Related papers: A binomial formula for evaluating integrals
The main objective of this paper is to evaluate six new Ap\'ery-like series of weight $5$ in closed form. These series involve harmonic numbers and exhibit the characteristic reciprocal central binomial coefficient structure. Generating…
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…
The calculation of one loop integrals at finite temperature requires the evaluation of certain series, which converge very slowly or can even be divergent. Here we review a new method, recently devised by the author, for obtaining…
Given two infinite sequences with known binomial transforms, we compute the binomial transform of the product sequence. Various identities are obtained and numerous examples are given involving sequences of special numbers: Harmonic…
Using appropriate power series evaluations, we determine all moments of arbitrary positive powers of the arcsine. As consequences we evaluate several doubly infinite classes of power series involving central binomial coefficients and…
In this paper I give an evaluation of a functional integral by means of a series in functional derivatives, first of all we propose a differential equation of first order and solve it by iterative methods, to obtain a series for the…
We study inverse factorial series and their relation to Stirling numbers of the first kind. We prove a special representation of the polylogarithm function in terms of series with such numbers. Using various identities for Stirling numbers…
We present a method using contour integration to derive definite integrals and their associated infinite sums which can be expressed as a special function. We give a proof of the basic equation and some examples of the method. The advantage…
We consider here a particular quadratic equation linking two elements of a C-Algebra. By analysing powers of the unknowns, it appears a double sequence of polynomials related to classical Bernoulli polynomials. We get the generating…
This paper describes algorithms to deal with nested symbolic sums over combinations of harmonic series, binomial coefficients and denominators. In addition it treats Mellin transforms and the inverse Mellin transformation for functions that…
In this note a general result is proved that can be used to evaluate exactly a class of highly oscillatory integrals.
Power series in which the summand satisfies a linear recurrence relation with polynomial coefficients are shown to be the solution of a linear differential or algebraic equation. Solving the associated differential or algebraic equation…
An integral formula is developed which applies to an essentially arbitrary function. An application is made to the Riemann zeta function.
A general integral expression to transform power series is applied to $\arcsin{x}$ and its positive integer powers. We concentrate on the first to the fourth powers and obtain infinite classes of new power series involving central binomial…
In this paper, we study some Euler-Ap\'ery-type series which involve central binomial coefficients and (generalized) harmonic numbers. In particular, we establish elegant explicit formulas of some series by iterated integrals and…
Splitting methods for the numerical integration of differential equations of order greater than two involve necessarily negative coefficients. This order barrier can be overcome by considering complex coefficients with positive real part.…
We evaluate several arctangent and logarithmic integrals depending on a parameter. This provides a closed form summation of certain series and also gives integral and series representation of some classical constants.
An analytical approach to convolution of functions, which appear in perturbative calculations, is discussed. An extended list of integrals is presented.
In this paper we investigate the properties of the Euler functions. By using the Fourier transform for the Euler function, we derive the interesting formula related to the infinite series. Finally we give some interesting identities between…
In this note, we develop transformation formulae and expansions for the log tangent integral, which are then used to derive series acceleration formulae for certain values of Dirichlet L-functions, such as Catalan's constant. The formulae…