[seqfan] Re: Closed Form Solution to Special Diff Eq?

[A.N.W.Hone at kent.ac.uk]

Dear Paul, 

Here are some quick observations. 

The functional differential equation can be rewritten as 

2+8x F(4x) - F(x)=0  

where F(x)=A'(x)/A(x). Suppose that A is an analytic function of x with a zero at x=x_0; then 
F has a simple pole there, and it is easy to see that it must then have a pole at 4x_0 and at 
x_0/4, from which it follows that the poles of F coalesce at x=0, hence the zeros of A coalesce 
there, so A cannot be analytic at x=0. 

So if A is analytic it must be without zeros, hence we can write A(x) =exp(g(x)) for some 
analytic function g(x), and then F(x)=g'(x). So we should seek an analytic solution of the equation 
for F, then get A by integrating and exponentiating. Substituting in a Taylor series 

F(x)=\sum c_n x^n 

around x=0 gives c_0=2 and the recursion c_n = 2 . 4^n c_{n-1}with solution 

c_n = 2^{n+1} 4^{n(n+1)/2}. 

However, by the ratio test it is clear that this is a divergent series. 

Therefore I would say the equaation has no solutions in terms of known functions.

All the best 
Andy 
    
----- Original Message -----
From: Paul D Hanna <pauldhanna at juno.com>
Date: Friday, February 27, 2009 10:57 am
Subject: [seqfan]  Closed Form Solution to Special Diff Eq?
To: seqfan at list.seqfan.eu

> Seqfans, 
>       Is there a closed form for the 
> solution A(x) to this differential equation?
>  
> 2*A(x)*A(4x) + 8*x*A(x)*A'(4x) - A'(x)*A(4x) = 0
>  
> A power series solution exists, but can it be expressed finitely 
> in terms of known functions? 
>       Paul 
> 
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