Related papers: On AZ-style identity
Schur's inequality for the sum of products of the differences of real numbers states that for $x,y,z,t\geq 0$, $x^t(x-y)(x-z) + y^t(y-z)(y-x) + z^t(z-x)(z-y) \geq 0$. In this paper we study a generalization of this inequality to more terms,…
Several combinatorial identities are presented, involving Stirling functions of the second kind with a complex variable. The identities involve also Stirling numbers of the first kind, binomial coefficients and harmonic numbers.
Leibniz algebras are certain generalization of Lie algebras. It is natural to generalize concepts in Lie algebras to Leibniz algebras and investigate whether the corresponding results still hold. In this paper we introduce the notion of…
Large language models (LLMs) remain broadly open and highly steerable: they imitate at scale, accept arbitrary system prompts, and readily adopt multiple personae. By analogy to human development, we hypothesize that progress toward…
We present and prove a general form of Vandermonde's identity and use it as an alternative solution to a classic probability problem.
In this article we will build a universal imbedding of a regular Hom- Lie triple system into a Lie algebra and show that the category of regular Hom-Lie triple systems is equivalent to a full subcategory of pairs of…
In this article, a short combinatorial proof of the Capelli's identity is given. It also leads to an easy proof of the Capelli--Cauchy--Binet identity, a more general form of Capelli's identity. With the technique introduced, the Turnbull's…
Let $A$ and $B$ be finite-dimensional simple algebras with arbitrary signature over an algebraically closed field. Suppose $A$ and $B$ are graded by a semigroup $S$ so that the graded identitical relations of $A$ are the same as those of…
We describe a bilinear identity satisfied by certain multidimensional q-hypergeometric integrals. The identity can be considered as a deformation of the Riemann bilinear relation for the twisted de Rham (co)homologies. The identity also…
We give common generalizations of the Menon-type identities by Sivaramakrishnan (1969) and Li, Kim, Qiao (2019). Our general identities involve arithmetic functions of several variables, and also contain, as special cases, identities for…
We prove, by the WZ-method, some hypergeometric identities which relate ten extended Ramanujan type series to simpler hypergeometric series. The identities we are going to prove are valid for all the values of a parameter $a$ when they are…
In this paper, an integral identity for twice differentiable functions is generalized. Then, by using convexity of |f''| or q-th power of |f''| and with the aid of power mean and Holder's inequalities we achieved some new results. We also…
The purpose of this paper is to provide a random version of Simons' inequality.
We define the notion of an almost polynomial identity of an associative algebra $R$, and show that its existence implies the existence of an actual polynomial identity of $R$. A similar result is also obtained for Lie algebras and Jordan…
In this article, we obtain two interesting general inequalities concerning Riemman sums of convex functions, which in particular, sharpen Alzer's inequality and give a suitable converse for it.
A generalization of the H\"older inequality is considered. Its relations with a previously obtained improvement of the Cauchy--Schwarz inequality are discussed.
Recently the second named author discovered a combinatorial identity in the context of vertex representations of quantum Kac-Moody algebras. We give a direct and elementary proof of this identity. Our method is to show a related identity of…
A key value proposition of machine learning is generalizability: the same methods and model architecture should be able to work across different domains and different contexts. While powerful, this generalization can sometimes go too far,…
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.
In this article we prove an algebraic identity which significantly generalizes the formula for sum of powers of consecutive integers involving Stirling numbers of the second kind. Also we have obtained a generalization of Newton-Girard…