Some Classes of Bi-Starlike and Bi-Convex Functions Associated with the Poisson-Charlier Polynomials

Motivated by the interplay between discrete orthogonal polynomials and geometric function theory, we introduce and investigate some new Ma-Minda-type subclasses of bi-univalent functions generated by the Poisson-Charlier polynomials through their following analytic generating function: Ea(x,z)=(1+z)xe−az(|z|<1). Within the classical class Σ of analytic and bi-univalent functions, we first define the Poisson-Charlier-generated bi-starlike and bi-convex families via the subordination relations involving Ea(x,·) for both a function f and its inverse f−1. By combining the Carathéodory representation with the series expansion of Ea(x,z) and the Lagrange inversion formula for f−1, we derive coefficient estimates for the initial Taylor-Maclaurin coefficients a2 and a3 of functions in the starlike class ΣS*E(a,x) and in the convex class ΣKE(a,x). In addition, we obtain corresponding Fekete-Szegö type inequalities of the form a3−μa22 for a real parameter μ in both settings, which are expressed explicitly in terms of the parameters a and x of the Poisson-Charlier framework. To the best of our knowledge, these Poisson-Charlier-generated bi-starlike and bi-convex classes have not previously been investigated, so the resulting coefficient and Fekete-Szegö estimates constitute a distinct contribution rather than direct special cases of previously studied Ma-Minda families.