Related papers: Cauchy Pairs
We present a broader framework for the Cauchy identity derived from the determinant expansion of collocation matrices. This approach yields an infinite family of identities, where the original Cauchy identity stands as a particular case. To…
Everyone knows that the Euler characteristic of a combinatorial manifold is given by the alternating sum of its numbers of simplices. It is shown that there are other linear combinations of the numbers of simplices which are combinatorial…
Given a finite simplicial complex L and a collection of pairs of spaces indexed by its vertex set, one can define their polyhedral product. We record a simple formula for its Euler characteristic. In special cases the formula simplifies…
An interplay between the Lambert series and Euler's Pentagonal Number Theorem gives an Euler-type recurrence relation for any given arithmetical function. As consequences of this, we present Euler-type recurrence relations for some…
We lift to the multivariate Eulerian polynomials the identity implying that univariate Eulerian polynomials are palindromic. As a consequence of this generalization, we obtain nice combinatorial identities that can be directly extracted…
A new family of integer-valued Cauchy-type distributions is introduced, the {\it Cauchy-Cacoullos family}. The characteristic function is evaluated, showing some interesting distributional properties, similar to the ordinary (continuous)…
The polynomial relationship between elementary symmetric functions (Cauchy enumeration formula) is formulated via a ``raising operator" and Fock space construction. A simple graphical proof of this relation is proposed. The new operator…
Using an elementary approach involving the Euler Beta function and the binomial theorem, we derive two polynomial identities; one of which is a generalization of a known polynomial identity. Two well-known combinatorial identities, namely…
By means of the generating function method, a linear recurrence relation is explicitly resolved. The solution is expressed in terms of the Stirling numbers of both the first and the second kind. Two remarkable pairs of combinatorial…
We provide a natural interpretation of the secondary Euler characteristic and introduce higher Euler characteristics. For a compact oriented manifold of odd dimension, the secondary Euler characteristic recovers the Kervaire…
We give conditions for when two Euler products are the same given that they satisfy a functional equation and their coefficients are not too large and do not differ from each other by too much. Additionally, we prove a number of…
In this paper, several differentiability criteria for real functions of multiple variables in n-dimensional Euclidean space are considered. Simple and easy-to-use Cauchy-like criterion is formulated and proven. Relaxed sufficient conditions…
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…
A new class of functions with a unique identification minor is introduced: functions determined by content and singletons. Relationships between this class with other known classes of functions with a unique identification minor are…
We extend the theory of Euler integration from the class of constructible functions to that of "tame" real-valued functions (definable with respect to an o-minimal structure). The corresponding integral operator has some unusual defects (it…
In this note we will present how Euler's investigations on various different subjects lead to certain properties of the Legendre polynomials. More precisely, we will show that the generating function and the difference equation for the…
We generalize the notion of harmonic conjugate functions and Hilbert transforms to higher dimensional euclidean spaces, in the setting of differential forms and the Hodge-Dirac system. These conjugate functions are in general far from being…
A series of bilinear identities on the Schur symmetric functions is obtained with the use of Pluecker relations.
In this paper we introduce the notion of $\mathcal I^{\mathcal K}$-Cauchy function, where $\mathcal I$ and $\mathcal K$ are ideals on the same set. The $\mathcal I^{\mathcal K}$-Cauchy functions are a generalization of $\mathcal I^*$-Cauchy…
The two function theories of monogenic and of slice monogenic functions have been extensively studied in the literature and were developed independently; the relations between them, e.g. via Fueter mapping and Radon transform, have been…