相关论文: On Basmajian's identities, and other stories

In this survey, we discuss four classes of identities due principally to Basmajian, McShane, Bridgeman-Kahn and Luo-Tan on hyperbolic manifolds and provide a unified approach for proving them. We also elucidate on the connections between…

We give a simple geometric argument to derive in a common manner orthospectrum identities of Basmajian and Bridgeman. Our method also considerably simplifies the determination of the summands in these identities. For example, for every odd…

For suitable metrics on the locally symmetric space associated to a maximal representation, we prove inequalities between the length of the boundary and the lengths of orthogeodesics that generalize the classical Basmajian's identity from…

In this paper we study McShane's identity in real and complex hyperbolic spaces and obtain various generalizations of the identity for representations of surface groups into the isometry groups of rank one symmetric spaces. Our methods…

We give a simple unified proof for all existing rational hypergeometric Ramanujan identities for $1/\pi$, and give a complete survey (without proof) of several generalizations: rational hypergeometric identities for $1/\pi^c$, Taylor…

See math.CV/0509030 which replaces this paper.

The lengths of geodesics on hyperbolic surfaces satisfy intriguing equations, known as identities, relating these lengths to geometric quantities of the surface. This paper is about a large family of identities that relate lengths of closed…

We introduce several new identities combining basic hypergeometric sums and integrals. Such identities appear in the context of superconformal index computations for three-dimensional supersymmetric dual theories. We give both analytic…

We prove an extension of Basmajian's identity to $n$-Hitchin representations of compact bordered surfaces. For $n=3$, we show that this identity has a geometric interpretation for convex real projective structures analogous to Basmajian's…

We describe a hyperbolic version of the Ambartzumian-Pleijel identity. We use this identity to prove the hyperbolic Crofton formula and the hyperbolic isoperimetric inequality. This identity also provides a way to compute the chord length…

The interaction between combinatorics and algebraic and differential geometry is very strong. While researching a problem of Hessian topology, we came across a series of identities of binomial coefficients, which are useful for proving a…

We give an account of some recent development that connects the concept of mass in general relativity to the geometry of large Riemannian polyhedra, in the setting of both asymptotically flat and asymptotically hyperbolic manifolds.

This paper describes a method to find a connection between combinatorial identities and hypergeometric series with a number of examples. Combinatorial identities can often be written as hypergeometric series with unit argument. In a number…

Let $\Sigma$ be a connected compact oriented surface with boundary and negative Euler characteristic. Let $k$ be a non-Archimedean local field. In this paper, we prove Basmajian's identity for projective Anosov representations $\rho \colon…

Several new identities for elliptic hypergeometric series are proved. Remarkably, some of these are elliptic analogues of identities for basic hypergeometric series that are balanced but not very-well-poised.

This part is a continuation of the Part I where we built resolutions of identity for certain non-Hermitian Hamiltonians constructed of biorthogonal sets of their eigen- and associated functions for the spectral problem defined on entire…

We present a different combinatorial interpretations of Lucas and Gibonacci numbers. Using these interpretations we prove several new identities, and simplify the proofs of several known identities. Some open problems are discussed towards…

We study two aspects of Hecke symmetry in this note: first, we conjecture a generalization of the Ramanujan identities to the case of automorphic forms of Hecke groups; second, we conjecture a generalization of an inversion formula from the…

Given two combinatorial identities proved earlier, a new set of variations of these combinatorial identities is listed and proved with the integral representation method. Some identities from literature are shown to be special cases of…

In this note, we present two new identities for derangements. As a corollary, we have a combinatorial proof of the irreducibility of the standard representation of symmetric groups.