimensional $ ({p}, {q}) $-Hermite-Appell polynomials.

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