Related papers: Number Identities and Integer Partitions

The aim of the paper is the proof of new identities for the constant in the Mertens product for arithmetic progressions. We deal with the problem of the numerical computation of these constants in another paper.

This manuscript introduces a general multisection identity expressed equivalently in terms of infinite double products and/or infinite double series, from which several new product or summation identities involving special functions…

Some finite series of harmonic numbers involving certain reciprocals are evaluated. Products of such reciprocals are expanded in a sum of the individual reciprocals, leading to a computer program. A list of examples is provided.

We prove a recursive identity involving formal iterated logarithms and formal iterated exponentials. These iterated logarithms and exponentials appear in a natural extension of the logarithmic formal calculus used in the study of…

Dirac delta function (delta-distribution) approach can be used as efficient method to derive identities for number series and their reciprocals. Applying this method, a simple proof for identity relating prime counting function…

We consider a special class of binomial sums involving harmonic numbers and we prove three identities by using the elementary method of the partial fraction decomposition. Some applications to infinite series and congruences are given.

We prove an identity about partitions with a very elementary formulation. We had previously conjectured this identity, encountered in the study of shifted Jack polynomials (math.CO/9901040). The proof given is using a trivariate generating…

A partition of a finite abelian group gives rise to a dual partition on the character group via the Fourier transform. Properties of the dual partitions are investigated and a convenient test is given for the case that the bidual partition…

The number of tuples with positive integers pairwise relatively prime to each other with product at most $n$ is considered. A generalization of $\mu^{2}$ where $\mu$ is the M\"{o}bius function is used to formulate this divisor sum and…

Let $B$ be an infinite subset of $\mathbf{N}$. When we consider partitions of natural numbers into elements of $B$, a partition number without a restriction of the number of equal parts can be expressed by partition numbers with a…

This paper has a two-fold purpose. First, by considering a reformulation of a deep theorem of G\"ollnitz, we obtain a new weighted partition identity involving the Rogers-Ramanujan partitions, namely, partitions into parts differing by at…

We derive an identity that relates a class of multiple integrals involving Vandermonde polynomials to divided differences. Alternatively the identity can be viewed as an integral formula for divided differences. As part of the derivation we…

It is shown how many of the partial theta function identities in Ramanujan's lost notebook can be generalized to infinite families of such identities. Key in our construction is the Bailey lemma and a new generalization of the Jacobi triple…

In this paper, we study various classes of partition functions such as those related to the parity of the number of parts, to differences of partition numbers, and to partitions with a repeated smallest part. We establish identities…

We exploit transformations relating generalized $q$-series, infinite products, sums over integer partitions, and continued fractions, to find partition-theoretic formulas to compute the values of constants such as $\pi$, and to connect sums…

In this paper, we use the Lambert series generating function for Euler's totient function to introduce a new identity for the number of $1$'s in the partitions of $n$. A new expansion for Euler's partition function $p(n)$ is derived in this…

This paper investigates some issues arising in categorical models of reversible logic and computation. Our claim is that the structural (coherence) isomorphisms of these categorical models, although generally overlooked, have decidedly…

The theory of partition congruences has been a fascinating and difficult subject for over a century now. In attempting to prove a given congruence family, multiple possible complications include the genus of the underlying modular curve,…

We prove a family of partition identities involving integer partitions in three colors. The conditions imposed on the types of partitions appearing in these identities involve constraints that arise in the Rogers-Ramanujan and…

Using a new graphical representation for partitions, the author obtains a family of partition identities associated with partitions into distinct parts of an arithmetic progression, or, more generally, with partitions into distinct parts of…