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

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.