https://nova.newcastle.edu.au/vital/access/ /manager/Index en-au 5 https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:13009 n²+nm+m²)³ = ([formula cannot be replicated] ω^n-mq^n²+nm+m²)³ + ([formula cannot be replicated] q^{(n+1/3)²+(n+1/3)(m+1/3)+(m+1/3)²})³. Here ω = exp(2π i/3). In this note we provide an elementary proof of this identity and of a related identity due to Ramanujan. We also indicate how to discover and prove such identities symbolically.]]> Wed 11 Apr 2018 15:27:13 AEST ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:13030 n+1 := a_n + 2b_n / 3 and b_n+1 := [formula cannot be replicated]. The limit of this iteration is identified in terms of the hypergeometric function ₂F₁ (1/3, 2/3; 1 ; ·), which supports a particularly simple cubic transformation.]]> Wed 11 Apr 2018 13:39:27 AEST ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:13015 Wed 11 Apr 2018 12:30:01 AEST ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:13087 η (a,b) + R_η(b, a) = 2R_η ((a + b)/2, √ab). Alas, for some parameters the continued fraction R_η does not converge; moreover, there are converging instances where the AGM relation itself does not hold. To unravel these dilemmas we herein establish convergence theorems, the central result being that R₁ converges whenever |a| ≠|b|. Such analysis leads naturally to the conjecture that divergence occurs whenever a = be^iϕ with cos^2ϕ ≠ 1 (which conjecture has been proven in a separate work) [Borwein et al. 04b.] We further conjecture that for a/b lying in a certain—and rather picturesque—complex domain, we have both convergence and the truth of the AGM relation.]]> Sat 24 Mar 2018 08:15:36 AEDT ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:6480 Sat 24 Mar 2018 07:47:13 AEDT ]]>