Abstract
In this paper we provide proofs of two new theorems that provide a broad class of partition inequalities and that illustrate a na¨ıve version of Andrews’ anti-telescoping technique quite well. These new theorems also put to rest any notion that including parts of size 1 is somehow necessary in order to have a valid irreducible partition inequality. In addition, we prove (as a lemma to one of the theorems) a rather nontrivial class of rational functions of three variables has entirely nonnegative power series coefficients.
Highlights
When examining two q-products Π1 and Π2 and their corresponding q-series, it sometimes happens that the coefficients in the q-series for Π1 are never less than the coefficients in the q-series for Π2
We say that Π1 is dominant and that Π2 is subordinate, and we express this relationship with the more succinct notation Π1 < Π2 . (Note that < yields a partial ordering on the set of q-products if we identify products that produce the same q-series; any given product may be dominant when paired with some products, subordinate when paired with others, neither when paired with still other products, and both dominant and subordinate only when paired with “itself”.)
From this definition it follows that if Π1 < Π2, the q-series determined by Π1 − Π2
Summary
When examining two q-products Π1 and Π2 and their corresponding q-series, it sometimes happens that the coefficients in the q-series for Π1 are never less than the coefficients in the q-series for Π2. If one is lucky enough that each addend in Equation (10) is < 0, that is all one needs to show in order to prove the desired inequality. Albeit more difficult, to use naıve anti-telescoping to yield the following new theorem.
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