# A new family of series expansions for $1/\pi$ and a binomial identity

## TL;DR

This paper introduces a novel family of series expansions for 1/π derived from gamma function ratios and Bessel functions, along with a new binomial identity, expanding mathematical tools for π approximation.

## Contribution

It presents a new family of series expansions for 1/π and a novel binomial identity based on gamma and Bessel functions, offering fresh analytical methods.

## Key findings

- Doubly infinite series expansions for 1/π are derived.
- A new binomial identity is discovered from gamma function ratios.
- The approach uses formal expansions involving Bessel functions.

## Abstract

A doubly infinite set of series expansion for $1/\pi$ are reported. They follow trivially from a formal expansion for the quotient of the values taken by the gamma function for two (complex) arguments differing by an integer plus one half, obtained by an alternative computation of the Wronskian of the modified Bessel functions. The same formal expansion allows to discover also a new binomial identity.