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.

Bourlets 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 difference equations

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.

On certain definite integrals and infinite series

A translation of Kummer`s paper On certain definite integrals and infinite series

Translation of the C.J. Malmstens paper De integralibus quibusdam definitis, seriebusque infinitis

We translate the paper De Integralibus quibusdam definitis, seriebusque infinitis

Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, sive solutio problematis isoperimetrici lattissimo sensu accepti

This translation has been withdrawn due to certain imperfections and mistakes, which are corrected in the version uploaded at The Euler Archive (see E65 at http://www.eulerarchive.org/)

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^n that, when divided by a numer N, prime to x, and that leaves a remainder of 1, is equal to the number of parts of N that are prime to n, that is to say, the number of distinct aliquot parts of N. The translation is presnted from Eulers Latin original into German.

Theorematum quorundam arithmeticorum demonstrationes

Euler proves that the sum of two 4th powers cant be a 4th power and that the difference of two distinct non-zero 4th powers cant be a 4th power and Fermats theorem that the equation x(x+1)/2=y^4 can only be solved in integers if x=1 and the final theorem y^3+1=x^2 can only be solves for x=3 and y=2 in integers. The paper is translated from Eulers Latin original into German.

De seriebus divergentibus

Euler gives a long introduction, giving all the arguments for and against the use of divergent series in calculus and then gives his own definition of the sum of a diverging series. Then in the second half of this paper he evaluates the the 1-1+2-6+24-120+720-... on several ways and gets the sum 0.5963473621372. The paper is translated from Eulers Latin original into German.

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 its application to partition numbers. The paper is translated from Eulers Latin original into German.