Related papers: An identity for derivatives

We prove a closed formula for the derivative, of any order, of a implicit function, in terms of some binomial building blocks, and explain the combinatorics behind the coefficients appearing in the formula.

Higher order derivatives of functions are structured high dimensional objects which lend themselves to many alternative representations, with the most popular being multi-index, matrix and tensor representations. The choice between them…

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…

The aim of this paper is to prove characterization theorems for higher order derivations. Among others we prove that the system defining higher order derivations is stable. Further characterization theorems in the spirit of N.~G.~de Bruijn…

An technically interesting proof of a known theorem.

We prove a uniqueness theorem for an entire function, which shares certain values with its higher order derivatives.

We propose an equivalent formula for the higher-order derivatives used in the study of Generalized Almost Perfect Nonlinear functions over an arbitrary finite field of characteristic $p$. The result is obtained by counting the number of…

We prove a new universal identity for umbral operators. This motivates the definition of a subclass satisfying a simplified identity, which we fully characterize. The results are illustrated with common examples of the theory of umbral…

This paper presents an algebraic approach to characterizing higher-order differential operators. While the foundational Leibniz rule addresses first-order derivatives, its extension to higher orders typically involves identities relating…

A simple heuristic proof of an integral identity recently derived (Glasser ML 2011 J. Phys. A: Math. Theor. 44 225202) is presented.

We prove some new results related to Tanaka's formula.

We establish a simple identity and using it we find a new proof of a result of Kloosterman.

A family of general integral identities is derived and several applications of physical interest are presented

We derive an identity that relates a class of multiple integrals involving Vandermonde polynomials to divided differences. Alternatively the identity can be viewed as an integral formula for divided differences. As part of the derivation we…

Motivated by previous work leveraging factorizations of second- and fourth-order differential operators, a general integral inequality involving higher order derivatives is proven by elementary means. It is then shown how this framework…

Based on the Liouville-Weyl definition of the fractional derivative, a new direct fractional generalization of higher order derivatives is presented. It is shown, that the Riesz and Feller derivatives are special cases of this approach.

In this paper, a new identity for convex functions is derived. A consequence of the identity is that we can derive new estimates for the remainder term of the midpoint, trapezoid, and Simpson formulae for functions whose derivatives in…

We present theory for general partial derivatives of matrix functions on the form $f(A(x))$ where $A(x)$ is a matrix path of several variables ($x=(x_1,\dots,x_j)$). Building on results by Mathias [SIAM J. Matrix Anal. Appl., 17 (1996), pp.…

In this paper, we give a short proof of a relation generalizing many identities for Bernoulli numbers.

Leibniz's rule for the $n$-th derivative of a product is a very well known and extremely useful formula. In this article, we introduce an analogous explicit formula for the $n$-th derivative of a quotient of two functions. Later, we use…