Choosing the simplest alternatives in http://mathworld.wolfram.com/ReciprocalFibonacciConstant.html leads to the remarkably simple http://gosper.org/recipfib.pdf . Jörg, did you know about this all along? (My Borwein&Borwein is still AWOL.) Wikipedia still has two articles maintaining no closed form. Are they really stodgy with their definition, or just uninformed? --rwg To: math-fun@mailman.xmission.com Sent: Friday, May 23, 2014 3:00:22 AM Subject: sum 1/Fib[n] Argh! From: rwg@sdf.lonestar.org < rwg@sdf.lonestar.org > [Jul 12. 2008 16:01]: [...] but still nothing about plain old inf ==== n \ q
------, / n ==== q - 1 n = 1
with which we could sum the reciprocal Fibonacci numbers. Jörg>Have you checked Borwein/Borwein "Pi and the AGM"? pp.91-101 might make you happy. rwg>Sadly not)-: They denote this Lambert series -L(q), but never give it in terms of Thetas or ThetaPrimes. Despite the section heading, they never deliver Sum 1/Fib(n) except in terms of L. It may well be inexpressible without some new special function. (They do give Sum 1/Fib(2*n+1) in Thetas, which is easy.)<rwg But did they give it in terms of q-digamma and I somehow missed or rejected it?http://mathworld.wolfram.com/ReciprocalFibonacciConstant.html claims they do! Could it have been added in a later edition? My (autographed) first edition is unhandy. Here I thought I had capped my 40 yr search, and the answer is sitting in Mathworld. The only consolation is that my answer doesn't use 𝜗s, so we get this peculiarity: EllipticTheta[2, 0, ϕ^-2]^2 = 2 Im[QPolyGamma[0, 1/2 + I π/(4 Log[ϕ]), ϕ^-2]]/Log[ϕ] ~ 3.26379 Can this generalize, or is it only about the Golden Ratio? --rwg