New proofs of Borwein-type algorithms for Pi

ABSTRACT

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 use neither the modular theory nor the formula of Gauss that we used in another paper. Our proofs are short and self-contained.

KEYEORDS:

• Hypergeometric series
• algebraic transformations
• Borwein-type algorithms for Pi

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 11Y60
• 33F05
• 33C75
• 33C05
• 33F10

Acknowledgments

Many thanks to Jonathan Sondow for improving my sometimes awkward English.

Disclosure statement

No potential conflict of interest was reported by the author.