A computer proof of a polynomial identity implying a partition theorem of Goellnitz

A double bounded key identity for Goellnitz's (big) partition theorem

Given integers i,j,k,L,M, we establish a new double bounded q-series identity from which the three parameter (i,j,k) key identity of Alladi-Andrews-Gordon for Goellnitz's (big) theorem follows if L, M tend to infinity. When L = M, the identity yields a strong refinement of Goellnitz's theorem with a bound on the parts given by L. This is the first time a bounded version of Goellnitz's (big) theorem has been proved. This leads to new bounded versions of Jacobi's triple product identity for theta functions and other fundamental identities.

A new four parameter q-series identity and its partition implications

We prove a new four parameter q-hypergeometric series identity from which the three parameter key identity for the Goellnitz theorem due to Alladi, Andrews, and Gordon, follows as a special case by setting one of the parameters equal to 0. The new identity is equivalent to a four parameter partition theorem which extends the deep theorem of Goellnitz and thereby settles a problem raised by Andrews thirty years ago. Some consequences including a quadruple product extension of Jacobi's triple product identity, and prospects of future research are briefly discussed.

A limiting form of the q-Dixon_4ϕ_3 summation and related partition identities

By considering a limiting form of the q-Dixon_4ϕ_3 summation, we prove a weighted partition theorem involving odd parts differing by >= 4. A two parameter refinement of this theorem is then deduced from a quartic reformulation of Goellnitz's (Big) theorem due to Alladi, and this leads to a two parameter extension of Jacobi's triple product identity for theta functions. Finally, refinements of certain modular identities of Alladi connected to the Goellnitz-Gordon series are shown to follow from a limiting form of the q-Dixon_4ϕ_3 summation.

New polynomial analogues of Jacobi's triple product and Lebesgue's identities

In a recent paper by the authors, a bounded version of Goellnitz's (big) partition theorem was established. Here we show among other things how this theorem leads to nontrivial new polynomial analogues of certain fundamental identities of Jacobi and Lebesgue. We also derive a two parameter extension of Jacobi's famous triple product identity.

A Polynomial Identity Implying Schur's Partition Theorem

We propose and prove a new polynomial identity that implies Schur's partition theorem. We give combinatorial interpretations of some of our expressions in the spirit of Kurşungöz. We also present some related polynomial and $q$-series identities.

多重变量欧拉多项式的回文性

Palindromicity of multivariate Eulerian polynomials

本文研究了多重变量欧拉多项式的回文性，并由此推导出新的组合恒等式。