Related papers: Cumulants as iterated integrals

Recently, Dil and Boyadzhiev \cite{AD2015} proved an explicit formula for the sum of multiple harmonic numbers whose indices are the sequence $\left( {{{\left\{ 0 \right\}}_r},1} \right)$. In this paper we show that the sums of multiple…

The cluster cumulant formula of Kubo is derived by appealing only to elementary properties of subsets and binomial coefficients. It is shown to be a binomial transform of the grand potential. Extensivity is proven without introducing…

Resultants are important special functions used in description of non-linear phenomena. Resultant $R_{r_1, ..., r_n}$ defines a condition of solvability for a system of $n$ homogeneous polynomials of degrees $r_1, ..., r_n$ in $n$…

An integer sequence that is defined by initial values and a linear recurrence with constant integer coefficients, can be represented by the difference of two arithmetic terms containing exponentiation. All constants occuring in the term are…

This paper gives new explicit formulas for sums of powers of integers and their reciprocals.

We consider Tuenter polynomials as linear combinations of descending factorials and show that coefficients of these linear combinations are expressed via a Catalan triangle of numbers. We also describe a triangle of coefficients in terms of…

In this paper, we consider representations of integers as sums of at most four distinct $m$-gonal numbers (allowing a fixed number of repeats of each polygonal number occurring in the sum). We show that the number of such representations…

We give an algorithm to compute the series expansion for the inverse of a given function. The algorithm is extremely easy to implement and gives the first $N$ terms of the series. We show several examples of its application in calculating…

We study an extension of the Distributive Full Non-associative Lambek Calculus with iterative division operators. The iterative operators can be seen as representing iterative composition of linguistic resources or of actions. A complete…

A solution is proposed for the problem of composition of ordinary generating functions. A new class of functions that provides a composition of ordinary generating functions is introduced; main theorems are presented; compositae are written…

We study how Boolean cumulants can be used in order to address operations with freely independent random variables, particularly in connection to the $*$-distribution of the product of two selfadjoint freely independent random variables,…

We consider annihilating random walks on the finite one-dimensional integer torus with deposition of pairs of particles, conditioned on an atypical jump activity. All cumulants of the activity, defined as the number of particle jumps up to…

This paper presents a reinterpretation of a second-order linear recurrence sequence as a sequence of continuants derived from the convergents to a continued fraction. As a result, we are able to derive the generating function and Binet…

We obtain a new decomposition of the Riemann-Liouville operators of fractional integration as a series involving derivatives (of integer order). The new formulas are valid for functions of class $C^n$, $n \in \mathbb{N}$, and allow us to…

We obtain exact formulas for the cumulative distribution function of the variance-gamma distribution, as infinite series involving the modified Bessel function of the second kind and the modified Lommel function of the first kind. From…

In this paper, we introduce a new method for calculating fractional integrals and differentials. The method involves an equation that we have obtained from infinite applied integration by parts. The equation works for special class of…

Using a recently derived integral in terms of elementary functions, we derive new asymptotic expansions of the normal inverse Gaussian cumulative distribution function. One of the asymptotic representations is in terms of the normal…

By the theory of elliptic curves, we study the integers representable as the product of the sum of four integers with the sum of their reciprocals and give a sufficient condition for the integers with a positive representation.

Using the notion of the composita, we obtain a method of solving iterative functional equations of the form $A^{2^n}(x)=F(x)$, where $F(x)=\sum_{n>0} f(n)x^n$, $f(1)\neq 0$. We prove that if $F(x)=\sum_{n>0} f(n)x^n$ has integer…

This paper offers a mathematical invention that shows how to convert integrated quantiles, which often appear in risk measures, into integrated cumulative distribution functions, which are technically more tractable from various…