$\pi$-formulas with free parameters

TL;DR

This paper extends known series for pi using hypergeometric methods and derives 29 new summation formulas involving pi, pi squared, and 1/pi with free parameters.

Contribution

It introduces new summation formulas for pi and related constants using hypergeometric techniques, expanding the mathematical toolkit for series representations.

Findings

Extended two known pi series using hypergeometric methods

Derived 29 new summation formulas involving pi, pi^2, and 1/pi with free parameters

Provided a unified approach to generating series for pi and related constants

Abstract

In terms of the hypergeometric method, we give the extensions of two known series for $\pi$. Further, other twenty-nine summation formulas for $\pi$, $\pi^2$ and $1/\pi$ with free parameters are also derived in the same way.