Ramanujan continued fractions of order sixteen

Abstract

We study the continued fractions [Formula: see text] and [Formula: see text] of order sixteen by adopting the theory of modular functions. These functions are analogues of Rogers–Ramanujan continued fraction [Formula: see text] with modularity and many interesting properties. Here we prove the modularities of [Formula: see text] and [Formula: see text] to find the relation with the generator of the field of modular functions on [Formula: see text]. Moreover we prove that the values [Formula: see text] and [Formula: see text] are algebraic integers for certain imaginary quadratic quantity [Formula: see text].