On discrete Appell polynomials of Apostol-Bernoulli-type polynomials and their patterns of distribution of zeros

This study introduces a new category of discrete Appell polynomials derived from Apostol-Bernoulli-type polynomials. We derive their generating function, provide explicit formulae, and represent them in matrix form to enhance computational efficiency and explore deeper algebraic properties. Using algebraic structure and recurrence relations to emphasize these recurrences, we develop the principle of monomiality for these polynomials. The study also approximates the roots of Apostol-Bernoulli-type discrete polynomials using numerical methods, underscoring their relevance in various applications. The paper concludes by discussing potential extensions and the broader impact of this work in combinatorial mathematics and numerical analysis.