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Abstract

We introduce factorial analogues of the ordinary Hall–Littlewood $P$- and $Q$-polynomials, which we call the factorial Hall–Littlewood $P$- and $Q$-polynomials. Using the universal formal group law, we further generalize these polynomials to the universal factorial Hall–Littlewood $P$- and $Q$-functions. We show that these functions satisfy the vanishing property which the ordinary factorial Schur $S$-, $P$-, and $Q$-polynomials have. By the vanishing property, we derive the Pieri-type formula and a certain generalization of the classical hook formula. We then characterize our functions in terms of Gysin maps from flag bundles in complex cobordism theory. Using this characterization and Gysin formulas for flag bundles, we obtain generating functions for the universal factorial Hall–Littlewood $P$- and $Q$-functions. Using our generating functions, we show that our factorial Hall–Littlewood $P$- and $Q$-polynomials have a certain cancellation property. Further applications such as Pfaffian formulas for $K$-theoretic factorial $Q$-polynomials are also given.