A four parameter generalization of Gollnitz's (BIG) partition theorem

Recently, $4$-regular partitions into distinct parts are connected with a family of overpartitions. In this paper, we provide a uniform extension of two relations due to Andrews for the two types of partitions. Such an extension is made possible with recourse to a new trivariate Rogers--Ramanujan type identity, which concerns a family of quadruple summations appearing as generating functions for the aforementioned overpartitions. More interestingly, the derivation of this Rog...

In 2003, Alladi, Andrews and Berkovich proved an identity for partitions where parts occur in eleven colors: four primary colors, six secondary colors, and one quaternary color. Their work answered a longstanding question of how to go beyond a classical theorem of G\"ollnitz, which uses three primary and three secondary colors. Their main tool was a deep and difficult four parameter $q$-series identity. In this paper we take a different approach. Instead of adding an eleventh...

We focus on writing closed forms of generating functions for the number of partitions with gap conditions as double sums starting from a combinatorial construction. Some examples of the sets of partitions with gap conditions to be discussed here are the set of Rogers--Ramanujan, G\"ollnitz--Gordon, and little G\"ollnitz partitions. This work also includes finding the finite analogs of the related generating functions and the discussion of some related series and polynomial id...

The partition functions $P(n,m,p)$, the number of integer partitions of $n$ into exactly $m$ parts with each part at most $p$, and $Q(n,m,p)$, the number of integer partitons of $n$ into exactly $m$ distinct parts with each part at most $p$, are related by double summation identities which follow from their generating functions. From these identities and some identities from an earlier paper, some other identities involving distinct partitions and some q-binomial summation id...

Recently, George Andrews has given a Glaisher style proof of a finite version of Euler's partition identity. We generalise this result by giving a finite version of Glaisher's partition identity. Both the generating function and bijective proofs are presented.

In this paper we give a computer proof of a new polynomial identity, which extends a recent result of Alladi and the first author. In addition, we provide computer proofs for new finite analogs of Jacobi and Euler formulas. All computer proofs are done with the aid of the new computer algebra package qMultiSum developed by the second author. qMultiSum implements an algorithmic refinement of Wilf and Zeilberger's multi-q-extension of Sister Celine's technique utilizing additio...

We use an injection method to prove a new class of partition inequalities involving certain $q$-products with two to four finitization parameters. Our new theorems are a substantial generalization of work by Andrews and of previous work by Berkovich and Grizzell. We also briefly discuss how our products might relate to lecture hall partitions.

We propose a method to construct a variety of partition identities at once. The main application is an all-moduli generalization of some of Andrews' results in [5]. The novelty is that the method constructs solutions to functional equations which are satisfied by the generating functions. In contrast, the conventional approach is to show that a variant of well-known series satisfies the system of functional equations, thus reconciling two separate lines of computations.

In this paper, we prove a theorem which adds a new member to the famous G\"oellnitz-Gordon identities. We construct a "new system of recurrence formulas" in order to prove it.

In this note we give three identities for partitions with parts separated by parity, which were recently introduced by Andrews.