51 citations · 83 across the 22 of their papers we have counts for
23 papers
Euler and the Legendre Polynomials
Alexander Aycock
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…
Euler's First Proof of Stirling's Formula
Alexander Aycock
We present a proof given by Euler in his paper {\it ``De serierum determinatione seu nova methodus inveniendi terminos generales serierum"} \cite{E189} (E189:``On the determination…
On Euler's Solution of the simple Difference Equation
Alexander Aycock
In this note we will discuss Euler's solution of the simple difference equation that he gave in his paper{\it ``De serierum determinatione seu nova methodus inveniendi terminos gen…
Euler and the Duplication Formula for the Gamma-Function
Alexander Aycock
We show how the formulas in paper Variae considerationes circa series hypergeometricas written by Euler imply the duplication formula for the Gamma-function. This paper can be seen…
Answer to a question concerning Euler's paper "Variae considerationes circa series hypergeometricas"
Alexander Aycock
We solve a problem concerning Euler's paper "Variae considerationes circa series hypergeometricas" (\cite{E661}) suggested by G. Faber in the preface to Volume 16,2 of the first se…
Euler and the Gammafunction
Alexander Aycock
We review Euler's idea on the Gammafunction. We will explain, how Euler obtained them and how Euler's ideas anticipate more modern approaches and theories. Furthermore, some questi…
Institutiones calculi differentialis cum eius usu in analysi finitorum ac doctrina serierum
Leonhard Euler, Alexander Aycock
This is the translation of Euler's Latin textbook Institutiones calculi differentialis cum eius usu in analysi finitorum ac doctrina serierum (second volume) into English.
Euler and the Multiplication Formula for the Gamma Function
Alexander Aycock
We show that an apparently overlooked result of Euler from \cite{E421} is essentially equivalent to the general multiplication formula for the -function that was proven by Gauss…
On solids whose (entire) surface can be unfolded onto a plane
Leonhard Euler, Alexander Aycock
This is the English translation of Leonhard Euler's Latin paper "De solidis quorum superficiem in planum explicare licet". Euler explains several methods to obtain equations for de…
Observations on continued fractions
Leonhard Euler, Alexander Aycock
This is a translation of Euler's Latin paper "De fractionibus continuis observationes" into English. In this paper Euler describes his theory of continued fractions. He teaches, ho…
On divergent Series
Leonhard Euler, Alexander Aycock
This is the translation of Leonhard Euler's paper "De Seriebus divergentibus" written in Latin into English. Leonhard Euler defines and discusses divergent series. He is especially…
Explicit formulas for orthogonal polynomials derived from their difference equation
Alexander Aycock
We solve the difference equation with linear coefficients by the Momentenansatz to obtain explicit formulas for orthogonal polynomials.
Bourlet's Theorem for the product of differential operators, an application of the operator method and a proof for , that Euler missed, derived from difference equations
Alexander Aycock
We give another proof for \[ \sum_{n=1}^{\infty}\frac{1}{n^2}=\frac{π^2}{6} \] that basically follows from the theory of difference equations.
Translation of the C.J. Malmstén's paper De integralibus quibusdam definitis, seriebusque infinitis
Alexander Aycock
We translate the paper De Integralibus quibusdam definitis, seriebusque infinitis
On proving some of Ramanujan's formulas for with an elementary method
Alexander Aycock
In this paper we want to prove some formulas listed by S. Ramanujan in his paper "Modular equations and approximations to " \cite{24} with an elementary method.
Notes and Remarks on certain logarithmic integrals
Alexander Aycock
Logarithmic integrals revisited. We consider integrals of the form again, where is a rational function, and we will explai…
Note on Malmstèn's paper De Integralibus quibusdam definitis seriebusque infinitis
Alexander Aycock
We present a proof of the functional equation of the Riemann zeta-function or more precisely the Dirichlet eta-function, which proof seems to be new but follows almost immediately…
Theoremata arithmetica nova methodo demonstrata
Leonhard Euler, Artur Diener, Alexander Aycock
Euler presents a third proof of the Fermat theorem, the one that lets us call it the Euler-Fermat theorem. This seems to be the proof that Euler likes best. He also proves that the…
Evolutio producti infiniti (1-x)(1-xx)(1-x^3)(1-x^4)(1-x^5) etc. in seriem simplicem
Leonhard Euler, Artur Diener, Alexander Aycock
This paper does exactly what the title says it does. It expands the given series to arrive at the familiar "pentagonal number" expansion, also known as the pentagonal number theore…
Methodus succincta summas serierum infinitarum per formulas differentiales investigandi
Leonhard Euler, Artur Diener, Alexander Aycock
This paper has more formal series with Bernoulli numbers and the Riemann zeta function. He gives the generating function for the Bernoulli numbers. The paper is translated from the…
Variae considerationes circa series hypergeometricas
Leonhard Euler, Artur Diener, Alexander Aycock
Euler gives an asymptotic approximation for the function f(x) and recognizes that he is trying to interpolate the factorial function introduced in E19 "De progressionibus transcend…
Nova series infinita maxime convergens perimetrum ellipsis exprimens
Leonhard Euler, Artur Diener, Alexander Aycock
The title says what is done here. Euler finds a news series for the arc of an ellipse. The paper is translated from the Latin original into German.
Proof of some conjectured formulas for 1/pi by Z.-W. Sun
Gert Almkvist, Alexander Aycock, appendix by Arne Meurman
Recently Z.W.Sun found over hundred conjectured formulas for 1/pi. Many of them were proved by H.H.Chan, J.Wan andW.Zudilin (see [3], [8] in the paper). Here we show that several o…