Discrete Multiplier Sequences: Infinite Support and Low Finite Rank

A discrete multiplier sequence acts diagonally in the falling-factorial basis and preserves polynomials whose zeros are nonnegative and separated by at least one. We characterize the sequences of infinite support as the nonnegative, nondecreasing classical multiplier sequences after a common sign normalization. This establishes the infinite-support case of a conjecture of Brändén, Krasikov, and Shapiro. A three-term sequence gives a counterexample to the unrestricted statement. We classify sequences of ranks three and four, give sharp rank-three tests for bounded input degree, and describe the interval of admissible values of the last coefficient at finite rank. We also prove sharp Wasserstein bounds for the ordered zero map and characterize the sequences whose entrywise powers preserve the class for arbitrarily small positive exponents. The deposit includes the article, its LaTeX source, and reproducible exact computations.