Related papers: Siladi\'c's theorem: weighted words, refinement an…
In most text books on number theory Wilson Theorem is proved by applying Lagrange theorem concerning polynomial congruences.Hardy and Wright also give a proof using cuadratic residues. In this article Wilson theorem is derived as a…
We give a combinatorial proof of the first Rogers-Ramanujan identity by using two symmetries of a new generalization of Dyson's rank. These symmetries are established by direct bijections.
We present a new proof of the Rogers-Ramanujan identities. Surprisingly, all its ingredients are available already in Rogers seminal paper from 1894, where he gave a considerably more complicated proof.
In this paper, we prove a new Rogers-Ramanujan-type identity, involving grounded partitions, by computing a character of the affine Kac-Moody algebra $D_4^{(3)}$ in two different ways. The product side is derived using Lepowsky's product…
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.
We continue to investigate which polynomials can possibly occur as factors in the denominators of rational solutions of a given partial linear difference equation. In an earlier article we had introduced the distinction between periodic and…
Motivated by Alladi's recent multi-dimensional generalization of Sylvester's classical identity, we provide a simple combinatorial proof of an overpartition analogue, which contains extra parameters tracking the numbers of overlined parts…
We give several expansion and identities involving the Ramanujan function $A_q$ and the Stieltjes--Wigert polynomials. Special values of our idenitities give $m$-versions of some of the items on the Slater list of Rogers-Ramanujan type…
The famous Rogers-Ramanujan and Andrews--Gordon identities are embedded in a doubly-infinite family of Rogers-Ramanujan-type identities labelled by positive integers m and n. For fixed m and n the product side corresponds to a specialised…
Lucy Slater used Bailey's $_6\psi_6$ summation formula to derive the Bailey pairs she used to construct her famous list of 130 identities of the Rogers-Ramanujan type. In the present paper we apply the same techniques to Chu's…
The product sides of the Rogers--Ramanujan identities and alike often appear to be "transparently modular" (functions). The old work by Rogers (1894) and recent work by Rosengren make use (somewhat implicitly) of this fact for proving the…
Jarnik's identity plays a major role in classical simultaneous approximation to two real numbers. O. German [2] has shown a generalization to the weighted setting in which the identity has to be replaced by two inequalities. His methods…
Regarding Euler's odd-strict theorem, which is the most basic partition identity, A refinement was done by Sylvester, and it was generalized by Bessenrodt to the r-regular and r-class regular cases. In this paper, we focus on the…
We utilize false theta function results of Nathan Fine to discover three new partition identities involving weights. These relations connect G\"ollnitz--Gordon type partitions and partitions with distinct odd parts, partitions into distinct…
In $2003$, Alladi, Andrews and Berkovich proved a four-parameter partition identity lying beyond a celebrated identity of G\"ollnitz. Since then it has been an open problem to extend their work to five or more parameters. In part I of this…
In a famous paper, R. A. Gordon proved a dozen theorems using tagged partitions and Cousin's theorem. The purpose of this paper is to present several classical results using the key-lemma underlying Cousin's theorem.
We study the Andrews-Gordon-Bressoud (AGB) generalisations of the Rogers-Ramanujan q-series identities in the context of cylindric partitions. We recall the definition of r-cylindric partitions, and provide a simple proof of Borodin's…
We derive relations between geometric means of the Fourier moduli of a refinable distribution and of a related polynomial. We use Pisot-Vijayaraghavan numbers to construct families of one dimension quasilattices and multiresolution analyses…
The partition function $p(n)$, which counts the number of partitions of a positive integer $n$, is widely studied. Here, we study partition functions $p_S(n)$ that count partitions of $n$ into distinct parts satisfying certain congruence…
We explore some connections between vectors of integers and integer partitions seen as bi-infinite words. This methodology enables us to give a combinatorial interpretation of the Macdonald identities for affine root systems of the seven…