Mick's differential equation

[tommy1729]

Mick posted an intresting question here :

http://math.stackexchange.com/questions/...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

[tommy1729]

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

[tommy1729]

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

[tommy1729]

About post 2 i use truncated carleman matrices for (fake) entire functions.
And then simply solve THE equation.

Regards

Tommy1729