Related papers: On Amicable Numbers
At a crossroads of calculus and combinatorics, the generating function of secant and tangent numbers (Euler numbers) provides enumeration of alternating permutations. In this article, we present a new refinement of Euler numbers to answer…
Translated from the Latin original, "Observationes circa bina biquadrata quorum summam in duo alia biquadrata resolvere liceat" (1772). E428 in the Enestroem index. This paper is about finding A,B,C,D such that $A^4+B^4=C^4+D^4$. In sect.…
In the paper, the author elementarily unifies and generalizes eight identities involving the functions $\frac{\pm1}{e^{\pm t}-1}$ and their derivatives. By one of these identities, the author establishes two explicit formulae for computing…
In this paper we extend the approach of M. Cavaleri to effective amenability to the class of computably enumerable groups, i.e. in particular we do not assume that groups are finitely generated. In the case of computable groups we also…
The Bernoulli numbers are fascinating and ubiquitous numbers, they occur in several domains of Mathematics like Number theory (FLT), Group theory, Calculus and even in Physics. Since Bernoulli's work, they are yet studied to understand…
Recently, Araci-Acikgoz-Sen derived some interesting identities on weighted q-Euler polynomials and higher-order q-Euler polynomials from the applications of umbral calculus (See [1]). In this paper, we develop the new method of q-umbral…
In this paper, we introduce a novel identity for generalized Euler polynomials, leading to further generalizations for several relations involving classical Euler numbers, Euler polynomials, Genocchi polynomials, and Genocchi numbers.
Euler explored the problem of finding three numbers such that the sum or difference of any two of them is a perfect square. He discovered a parametric solution represented by polynomials of degree 18 and identified the smallest of these…
The number of tuples with positive integers pairwise relatively prime to each other with product at most $n$ is considered. A generalization of $\mu^{2}$ where $\mu$ is the M\"{o}bius function is used to formulate this divisor sum and…
In this note we will present how Euler's investigations on various different subjects lead to certain properties of the Legendre polynomials. More precisely, we will show that the generating function and the difference equation for the…
We discuss heuristic asymptotic formulae for the number of pairing-friendly abelian varieties over prime fields, generalizing previous work of one of the authors arXiv:math1107.0307
The notion of pairable functions is introduced and some of its properties are developed. In this connection the famous Euler identity is interpreted as a property of certain pairable functions and finite cyclic groups.
The subject of our discussion is the theory of differential equations as set out in two classical Euler's textbooks "Institutiones Calculi Differentialis" and "Institutiones Calculi Integralis".
A sequence inverse relationship can be defined by a pair of infinite inverse matrices. If the pair of matrices are the same, they define a dual relationship. Here presented is a unified approach to construct dual relationships via…
This is an English translation of E579 in which the introductory remarks are in French, while Euler's original text is in Latin. By considering the balance of forces acting on a raising balloon on an isothermal atmosphere, namely the weight…
Euler starts with a hypergeometric series F(a, b, c, x), and differentiates it to get a functional relation. This relation is today known as Euler's identity. Then he integrates to get another and ends up with something like Legendre…
Recently the new q-Euler numbers are defined. In this paper we derive the the Kummer type congruence related to q-Euler numbers and we introduce some interesting formulae related to these q-Euler numbers.
Following an idea due to Euler, we evaluate the alternating sums of powers of consrcutive integers.
Euler had considered the problem of finding three integers whose sum, product, and also the sum of the products of the integers, taken two at a time, are all perfect squares. Euler's methods of solving the problem lead to parametric…
We establish some new combinatorial identities involving Euler polynomials and balancing (Lucas-balancing) polynomials. The derivations use elementary techniques and are based on functional equations for the respective generating functions.…