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

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.