# On Entry II.16.12: A continued fraction of Ramanujan

## TL;DR

This paper investigates Ramanujan's continued fraction Entry 12, providing two proofs and exploring a natural generalization through Euler's method and orthogonal polynomial theory.

## Contribution

It offers two new proofs of Ramanujan's Entry 12 and introduces a generalization based on orthogonal polynomial theory.

## Key findings

- Two alternative proofs of Ramanujan's Entry 12
- A natural generalization of the continued fraction
- Connections between continued fractions and orthogonal polynomials

## Abstract

We study a continued fraction due to Ramanujan, that he recorded as Entry 12 in Chapter 16 of his second notebook. It is presented in Part III of Berndt's volumes on Ramanujan's notebooks. We give two alternate approaches to proving Ramanujan's Entry 12, one using a method of Euler, and another using the theory of orthogonal polynomials. We consider a natural generalization of Entry 12 suggested by the theory of orthogonal polynomials.