Related papers: Jacobi's Identity and Synchronized Partitions
In this note, we will give a short proof of an identity for cubic partitions.
Making use of the theory of infinitesimal canonical transformations, a concise proof is given of Jacobi's identity for Poisson brackets.
We introduce two remarkable identities written in terms of single commutators and anticommutators for any three elements of arbitrary associative algebra. One is a consequence of other (fundamental identity). From the fundamental identity,…
Recently Andrews and Bachraoui proved identities relating certain restricted partitions into distinct even parts with restricted 4-regular partitions by the theory of basic hypergeometric series. They also posed a question regarding…
An identity is proved connecting two finite sums of inverse tangents. This identity is discretized version of Jacobi's imaginary transformation for the modular angle from the theory of elliptic functions. Some other related identities are…
We present a constructive proof of Jacobi's identity for the sum of two squares. We present a combinatorial proof of the Jacobi Triple Product and combine with a proof of Hirschhorn to define an algorithm. The input is a factorization…
We prove an identity about partitions, previously conjectured in the study of shifted Jack polynomials (math.CO/9903020). The proof given is using $\lambda$-ring techniques. It would be interesting to obtain a bijective proof.
We translate Uchimura's identity for the divisor function and whose generalizations into combinatorics of partitions, and give a combinatorial proof of them. As a by-product of their proofs, we obtain some combinatorial results.
The Jacobi identities play an important role in constructing the explicit exact solutions of a broad class of integrable systems in soliton theory. In the paper, a direct and simple proof of the Jacobi identities for determinants is…
Recently, Andrews and EI Bachraoui discovered several companions for some famous $q$-series formulas, and derived some new identities involving partitions and overpartitions with distinct parts. In this paper, we shall refine their results…
In this paper we present a systematic way to describe exceptional Jacobi polynomials via two partitions. We give the construction of these polynomials and restate the known aspects of these polynomials in terms of their partitions. The aim…
On the basis of the generalizations of the Jacobi identity found by the author some identities satisfied by the curvature and torsion of a covariant differentiation are derived. A kind of the generalized covariant differentiation is…
Partitions with distinct even parts have long been the subject of extensive research. In this paper, We present some new perspectives on such partitions from a combinatorial viewpoint, and connect them with signed partitions and bicolored…
In this paper we present our variant of quantum antisymmetry and quantum Jacobi identity.
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…
It is demonstrated that the knowledge of a single and arbitrary solution of the three-dimension\-al Jacobi equations allows determining infinite families of new solutions, which are generally and explicitly constructed in what follows.…
In this note, we show how a combinatorial identity of Frisch can be applied to prove and generalize some well-known identities involving harmonic numbers. We also present some combinatorial identities involving odd harmonic numbers which…
In his paper, "On a Partition Function of Richard Stanley," George Andrews proves a certain partition identity analytically and asks for a combinatorial proof. This paper provides the requested combinatorial proof.
In this brief note a straightforward combinatorial proof for an identity directly connecting rooted forests and unordered set partitions is provided. Furthermore, references that put this type of identity in the context of forest volumes…
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…