Related papers: Series Crimes
In this thesis we show that the partial sums of the Maclaurin series for a certain class of entire functions possess scaling limits in various directions in the complex plane. In doing so we obtain information about the zeros of the partial…
Factorial series played a major role in Stirling's classic book "Methodus Differentialis" (1730), but now only a few specialists still use them. This article wants to show that this neglect is unjustified, and that factorial series are…
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…
Monte Carlo simulations of diffusion processes often introduce bias in the final result, due to time discretization. Using an auxiliary Poisson process, it is possible to run simulations which are unbiased. In this article, we propose such…
Fractional calculus is the calculus of differentiation and integration of non-integer orders. In a recently paper (Annals of Physics 323 (2008) 2756-2778), the Fundamental Theorem of Fractional Calculus is highlighted. Based on this…
We estimate the truncated double trigonometric series $\sum_{n=0}^{N}\sum_{m=0}^{M}a_{mn} {e}^{2\pi \imath \left(m x+n y\right)}$, $a_{mn} \in\mathbb{C}$, in Lebesgue spaces with mixed norms in terms of the $p^{th}-q^{th}$ power finite…
Many of the fastest known algorithms to compute $\pi$ involve generalized hypergeometric series, such as the Ramanujan-Sato series. In this paper, we investigate the rates of convergence for several such series and we give asymptotic…
An Engel series is a sum of reciprocals of a non-decreasing sequence $(x_n)$ of positive integers, which is such that each term is divisible by the previous one, and a Pierce series is an alternating sum of the reciprocals of a sequence…
Holonomic functions play an essential role in Computer Algebra since they allow the application of many symbolic algorithms. Among all algorithmic attempts to find formulas for power series, the holonomic property remains the most important…
For a regular chain $R$, we propose an algorithm which computes the (non-trivial) limit points of the quasi-component of $R$, that is, the set $\bar{W(R)} \setminus W(R)$. Our procedure relies on Puiseux series expansions and does not…
The vast use of computers on scientific numerical computation makes the awareness of the limited precision that these machines are able to provide us an essential matter. A limited and insufficient precision allied to the truncation and…
The problem of extrapolating the series in powers of small variables to the region of large variables is addressed. Such a problem is typical of quantum theory and statistical physics. A method of extrapolation is developed based on…
Power series representations for special functions are computationally satisfactory only in the vicinity of the expansion point. Thus, it is an obvious idea to use instead Pad\'{e} approximants or other rational functions constructed from…
A novel method of summation for power series is developed. The method is based on the self-similar approximation theory. The trick employed is in transforming, first, a series expansion into a product expansion and in applying the…
We consider the linear combinations of elements of two sequences: the first one a priory given nonnegative sequence and the second random sequence from the unit interval. We investigate the expected value of the smallest natural number such…
We study formal power series which can be interpreted as interpolations of Fibonacci and Lucas polynomials with even (or odd) indices.
We discuss certain special cases of algebraic approximants that are given as zeroes of so-called "effective characteristic polynomials" and their generalization to a multiseries setting. These approximants are useful for the convergence…
Functions that are smooth but non-periodic on a certain interval possess Fourier series that lack uniform convergence and suffer from the Gibbs phenomenon. However, they can be represented accurately by a Fourier series that is periodic on…
Asymptotic expansions of series $\sum_{k=0}^\infty \epsilon^k(k+a)^\gamma e^{-(k+a)^\alpha x}$ and $\sum_{k=0}^\infty \epsilon^k(k+a)^\gamma / (x(k+a)^\alpha+1)^\mu}$ in powers of $x$ as $x\to+0$ are found, where $\epsilon=1$ or…
Recent results in the theory and application of Newton-Puiseux expansions, i.e. fractional power series solutions of equations, suggest further developments within a more abstract algebraic-geometric framework, involving in particular the…