An exploratory study on bivariate extended $ q $-Laguerre-based Appell polynomials with some applications

Mohra Zayed; Waseem Ahmad Khan; Cheon Seoung Ryoo; Ugur Duran; Mohra Zayed; Waseem Ahmad Khan; Cheon Seoung Ryoo; Ugur Duran

doi:10.3934/math.2025577

AIMS Mathematics

2025, Volume 10, Issue 6: 12841-12867. doi: 10.3934/math.2025577

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An exploratory study on bivariate extended $ q $-Laguerre-based Appell polynomials with some applications

1.
Mathematics Department, College of Science, King Khalid University, Abha 61413, Saudi Arabia
2.
Department of Electrical Engineering, Prince Mohammad Bin Fahd University, P.O Box 1664, Al Khobar 31952, Saudi Arabia
3.
Department of Mathematics, Hannam University, Daejeon 34430, South Korea
4.
Department of Basic Sciences of Engineering, Iskenderun Technical University, Hatay 31200, Turkiye

Received: 04 March 2025 Revised: 27 April 2025 Accepted: 27 May 2025 Published: 03 June 2025
MSC : 33C45, 05A30, 11B83

In this paper, we employed the $ q $-Bessel Tricomi functions of zero-order to introduce bivariate extended $ q $-Laguerre-based Appell polynomials. Then, the bivariate extended $ q $-Laguerre-based Appell polynomials were established in the sense of quasi-monomiality. We examined some of their properties, such as $ q $-multiplicative operator property, $ q $-derivative operator property and two $ q $-integro-differential equations. Additionally, we acquired $ q $-differential equations and operational representations for the new polynomials. Moreover, we drew the zeros of the bivariate extended $ q $-Laguerre-based Bernoulli and Euler polynomials, forming 2D and 3D structures, and provided a table including approximate zeros of the bivariate extended $ q $-Laguerre-based Bernoulli and Euler polynomials.
- quasi monomiality,
- extension of monomiality principle,
- quantum calculus,
- $ q $-Laguerre polynomials,
- $ q $-Dilatation operator,
- $ q $-Laguerre-based Appell polynomials
Citation: Mohra Zayed, Waseem Ahmad Khan, Cheon Seoung Ryoo, Ugur Duran. An exploratory study on bivariate extended $ q $-Laguerre-based Appell polynomials with some applications[J]. AIMS Mathematics, 2025, 10(6): 12841-12867. doi: 10.3934/math.2025577

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Abstract

In this paper, we employed the $ q $-Bessel Tricomi functions of zero-order to introduce bivariate extended $ q $-Laguerre-based Appell polynomials. Then, the bivariate extended $ q $-Laguerre-based Appell polynomials were established in the sense of quasi-monomiality. We examined some of their properties, such as $ q $-multiplicative operator property, $ q $-derivative operator property and two $ q $-integro-differential equations. Additionally, we acquired $ q $-differential equations and operational representations for the new polynomials. Moreover, we drew the zeros of the bivariate extended $ q $-Laguerre-based Bernoulli and Euler polynomials, forming 2D and 3D structures, and provided a table including approximate zeros of the bivariate extended $ q $-Laguerre-based Bernoulli and Euler polynomials.

References

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