A family of Ramanujan–Orr formulas for 1/π

Abstract

We use a variant of Wan's method to prove two Ramanujan–Orr type formulas for $1/\pi$. This variant needs to know in advance the formulas for $1/\pi$ that we want to prove, but avoids the need of solving a system of equations.

Keywords:

• hypergeometric series
• Ramanujan–Orr type formulas
• complete elliptic integral
• Legendre's relation
• Zeilberger's algorithm

AMS Subject Classifications:

• Primary: 33E05
• 33C20
• Secondary: 11F03
• 33C75
• 33F10

Acknowledgements

I thank Wadim Zudilin for encouraging me to extend these investigations with the use of modular equations in alternative bases.

Disclosure statement

No potential conflict of interest was reported by the authors.