Related papers: Beyond the Gaussian II : A Mathematical Experiment
Computational approaches to finding non-trivial integer solutions of the equation in the title are discussed. We summarize previous work and provide several new solutions.
The classical quadratic formula and some of its lesser known variants for solving the quadratic equation are reviewed. Then, a new formula for the roots of a quadratic polynomial is presented.
In the first part, by the first author's work of 1972, an integral representation for an ultraspherical polynomial of higher index in terms of one of lower index and an infinite series was obtained. While this representation works well from…
The Hermite polynomials are ubiquitous but can be difficult to work with due to their unwieldy definition in terms of derivatives. To remedy this, we showcase an underappreciated Gaussian integral formula for the Hermite polynomials, which…
A counter-intuitive result of Gauss (formulae (1.6), (1.7) below) is made less mysterious by virtue of being generalized through the introduction of an additional parameter.
Comment of the very interesting paper by Hilhorst & Schehr, J. Stat. Mech. P06003 (2007). The main point is that one should be extremely careful when interpreting non-Gaussian data in terms of q-Gaussians.
Up-down permutations are counted by tangent resp. secant numbers. Considering words instead, where the letters are produced by independent geometric distributions, there are several ways of introducing this concept; in the limit they all…
The Fundamental Theorem of Integral Calculus links the integrand and its antiderivative via a simple first order differential equation. A numerical solution of this ode yields the antiderivative and hence the required integral. This…
Highly oscillatory integrals of composite type arise in electronic engineering and their calculations is a challenging problem. In this paper, we propose two Gaussian quadrature rules for computing such integrals. The first one is…
This paper provides a technique for evaluating some nonlinear Gaussian sums in closed forms. The evaluation is obtained from the known values of simpler exponential sums.
In this paper we continue our study, begun in part I, of the exceptional set of integers, not restricted by elementary congruence conditions, which cannot be represented as sums of three or four squares of primes. We correct a serious…
In this paper, we obtain some formulas for double nonlinear Euler sums involving harmonic numbers and alternating harmonic numbers. By using these formulas, we give new closed form sums of several quadratic Euler series through Riemann zeta…
Many statistical models are algebraic in that they are defined by polynomial constraints or by parameterizations that are polynomial or rational maps. This opens the door for tools from computational algebraic geometry. These tools can be…
We prove some special cases of Bergeron's inequality involving two Gaussian polynomials (or $q$-binomials).
These notes are intended as an introduction to a study of applications of noncommutative calculus to quantum statistical Physics. Centered on noncommutative calculus we describe the physical concepts and mathematical structures appearing in…
We consider the ratio of two Gauss hypergeometric functions, in which the parameters of the numerator function differ from the respective parameters of the denominator function by integers. We derive explicit integral representations for…
This survey paper discusses the history of approximation formulas for n-th order derivatives by integrals involving orthogonal polynomials. There is a large but rather disconnected corpus of literature on such formulas. We give some results…
Statistical inference more often than not involves models which are non-linear in the parameters thus leading to non-Gaussian posteriors. Many computational and analytical tools exist that can deal with non-Gaussian distributions, and…
Using the diagrammatic approach to integrals over Gaussian random matrices, we find a representation for polynomial Lie group integrals as infinite sums over certain maps on surfaces. The maps involved satisfy a specific condition: they…
We consider mixtures of $k\geq 2$ Gaussian components with unknown means and unknown covariance (identical for all components) that are well-separated, i.e., distinct components have statistical overlap at most $k^{-C}$ for a large enough…