Related papers: Partitions related to positive definite binary qua…
In this paper we study generating functions resembling the rank of strongly unimodal sequences. We give combinatorial interpretations, identities in terms of mock modular forms, asymptotics, and a parity result. Our functions imitate a…
We present a new partition identity and give a combinatorial proof of our result. This generalizes a result of Andrew's in which he considers the generation function for partitions with respect to size, number of odd parts, and number of…
In this work, we construct a new Bailey pairs for the integral pentagon identity in terms of q-hypergeometric functions. The pentagon identity considered here represents equality of the partition functions of a certain three-dimensional…
We study cylindric partitions with two-element profiles using MacMahon's partition analysis. We find explicit formulas for the generating functions of the number of cylindric partitions by first finding the recurrences using partition…
In this article we raise some new questions about positive definite functions on free groups, and explain how these are related to more well-known questions. The article is intended as a survey of known results that also offers some new…
This is a compendium of generating functions involving single, double sums and definite integrals. These generating functions also involve special functions in both the summand function and closed form solution.
The relation between representations and positive definite functions is a key concept in harmonic analysis on topological groups. Recently this relation has been studied on topological groupoids. This is the second in a series of papers in…
The number of parts in the partitions (resp. distinct partitions) of $n$ with parts from a set were considered. Its generating functions were obtained. Consequently, we derive several recurrence identities for the following functions: the…
Positive definite forms $f$ which are sums of squares are constructed to have the additional property that the members of any collection of forms whose squares sum to $f$ must share a nontrivial complex root.
We investigate generalized quadratic forms with values in the set of rational integers over quadratic fields. We characterize the real quadratic fields which admit a positive definite binary generalized form of this type representing every…
The aim of this paper is to organize some known mass formulas arising from a definite central division algebra over a global field and to deduce some more new ones.
In this work, we offer a historical stroll through the vast topic of binary quadratic forms. We begin with a quick review of their history and then an overview of contemporary algebraic developments on the subject.
We construct a $k$-fold $q$-series as a generating function of $k$-regular partitions for each positive integer $k$. The $k=1$ case is one of Euler's $q$-series identities pertaining to the partitions into distinct parts. The construction…
We investigate factorizability of a quadratic split quaternion polynomial. In addition to inequality conditions for existence of such factorization, we provide lucid geometric interpretations in the projective space over the split…
In this paper we determine the group of rational automorphisms of binary cubic and quartic forms with integer coefficients and non-zero discriminant in terms of certain quadratic covariants of cubic and quartic forms. This allows one to…
A classical method for partition generating functions is developed into a tool with wide applications. New expansions of well-known theorems are derived, and new results for partitions with n copies of n are presented.
We obtain some Bailey pairs associated with indefinite quadratic forms with the $\beta_n$ connected to a finite sum. A new general identity is given, which provides identities for $q$-hypergeometric series, including mock theta functions.
New identities and congruences involving the ranks and cranks of partitions are proved. The proof depends on a new partial differential equation connecting their generating functions.
The use of quadratic residues to construct matrices with specific determinant values is a familiar problem with connections to many areas of mathematics and statistics. Our research has focused on using cubic residues to construct matrices…
We define the category of partitioned binary relations and show that it contains many classical diagram categories, including categories of binary relations, maps, injective maps, partitions, (oriented) Brauer diagrams and (oriented)…