Mick's differential equation - 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: Mick's differential equation (/showthread.php?tid=981) Mick's differential equation - tommy1729 - 04/05/2015 Mick posted an intresting question here : http://math.stackexchange.com/questions/1221034/f-x-fx-x1t-1 It reminds of the binary partition function we discussed before which was strongly related by the similar equation F ' (x) = F(x/2). Maybe the method to solve F ' (x) = F(x/2) can be used/modified to solve Mick's differential equation. Anyway I think its intresting. regards tommy1729 RE: Mick's differential equation - tommy1729 - 04/06/2015 Although far from an answer , using the Mittag-Leffler function to Get a fake (1+x)^t will probably get us a good approximation in terms of a Taylor series. Although getting these Taylor coëfficiënts is Nice , its not a closed form asymtotic. Unless we get a simple rule for these coef , its hard to get THE asym from the Taylor. Regards Tommy1729 RE: Mick's differential equation - tommy1729 - 04/10/2015 Clearly this function grows slower than any exponential but faster than any polynomial or even exp(x^t). This implies that if our asymptotic is entire - or the function itself ?? - then it is determined by its zero's completely. Otherwise fake function theory can be applied. And then the fake is compl determined by its zero's. A quick brute estimate is exp(x^T + T x^{tT}) , where T is between t and 1. ITS not accurate i know. All of this is ofcourse Up to a multip constant. Regards Tommy1729 RE: Mick's differential equation - tommy1729 - 04/10/2015 About post 2 i use truncated carleman matrices for (fake) entire functions. And then simply solve THE equation. Regards Tommy1729