Bi-univalent functions of order ζ connected with ( m , n ) -Lucas polynomials

With the aid of the q -binomial coefficients and utilizing the convolution, we define a new q -convolution operator that helps us introduce two new families of bi-univalent functions. These classes are connected by subordination with a function G m , n . We give upper bounds for the coefficients estimate | a j | ( j = 2,3 ) of the functions that belong to these families, followed by some special cases. Moreover, we found estimates for the Fekete-Szeg¨o inequality for both of these families, followed by simple particular results. We emphasize that the defined convolution q -difference operator generalizes some other operators given by several authors. As an application of this study, Fekete-Szeg¨o inequalities for these classes of functions defined by Pascal distribution are investigated.