Related papers: Cauchy-type identities through collocation matrice…
New family of extended Cauchy type identities is found and related Fermat type matrices are provided ready for applications in extended scope. This is achieved due to the use specifically non-commuting variables of extended finite operator…
Some identities that involve the elliptic version of the Cauchy matrices are presented and proved. They include the determinant formula, the formula for the inverse matrix, the matrix product identity and the factorization formula.
A new integral representation is derived using a definite integral given by Cauchy and used to evaluate a number of integrals containing the finite series of special functions.
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…
The Capelli identities claim $det(A)det(B) = det(AB+correction)$ for certain matrices with noncommutative entries. They have applications in representation theory and integrable systems. We propose new examples of these identities,…
We introduce poly-Cauchy permutations that are enumerated by the poly-Cauchy numbers. We provide combinatorial proofs for several identities involving poly-Cauchy numbers and some of their generalizations. The aim of this work is to…
In this series of articles we study connections between combinatorics of multidimensional generalizations of Cauchy identity and continuous objects such as multidimensional Brownian motions and Brownian bridges. In Part I of the series we…
The celebrated Cauchy identity expresses the product of terms $(1 - x_i y_j)^{-1}$ for $(i,j)$ indexing entries of a rectangular $m\times n$-matrix as a sum over partitions $\lambda$ of products of Schur polynomials:…
A family of general integral identities is derived and several applications of physical interest are presented
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.…
We give a short proof of the Cauchy-Binet determinantal formula using multilinear algebra by first generalizing it to an identity {\em not} involving determinants. By extending the formula to abstract Hilbert spaces we obtain, as a…
We present identities for permutations with fixed points. The formulas are based on successive derivations or integrations of the determinant of a particular matrix.
In this paper, we clarified the relationship between continued fractions, determinants, and identities, making it easier to apply these methods systematically in other settings. In particular, we studied finite continued fractions from the…
We introduce a natural generalization of Maya diagrams -- the space of infinite Fibonacci configurations, which are specified functions on $\mathbb{Z}$ with values $1$ and $0$. Infinite Fibonacci configurations are particularly interesting…
In this paper we present a new identity and some of its variants which can be used for finding solutions while solving fractional infinite and finite series. We introduce another simple identity which is capable of generating solutions for…
We present an analogue of the differential calculus in which the role of polynomials is played by certain ordered sets and trees. Our combinatorial calculus has all nice features of the usual calculus and has an advantage that the elements…
The notion of pairable functions is introduced and some of its properties are developed. In this connection the famous Euler identity is interpreted as a property of certain pairable functions and finite cyclic groups.
Motivated by statistical applications, this paper introduces Cauchy identities for characters of the compact classical groups. These identities generalize the well-known Cauchy identity for characters of the unitary group, which are Schur…
We prove an interesting identity for the sum of determinants, which is a generalization of the sum of a geometric progression. The proof is quite long and a number of other identities are proved along the way. Some of the more elementary…
We introduce the concept of indexed identity, where the usual notion of identity is a particular case. Our mathematical framework allows us a generalized method for `indexing' predicates, which corresponds to `fuzzification' of properties,…