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

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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.

Keywords

(p,q)-polynomials / (p,q)-Appell polynomials / (p,q)-Hermite polynomials / determinant approach

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References

[1] A. Benahmadi, A. Ghanmi, Post-quantum complex Hermite polynomials, Bol. Soc. Mat. Mex., 30(2024). https://doi.org/10.1007/s40590-023-00586-0

[2] R. Chakrabarti, R. Jagannathan, A (p,q)-oscillator realization of two-parameter quantum algebras, J. Phys. A, 24(1991), L711-L718. https://doi.org/10.1088/0305-4470/24/13/002

[3] R. B. Corcino, On (p,q)-binomial coefficients, Electron. J. Combin. Number Theory, 8(2008).

[4] U. Duran, M. Acikgoz, A. Esi, S. Araci, A note on the (p,q)-Hermite polynomials, J. Appl. Math. Inf. Sci., 12(2018), 227-231. https://doi.org/10.18576/amis/120122

[5] U. Duran, M. Acikgoz, S. Araci, On higher order (p,q)-Frobenius-Euler polynomials, TWMS J. Pure Appl. Math., 8(2017), 198-208.

[6] N. Raza, M. Fadel, W. S. Du, New summation and integral representations for 2-variable (p,q)- Hermite polynomials, Axioms, 13(2024), 196. https://doi.org/10.3390/axioms13030196

[7] P. N. Sadjang, On (p,q)-Appell polynomials, Anal. Math., 45(2019), 583-598. https://doi.org/10.1007/s10476-019-0826-z

[8] G. Yasmin, A. Muhyi, On a family of (p,q)-hybrid polynomials, Kragujevac J. Math., 45(2019), 409-426. https://doi.org/10.46793/kgjmat2103.409y

[9] N. Raza, M. Fadel, S. Khan, On monomiality property of q-Gould-Hopper-Appell polynomials, Arab. J. Basic Appl. Sci., 32(2025), 21-29. https://doi.org/10.1080/25765299.2025.2457206

[10] W. Ramírez, D. Bedoya, A. Urieles, C. Cesarano, M. Ortega, New U-Bernoulli, U-Euler and U-Genocchi polynomials and their matrices, Carpathian Math., 15(2023), 449-467. https://doi.org/10.15330/cmp.15.2.449-467

[11] W. Ramírez, C. Kızılateş, D. Bedoya, C. Cesarano, C. S. Ryoo, On certain properties of three parametric kinds of Apostol-type unified Bernoulli-Euler polynomials, AIMS Math., 10(2025), 137-158. https://doi.org/10.3934/math.2025008

[12] A. Urieles, M. Ortega, W. Ramírez, D. Bedoya, Fourier expansion and integral representation for generalized Apostol-type Frobenius-Euler polynomials, Adv. Differ. Equ., 534(2020), 1-14. https://doi.org/10.1186/s13662-020-02988-0

[13] N. Raza, M. Fadel, C. Cesarano, On 2-variable q-Legendre polynomials: The view point of the q-operational technique, Carpathian Math. Publ., 17(2025), 14-26. https://doi.org/10.15330/cmp.17.1.14-26

[14] M. Fadel, N. Raza, A. Al-Gonah, U. Duran, Bivariate q-Laguerre-Appell polynomials and their applications, Appl. Math. Sci. Eng., 32(2024). https://doi.org/10.1080/27690911.2024.2412545

[15] I. M. Burban, A. U. Klimyk, (p,q)-Differentiation, (p,q)-Integration, and (p,q)-Hypergeometric functions related to quantum groups, Integr. Transforms Spec. Funct. 2(1994), 15-36. https://doi.org/10.1080/10652469408819035

[16] F. A. Costabile, E. Longo, ∆h−Appell sequences and related interpolation problem, Numer. Algorithms, 63(2013), 165-186. https://doi.org/10.1007/s11075-012-9619-1

[17] P. N. Sadjang, U. Duran, On two bivariate kinds of (p,q)-Bernoulli polynomials, Miskolc Math. Notes, 20(2019), 1185-1199. https://doi.org/10.18514/mmn.2019.2587

[18] F. A. Costabile, E. Longo, A determinantal approach to Appell polynomials, J. Comput. Appl. Math., 234(2010), 1528-1542. https://doi.org/10.1016/j.cam.2010.02.033

[19] F. A. Costabile, E. Longo, An algebraic approach to Sheffer polynomial sequences, Integr. Transforms Spec. Funct., 25(2014), 295-311. https://doi.org/10.1080/10652469.2013.842234

[20] M. E. Keleshteri, N. I. Mahmudov, A study on q-Appell polynomials from determinantal point of view, Appl. Math. Comput., 260(2015), 351-369. https://doi.org/10.1016/j.amc.2015.03.017