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Thursday, September 22, 2016

Entry 4

These q-continued fractions were "missed(?)" by Ramanujan. Given the golden ratio ϕ=1+52, then 51+ϕ1(v+1+v2)1/5ϕ=e2π5/51+e2π51+e4π51+e6π51+wherev=ϕ5
51+ϕ1(v+1+v2)1/5ϕ=e2π10/51+e2π101+e4π101+e6π101+wherev=1855
51+ϕ1(v+1+v2)1/5ϕ=e2π15/51+e2π151+e4π151+e6π151+wherev=1475554
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(in)=51+ϕ1R(i/n)ϕ
for some positive real n, though I have no proof of this assertion.

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