(09/13/2014, 07:15 PM)sheldonison Wrote: Have you seen this excellent asymptotic entire series?

Yes Ive seen this before.

Have you heard of hypergeometric functions or generalized hypergeometric functions ?

When I was still active at sci.math I mentioned the underrated and insuffiently known properties and powers of the hypergeometric functions and more particular their INVERSES.

Inverse hypergeometric functions is one those other " crazy math concepts " apart from tetration , collatz and prime twins.

Bill Dubuque and a few others recognized / remembered the power of these functions.

But its far from mainstraim.

It is amazing how unexpectedly these inverse hypergeometric functions can occur.

Literature is very very rare ( like tetration ) and it is not introduced to younger students.

From " entire function theory " comes the idea that hidden recursions

are often of hypergeometric nature.

hypergeometric and their inverse occur in for instance closed form solutions ( with integral or sum ) for half-iterates.

They are also very good approximations to other functions.

A good understanding of the gamma function is essential.

Id say Euler and Gauss were the first to recognise its importance , but that is open for debate.

The theory is very uncomplete ( as for tetration I guess ) , and that might be the reason for the avoidance in more classical math.

That's enough background.

I have to think about this , I have seen it before ...

But that is probably over 10 years ago and I have to dig in my memory.

Maybe its easy , but I lack time and concentration now.

Perhaps this matters or helps :

http://www.math.upenn.edu/~wilf/AeqB.pdf

edit :

this appears as the fake sqrt (after division by exp).

I conjecture for x > 2 :

regards

tommy1729