Related papers: A binomial formula for evaluating integrals

We give a formula for the inverse matrix to an infinite matrix with possibly noncommutative entries, generalizing the Newton interpolation formula and the Taylor formula.

We investigate the special class of formulas made up of arbitrary but finite com- binations of addition, multiplication, and exponentiation gates. The inputs to these formulas are restricted to the integral unit 1. In connection with such…

We give operational formulae of Burchnall type involving complex Hermite polynomials. Short proofs of some known formulae are given and new results involving these polynomials, including Nielsen's identities and Runge addition formula, are…

We give systematic method to evaluate a large class of one-dimensional integral relating to multiple zeta values (MZV) and colored MZV. We also apply the technique of iterated integrals and regularization to elucidate the nature of some…

In this research, the Bernoulli polynomials are introduced. The properties of these polynomials are employed to construct the operational matrices of integration together with the derivative and product. These properties are then utilized…

An index transform, involving the square of Whittaker's function is introduced and investigated. The corresponding inversion formula is established. Particular cases cover index transforms of the Lebedev type with products of the modified…

We propose a novel way to study numerical methods for ordinary differential equations in one dimension via the notion of multi-indice. The main idea is to replace rooted trees in Butcher's B-series by multi-indices. The latter were…

Multiple Mellin-Barnes integrals are often used for perturbative calculations in particle physics. In this context, the evaluation of such objects may be performed through residues calculations which lead to their expression as multiple…

A short pedagogical introduction to a differential method used to calculate multi-loop scalar integrals is presented. As an example it is shown how to obtain, using the method, large mass expansion of the two loop sunrise master integrals.

We develop a method for calculating Riemann sums using Fourier analysis.

In this note we augment the poly-Bernoulli family with two new combinatorial objects. We derive formulas for the relatives of the poly-Bernoulli numbers using the appropriate variations of combinatorial interpretations. Our goal is to show…

In this paper, we exploit the r-Stirling numbers of both kinds in order to give explicit formulae for the values of the high order Bernoulli numbers and polynomials of both kinds at integers. We give also some identities linked the…

We study the properties of different type of transforms by means of operational methods and discuss the relevant interplay with many families of special functions. We consider in particular the binomial transform and its generalizations. A…

In this paper, we study the formulae for a product of two product Euler polynomials. From this study, we derive some formulae for the integral of the product of two or more Ruler polynomials.

We give an explicit formula for the Hankel transform of a regular sequence in terms of the coefficients of the associated orthogonal polynomials and the sequence itself. We apply this formula to some sequences of combinatorial interest,…

In this paper, we present a novel method to compute an explicit formula for the inverse of the confluent Vandermonde matrices. Our proposed results may have many interesting perspectives in diverse areas of mathematics and natural sciences,…

We survey various classical results on invariants of polynomials, or equivalently, of binary forms, focussing on explicit calculations for invariants of polynomials of degrees 2, 3, 4.

While teaching a course on integral equations, I noticed that a straightforward combination of Neumann series and Fourier series for the resolvent (or the solution) of an integral equation has good approximation qualities. This short…

We present an extension of a previously developed method employing the formalism of the fractional derivatives to solve new classes of integral equations. This method uses different forms of integral operators that generalizes the…

We analyze the conditions on the Taylor coefficients of an analytic function to admit global analytic continuation, complementing a recent paper of Breuer and Simon on general conditions for natural boundaries to form. A new summation…