The level 13 analogue of the Rogers–Ramanujan continued fraction and its modularity

Abstract

We prove the modularity of the level 13 analogue r13(τ) of the Rogers–Ramanujan continued fraction. We establish some properties of r13(τ) using the modular function theory. We first prove that r13(τ) is a generator of the function field on Γ0(13). We then find modular equations of r13(τ) of level n for every positive integer n by using affine models of modular curves; this is an extension of Cooper and Ye's results with levels n=2,3 and 7 to every level n. We further show that the value r13(τ) is an algebraic unit for any τ∈K−Q, where K is an imaginary quadratic field. © 2016 Elsevier Inc.