Some $q$-hypergeometric identities associated with partition theorems of Lebesgue, Schur and Capparelli

TL;DR

This work develops a unified q-hypergeometric framework that ties together Lebesgue, Schur, and Capparelli partition identities through a finite three-parameter polynomial identity and its corollaries. By constructing finite and infinite analogues (including a three-parameter Schur unification and a Capparelli-driven hierarchy), the authors derive key identities as special cases and provide polynomial versions that interpolate between classical generating functions and refined partition theorems. The approach leverages q-binomial and q-multinomial combinatorics, level parities, and color-graded partitions to connect distinct partition theorems and to propose an overarching hierarchy of identities with potential combinatorial and algebraic interpretations. The results illuminate deep connections among classical partition theory, finite polynomial identities, and refined q-hypergeometric structures, with links to recent work on polynomial identities by Berkovich-Uncu and others.

Abstract

Here, we establish a polynomial identity in three variables $a, b, c$, and with the degree of the polynomial given in terms of two integers $L, M$. By letting $L$ and $M$ tend to infinity, we get the 1993 Alladi-Gordon $q$-hypergeometric key-identity for the generalized Schur Theorem as well as the fundamental Lebesgue identity by two different choices of the variables. This polynomial identity provides a generalization and a unified approach to the Schur and Lebesgue theorems. We discuss other analytic identities for the Lebesgue and Schur theorems and also provide a key identity ($q$-hypergeometric) for Andrews' deep refinement of the Alladi-Schur theorem. Finally, we discuss a new infinite hierarchy of identities, the first three of which relate to the partition theorems of Euler, Lebesgue, and Capparelli, and provide their polynomial versions as well.