The Umbral-Algebraic Approach to Study the General Appell–λ Polynomials

We introduce and comprehensively study a new class of hybrid special polynomials in three variables, which we call the three-variable general-Appell–λ polynomials, which are denoted as PAλn(u,v,w:β). These polynomials are constructed through a symbolic convolution of the Appell polynomial class, the two-variable general polynomials, and the λ-polynomials. Using the symbolic-umbral framework together with the monomiality principle, we derive an exponential generating function, an explicit series representation, three differential recurrence relations, a generalized heat-type partial differential equation, and an operational identity. Further, displacement and double-index summation formulae, a quasi-monomiality analysis with explicit multiplicative and derivative operators, and a determinant characterization via Cauchy-product inversion are also established. Three distinguished special cases the Bernoulli-general-λ polynomials, the Euler-general-λ polynomials, and the Genocchi-general-λ polynomials are examined in detail. Number-sequence tables are presented for the Bernoulli and Euler subfamilies, and zero-distribution plots are provided for both cases with a thorough graphical significance analysis.