New theta-function identities and general theorems for the explicit evaluations of Ramanujan’s continued fractions

Nipen Saikia

doi:10.1007/s40065-016-0149-x

Abstract

We prove some new theta-function identities for two continued fractions of Ramanujan which are analogous to those of Ramanujan–Gollnitz–Gordon continued fraction. Then these identities are used to prove new general theorems for the explicit evaluations of the continued fractions.

Highlights

Throughout the paper, we assume |q| < 1 and ∞(a; q)∞ := (1 − aqn). n=0Ramanujan’s general theta-function f (a, b) is given by f (a, b) =ak(k+1)/2bk(k−1)/2, |ab| < 1. k=−∞ (1.1)Three special cases of f (a, b) are φ(q) := f (q, q) = qn2 n=−∞
We prove some theta-function identities analogous to (1.7)–(1.10) for the continued fractions
We define some parameters of theta-functions which will be used in the explicit evaluations of T (q) and W (q)

Summary

Introduction

We prove some theta-function identities analogous to (1.7)–(1.10) for the continued fractions. Use them to prove new general theorems for the explicit evaluations of T (q) and W (q). 3 and 4, we prove new theta-function identities for the continued fractions T (q) and W (q), respectively. To end this introduction, we define some parameters of theta-functions which will be used in the explicit evaluations of T (q) and W (q). Since modular equations are key in the proofs of P-Q theta-function identities, first we define Ramanujan’s modular equation.

Assume that for some integer n n β

Replacing q by

Setting q