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Euler and the Gammafunction.
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 questions asked by EulerExpand
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Theoremata arithmetica nova methodo demonstrata
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 smallest power x^nExpand
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Translation of the C.J. Malmstén's paper De integralibus quibusdam definitis, seriebusque infinitis
In 1846 (published in the Crelle Journal für die reine und angewandte Mathematik) the nowadays widely unknown Swedish mathematician C. J. Malmstén wrote his quite remarkable paper "De integralibusExpand
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Institutiones calculi differentialis cum eius usu in analysi finitorum ac doctrina serierum.
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.
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Evolutio producti infiniti (1-x)(1-xx)(1-x^3)(1-x^4)(1-x^5) etc. in seriem simplicem
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 theorem, and recalls itsExpand
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On proving some of Ramanujan's formulas for $\frac{1}{\pi}$ with an elementary method
In this paper we want to prove some formulas listed by S. Ramanujan in his paper "Modular equations and approximations to $\pi$" \cite{24} with an elementary method.
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Proof of some conjectured formulas for 1/pi by Z.-W. Sun
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 other formulas inExpand
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Explicit formulas for orthogonal polynomials derived from their difference equation
We solve the difference equation with linear coefficients by the Momentenansatz to obtain explicit formulas for orthogonal polynomials.
Notes and Remarks on certain logarithmic integrals
Logarithmic integrals revisited. We consider integrals of the form $\int_0^1 \ln{\ln{(\frac{1}{x})}}R{(x)}{\rm d}x$ again, where $R{(x)}$ is a rational function, and we will explain a way to obtainExpand
Bourlet's Theorem for the product of differential operators, an application of the operator method and a proof for $\sum_{n=1}^{\infty}\frac{1}{n^2}=\frac{\pi^2}{6}$, that Euler missed, derived from
We give another proof for \[ \sum_{n=1}^{\infty}\frac{1}{n^2}=\frac{\pi^2}{6} \] that basically follows from the theory of difference equations.
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