Related papers: An alternative to Vaughan's identity

An identity by Ramanujan is expressed using polar coordinates, so that its proof reduces to the verification of an elementary trigonometric identity. This approach produces a few variations on Ramanujan's original identity.

Recently Zagier proved a remarkable q-series identity. We show that this identity can also be proved by modifying Franklin's classical proof of Euler's pentagonal number theorem.

This paper proposes learning disentangled but complementary face features with minimal supervision by face identification. Specifically, we construct an identity Distilling and Dispelling Autoencoder (D2AE) framework that adversarially…

Using a probabilistic approach, we derive several interesting identities involving beta functions. Our results generalize certain well-known combinatorial identities involving binomial coefficients and gamma functions.

In a recent paper, Caracciolo, Sokal and Sportiello presented, inter alia, an algebraic/combinatorial proof for Cayley's identity. The purpose of the present paper is to give a "purely combinatorial" proof for this identity; i.e., a proof…

We prove some extensions of Andrews inequality.

We propose a simple model which possesses the chain fountain effect.

Young's integral inequality is complemented with an upper bound to the remainder. The new inequality turns out to be equivalent to Young's inequality, and the cases in which the equality holds become particularly transparent in the new…

The paper presents a counterexample to the Hodge conjecture.

We present a new and useful congruence identity satisfied by m-permutable varieties.

We prove a number of results involving the kernel of the identity minus the monodromy on the vanishing cycles.

Motivated by some binomial coefficients identities encountered in our approach to the enumeration of convex polyominoes, we prove some more general identities of the same type, one of which turns out to be related to a strange evaluation of…

A representation of heterogeneous stochastic populations that are composed of sub-populations with different levels of distinguishability is introduced together with an analysis of its properties. It is demonstrated that any instance of…

In their recent book on combinatorial identities, Quaintance and Gould devoted one chapter to Melzak's identity. We give new proofs for this identity and its generalization.

Identifiers (IDs) are pervasive throughout our modern life. We suggest that these IDs would be easier to manage and remember if they were easily readable, spellable, and pronounceable. As a solution to this problem we propose using…

A combinatorial construction proves an identity for the product of the Pfaffian of a skew-symmetric matrix by the Pfaffian of one of its submatrices. Several applications of this identity are followed by a brief history of Pfaffians.

We derive an identity connecting any two second-order linear recurrence sequences having the same recurrence relation but whose initial terms may be different. Binomial and ordinary summation identities arising from the identity are…

An polynomial identity is derived from the representation V_m(x)\otimes V_n(y) of U_q(\hat {sl_2}) and a new basis of V_m(x)\otimes V_n(y) is established under some condition.

In investigating the properties of a certain class of homogeneous polynomials, we discovered an identity satisfied by their coefficients which involves simple 2F1 Gauss hypergeometric functions. This result appears to be new and we supply a…

Mendes recently conjectured an identity simplifying the Poincar\'e series of the space of equivariant polynomial maps from $\mathbb{R}^{n}$ to a subrepresentation of $Sym^{2}(\mathbb{R}^{n})$. We show how to prove this identity using a…