Thursday, September 22, 2016
Entry 4
These q-continued fractions were "missed(?)" by Ramanujan. Given the golden ratio ϕ=1+√52, then √51+ϕ−1(−v+√1+v2)1/5−ϕ=e−2π√5/51+e−2π√51+e−4π√51+e−6π√51+⋱wherev=ϕ5 √51+ϕ−1(−v+√1+v2)1/5−ϕ=e−2π√10/51+e−2π√101+e−4π√101+e−6π√101+⋱wherev=18−5√5 √51+ϕ−1(−v+√1+v2)1/5−ϕ=e−2π√15/51+e−2π√151+e−4π√151+e−6π√151+⋱wherev=147−55√54 where, as √5n increases, then v is an algebraic number of generally increasing high degree. I found this family using Mathematica and the second cfrac in Entry 3. This implies that the Rogers-Ramanujan continued fraction R(τ) R(i√n)=√51+ϕ−1R(i/√n)−ϕ for some positive real n, though I have no proof of this assertion.
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