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.