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

A.N.W.Hone at kent.ac.uk A.N.W.Hone at kent.ac.uk
Fri Feb 27 12:59:50 CET 2009

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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