Related papers: A new algorithm for computing the multivariate Fa\…
We present a new variant of the Faa di Bruno formula with a simpler summation order.
We provide a novel representation of the total n-th derivative of the multivariate composite function $f \circ g$, i.e. a generalized Fa\`a di Bruno's formula. To this end, we make use of properties of the Kronecker product and the n-th…
How do we take repeated derivatives of composed multivariate functions? for one-dimensional functions, the common tools consist of the Fa\'a di Bruno formula with Bell polynomials; while there are extensions of the Fa\'a di Bruno formula,…
High-order derivatives of nested functions of a single variable can be computed with the celebrated Fa\`a di Bruno's formula. Although generalizations of such formula to multiple variables exist, their combinatorial nature generates an…
We propose new algorithms for generating $k$-statistics, multivariate $k$-statistics, polykays and multivariate polykays. The resulting computational times are very fast compared with procedures existing in the literature. Such speeding up…
Fa\`a di Bruno's formula gives an expression for the derivatives of the composition of two real-valued functions. In this paper we prove a multivariate and synthesized version of Fa\`a di Bruno's formula in higher dimensions, providing a…
We extend the multivariate Fa\`{a} di Bruno formula to the super case, where anticommuting odd coordinates are considered. The formula takes the same form as the classical case but contains some nontrivial signs, which essentially measure…
We give a one-sentence elementary proof of the combinatorial Fa\`a di Bruno's formula.
We give yet another proof for Fa\`{a} di Bruno's formula for higher derivatives of composite functions. Our proof technique relies on reinterpreting the composition of two power series as the generating function for weighted integer…
In this work, we derive numerous identities for multivariate q-Euler polynomials by using umbral calculus.
We introduce and compare three different Monte Carlo determinantal algorithms that allow one to compute dynamical quantities, such as the self-energy, of fermionic systems in their thermodynamic limit. We show that the most efficient…
In these lectures we present five interpretations of the Fa' di Bruno formula which computes the n-th derivative of the composition of two functions of one variable: in terms of groups, Lie algebras and Hopf algebras, in combinatorics and…
This paper introduces a simple and computationally efficient algorithm for conversion formulae between moments and cumulants. The algorithm provides just one formula for classical, boolean and free cumulants. This is realized by using a…
We present a new formula for umbral operators that yields three main insights. First, it makes explicit a connection between umbral calculus and iteration theory. Second, it leads naturally to a definition of fractional exponents of umbral…
A novel algorithm for computing the action of a matrix exponential over a vector is proposed. The algorithm is based on a multilevel Monte Carlo method, and the vector solution is computed probabilistically generating suitable random paths…
kStatistics is a package in R that serves as a unified framework for estimating univariate and multivariate cumulants as well as products of univariate and multivariate cumulants of a random sample, using unbiased estimators with minimum…
In this paper we use the viewpoint of the formal calculus underlying vertex operator algebra theory to study certain aspects of the classical umbral calculus and we introduce and study certain operators generalizing the classical umbral…
A very simple closed-form formula for Sheppard's corrections is recovered by means of the classical umbral calculus. By means of this symbolic method, a more general closed-form formula for discrete parent distributions is provided and the…
This thesis is intended to provide an account of the theory and applications of Operational Methods that allow the "translation" of the theory of special functions and polynomials into a "different" mathematical language. The language we…
We consider the problem of estimating the expected outcomes of Monte Carlo processes whose outputs are described by multidimensional random variables. We tightly characterize the quantum query complexity of this problem for various choices…