Operational matrix formulation for linear difference equations using ∇ _λ-appell polynomial bases

This paper develops an operational matrix formulation for nonhomogeneous linear difference equations with constant coefficients in powers of the backward difference operator. Using general ∇ λ ∇ λ -Appell polynomial bases, the action of ∇ λ ∇ λ is represented algebraically on finite-dimensional Appell spaces, while the numerical approximation is obtained through finite expansions and collocation. Structural identities, change-of-basis matrices, Gram matrices and operational matrices are derived, allowing equations of the form a₀y(x)+a₁∇ λy(x)+⋯ +aₘ∇ λᵐy(x)=g(x) a 0 y ( x ) + a 1 ∇ λ y ( x ) + ⋯ + a m ∇ λ m y ( x ) = g ( x ) to be reduced to finite-dimensional algebraic systems with initial or boundary conditions. Numerical examples with Bernoulli-type and Euler-type ∇ λ ∇ λ -Appell bases illustrate exact polynomial recovery, convergence for a non-polynomial test solution, and the influence of the chosen basis on conditioning.