Weighted partitions with interval restrictions: exact formulas and a bivariate master identity

Abstract

Let $a_2''(n)$ and $b_2''(n)$ be the signed partition functions introduced by Andrews and El Bachraoui for interval-restricted partitions whose parts greater than $1$ are controlled by the smallest even part and by the number of ones. We prove two conjectures for these functions. The first gives the generating function for $a_2''(n)$ as an elementary rational term plus a false theta series with periodic signs; the second asserts that the companion coefficients $b_2''(n)$ take only the values $-1,0,1,2$. The central structural result introduces an auxiliary variable $z$ recording the number of non-compulsory parts greater than $1$. We obtain closed forms for the two resulting generating functions and prove the master identity $(1+q^2)\mathcal B(z,q)-(1+q)\mathcal A(z,q)=-q^4/(1-q^3)$ using both analytic and combinatorial techniques. At $z=-1$, this identity, together with a Rogers--Fine evaluation, gives the false theta formula for $a_2''(n)$ and an explicit generating function for $b_2''(n)$. The latter formula implies the asserted coefficient range and leads to an exact coefficient description of $b_2''(n)$. We also include a direct Heine--Rogers--Fine proof of the false theta formula, ordinary and fixed-refinement consequences of the master identity, and the resulting quantum modular interpretation.