(untitled) (European Journal of Pure and Applied Mathematics, 2025)
This study investigates the development of polynomials, with a particular focus on the unique \(Δ_h\) Legendre polynomials. Explicit formulas for these polynomials are derived, along with summation formulae that provide further structural insights. Additionally, the monomiality principle is established, reinforcing the algebraic framework of these polynomials. Symmetric identities are also formulated, highlighting their fundamental properties. The study concludes with remarks summarizing the key findings and potential directions for future research.