相关论文: Giuga Numbers and the arithmetic derivative
A generalization of the Catalan numbers is considered. New results include binomial identities, recursive relations and a close formula for the multivariate generating function. A simple expression for the Catalan determinant is derived.
We show that Euler's relation and the Taxi-Cab relation are both solutions of the same equation. General solutions of sums of two consecutive cubes equaling the sum of two other cubes are calculated. There is an infinite number of relations…
In the present paper we generalize the Eulerian numbers (also of the second and third orders). The generalization is connected with an autonomous first-order differential equation, solutions of which are used to obtain integral…
A finite-dimensional unital and associative algebra over $\mathbb{R}$, or what we shall call simply "an algebra" in this paper for short, generalities the construction by which we derive the complex numbers by "adjoining an element $i$" to…
Recently, Jha (arXiv:2007.04243, arXiv:2011.11038) has found identities that connect certain sums over the divisors of $n$ to the number of representations of $n$ as a sum of squares and triangular numbers. In this note, we state a…
The generalized Riccati equation defined as an equation between first order derivative and the cubic polynomial is named Riccati-Abel equation. Unlike solutions of ordinary Riccati equation, the solutions of Riccati-Abel equation do not…
The identity j/n {kn}\choose{n+j} =(k-1) {kn-1}\choose{n+j-1}- {kn-1}\choose{n+j} shows that j/n {kn}\choose{n+j} is always an integer. Here we give a combinatorial interpretation of this integer in terms of lattice paths, using a uniformly…
We define a multiplication on the surreal numbers as higher inductive-inductive types.
We solve an open problem concerning the well-known $(\alpha,\beta,\gamma)$-derivations, proving that the spaces of $(\alpha,1,0)$-derivations of any Lie algebra are isomorphic ($\alpha\neq 0,1$). Also, we prove sharp bounds for the…
Let Omega(n) be the volume of the unit ball in R^n. We formulate as an infinite product the gamma function ratio gamma(x+1/2)/gamma(x),x>0, which allows us to reproduce and /or produce a variety of formulas and inequalities, some of them…
For a set $A$ of integers, the sumset $lA =A+...+A$ consists of those numbers which can be represented as a sum of $l$ elements of $A$ $$lA =\{a_1+... a_l| a_i \in A_i \}. $$ A closely related and equally interesting notion is that of…
Prime numbers are one of the most intriguing figures in mathematics. Despite centuries of research, many questions remain still unsolved. In recent years, computer simulations are playing a fundamental role in the study of an immense…
A base-$g$ Niven number is a natural number divisible by the sum of its base-$g$ digits. We show that, for any $g\geq 3$, all sufficiently large natural numbers can be written as the sum of three base-$g$ Niven numbers. We also give an…
We give a language for geometry which makes curves and number fields look alike.
For a composite $n$ and an odd $c$ with $c$ not dividing $n$, the number of solutions to the equation $n+a\equiv b\mod c$ with $a,b$ quadratic residues modulus $c$ is calculated. We establish a direct relation with those modular solutions…
By employing certain extended classical summation theorems, several surprising \pi and other formulae are displayed.
In this paper we give some interesting identities between Euler numbers and zeta functions. Finally we will give the new values of Euler zeta function at positive even integers.
We consider the derivatives of Horn hypergeometric functions of any number variables with respect to their parameters. The derivative of the function in $n$ variables is expressed as a Horn hypergeometric series of $n+1$ infinite summations…
An integer partition \lambda of n corresponds, via its Ferrers diagram, to an artinian monomial ideal I of colength n in the polynomial ring on two variables. If the partition \lambda corresponds to an integrally closed ideal we call…
We exhibit a numerical method to solve fractional variational problems, applying a decomposition formula based on Jacobi polynomials. Formulas for the fractional derivative and fractional integral of the Jacobi polynomials are proven. By…