 2015 Continuum sum conjecture - Printable Version +- Tetration Forum (https://math.eretrandre.org/tetrationforum) +-- Forum: Tetration and Related Topics (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=1) +--- Forum: Mathematical and General Discussion (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=3) +--- Thread: 2015 Continuum sum conjecture (/showthread.php?tid=1005) 2015 Continuum sum conjecture - tommy1729 - 05/25/2015 I noticed a possible pattern. next next define : then Now it appears The continuum sum up to z-1 of exp^[n](x) is given by CS [ exp^[n](x) , z-1] = P_n(x) + F_n(x) where F_n is 2pi i periodic and P_n is a polynomial of degree (at most) n. I guess there is a simple reason for it. Right ?? What if n is not an integer ? say n = 3/2 ? Does that imply P_{3/2}(x) =< O( x^{3/2} ) ?? regards tommy1729 RE: 2015 Continuum sum conjecture - tommy1729 - 05/26/2015 The answer appears to be simple. That is considering we work with the same continuum sum. Within an analytic region : Let r > 0. exp^[1+r](x) = exp^[r](0) + F(exp(x)) where F(0) = 0 and is analytic. Therefore the continuum sum CS is given by T_r(x) = CS exp^[1+r](x) = exp^[r](0) x + G(exp(x)). Therefore T(x) - T(x - 2pi i) = exp^[r](0) * 2 pi i. regards tommy1729 RE: 2015 Continuum sum conjecture - tommy1729 - 05/26/2015 One wonders what happens with r < 0. I will post a related thread. I have some ideas but little time. regards tommy1729 RE: 2015 Continuum sum conjecture - tommy1729 - 05/26/2015 Notice the counterintuitive nature sum [ f(x) - f(x - period f) ] =/= sum [ f(x) ] - sum[ f(x - period f) =/= 0. even if f(x) - f(x - period f) = 0 and sum [ 0 ] = 0. I have to think more about this. Any ideas ? I considered using tommysum[f(x)] := continuum sum [f(x) - f(0)]. regards tommy1729