In this article, we introduce and study a new family P Σ (δ, λ, k, γ, α, β, r) of normalized analytic and λ-pseudo-starlike bi-univalent functions by using the Horadam polynomials, which is associated with a certain convolution operator defined in the open unit disk U. We establish the bounds for |a 2 | and |a 3 |, where a 2 , a 3 are the initial Taylor-Maclaurin coefficients. Furthermore, we obtain the Fekete-Szegö inequality for functions in the class P Σ (δ, λ, k, γ, α, β, r), which we have introduced here. We indicate several special cases and consequences for our results. Finally, we comment on the recent usages, especially in Geometric Function Theory of Complex Analysis, of the basic (or q-) calculus and also of its trivial and inconsequentional (p, q)-variation involving an obviously redundant parameter p