相关论文: An Ap\'ery-like difference equation for Catalan's …
In recent decades, a growing number of discoveries in fields of mathematics have been assisted by computer algorithms, primarily for exploring large parameter spaces that humans would take too long to investigate. As computers and…
We first give a geometric construction of a 2-dimensional mixed motive over $\mathbb{Q}$ with the Catalan constant $\mathbf{G}=1-1/3^2+1/5^2-1/7^2+\cdots$ as a period. We then use this motive to obtain a supply of linear forms in 1 and…
We present several second-order linear differential equations that are associated to a particular Riccati equation with only one constant parameter in its coefficients through the technique of supersymmetric factorizations and through a…
We describe a simple method that produces automatically closed forms for the coefficients of continued fractions expansions of a large number of special functions. The function is specified by a non-linear differential equation and initial…
Fundamental mathematical constants like $e$ and $\pi$ are ubiquitous in diverse fields of science, from abstract mathematics to physics, biology and chemistry. For centuries, new formulas relating fundamental constants have been scarce and…
A new class of alternating convolutions concerning binomial coefficients and Catalan numbers are evaluated in closed forms.
We give an algebraic method to compute the fourth power of the quotient of any even theta constants associated to a given non-hyperelliptic curve in terms of geometry of the curve. In order to apply the method, we work out non-hyperelliptic…
We present a hypergeometric construction of rational approximations to $\zeta(2)$ and $\zeta(3)$ which allows one to demonstrate simultaneously the irrationality of each of the zeta values, as well as to estimate from below certain linear…
Creative telescoping algorithms compute linear differential equations satisfied by multiple integrals with parameters. We describe a precise and elementary algorithmic version of the Griffiths-Dwork method for the creative telescoping of…
A fast algorithm (linear in the degrees of freedom) for the solution of linear variable-coefficient rational-order fractional integral and differential equations is described. The approach is related to the ultraspherical method for…
In this paper, we develop new identities for the inverse tangent integral by connecting it to the dilogarithmic (polylogarithmic) structure and to a carefully designed auxiliary arctangent integral $Ti_2(a)$ with a tunable endpoint. The…
We explain in detail how to accelerate continued fractions (for constants as well as for functions) using the method used by R.~Ap\'ery in his proof of the irrationality of $\zeta(3)$. We show in particular that this can be applied to a…
We construct continued fraction expansions for several families of the Laurent series in $\mathbb{Q}[[t^{-1}]]$. To the best of the author's knowledge, this is the first result of this kind since Gauss derived the continued fraction…
The inspiral of two compact objects in gravitational wave astronomy is described by a post-Newtonian expansion in powers of $(v/c)$. In most cases, it is believed that the post-Newtonian expansion is asymptotically divergent. A standard…
In a previous escapade we gave a collection of continued fractions involving Catalan's constant. This paper provides more general formulae governing those continued fractions. Having distinguished different cases associated to regions in…
Li et al. give an integral formula for the Catalan-Qi number of the second kind. They show that this integral can be written as a summation with double factorials. In this paper the integral is reduced to a product of the Catalan number and…
In this work, we consider a class of substitutions on infinite alphabets and show that they exhibit a growth behaviour which is impossible for substitutions on finite alphabets. While for both settings the leading term of the tile counting…
We introduce a kind of $(p, q, t)$-Catalan numbers of Type A by generalizing the Jacobian type continued fraction formula, we proved that the corresponding expansions could be expressed by the polynomials counting permutations on…
We establish the strong unique continuation property of fractional orders of linear elliptic equations with Lipschitz coefficients by establishing monotonicity of some Almgren-type frequency functional via an extension procedure.
We propose a new second-order accurate lattice Boltzmann scheme that solves the quasi-static equations of linear elasticity in two dimensions. In contrast to previous works, our formulation solves for a single distribution function with a…