New proofs of Borwein-type algorithms for Pi
  We use a method of translation to recover Borweins' quadratic and quartic iterations. Then, by using the WZ-method, we obtain some initial values which lead to the limit $1/\pi$. We will not use the modular theory nor either the Gauss' formula that we used in another paper. Our proofs are short and self-contained. 
Jes\'us Guillera
math.NT