Asian Research Journal of Mathematics

Meromorphic functions and L-functions are central objects in value-distribution theory, where uniqueness questions are commonly studied through shared values and differential or difference expressions. This study investigates uniqueness relations between a nonconstant meromorphic function having finitely many poles and an L-function when associated difference-differential polynomials share a small function with a prescribed weight. The formulation incorporates the higher-order difference operator and considers expressions generated from powers of the functions, the factor involving the function minus one, and the corresponding difference term. The analysis is developed within Nevanlinna theory using weighted sharing, truncated counting functions, deficiency-type quantities, and growth properties of L-functions. Sufficient lower-bound conditions are obtained for the combined integer parameters under the three cases in which the sharing weight satisfies l ≥ 2, l = 1, or l = 0. Under the stated hypotheses, the relevant kth derivatives of the paired difference-differential polynomials either have product equal to the square of the shared small function, coincide, or yield a relation of the form f = tL subject to the corresponding algebraic restriction on the constant t. Corollaries for the cases n = 0 and m = 0 provide analogous conclusions for reduced forms. The results extend the manuscript’s earlier uniqueness formulations while remaining restricted to the stated assumptions on the functions, sharing weight, and integer parameters.