[seqfan] Re: Peculiar Continued Fractions

[Paul D Hanna]

Roland (and SeqFans), 
      Consider this formal power series: 
G(x) = exp( Sum_{n>=1} x^n/(n*A003500(n)) ) 
where A003500(n) = (2+sqrt(3))^n + (2-sqrt(3))^n.
 
Explicitly, 
G(x) = 1 + 1/4*x + 15/224*x^2 + 209/11648*x^3 + 608399/126543872*x^4 +...
which does not look very pretty at all. 
 
But the continued fraction expansion is nice and begins: 
G(x) = 1/(1 - (x/4)/(1 - (x/56)/(1 - (x/728)/(1 - (x/10088)/(1 - (x/140456)/(1 - (x/1956248)/(1 - (x/27246968)/(1 - (x/379501256)/(1 - (x/5285770568)/(1 - (x/73621286648)/(1 -...))))))))))))).
 
Wherein the denominators of x in the non-simple CF expansion begin: 
C=[1,4,56,728,10088,140456,1956248,27246968,379501256,5285770568,73621286648,...]
 
Let the offset of C be zero, then it appears that: 
C(n) = A003500(2n-1) + 4 for n>1. 
 
I'm sure that the other power series in this family also have predictable CF expansions. 
 
I wonder ... what is the general formula of the CF for the formal power series: 
   exp( Sum_{n>=1} 1/(n*D(n,b,c)) ) 
where D(n,b,c) = (b+sqrt(b^2-c))^n + (b-sqrt(b^2-c))^n. 
Setting c=1 is a good place to start. 
Something to explore ... 
    Paul 
 
---------- Original Message ----------
From: Roland Bacher <Roland.Bacher at ujf-grenoble.fr>
To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
Cc: Roland Bacher <Roland.Bacher at ujf-grenoble.fr>
Subject: [seqfan] Re: Peculiar Continued Fractions
Date: Tue, 23 Mar 2010 09:31:30 +0100
 
Your examples are perhaps specialisations of a formal power series having
nice continued fractions. 

If this is true, there are (at least three) tools for proving 
continued fractions expansions which work sometimes.

The first one are lattice walks based on Dyck paths
introduced in a famous paper of Flajolet. 

The second one is related to addition formulae for elliptic curves
(and goes back to the 19th century):
Writing an elliptic function as an exponential generating series,
the addition formula for the elliptic function provides (often) a continued
fraction expansion for the associated ordinary continued fraction expansion.

The last one is based on the existence of a differential equation.
(This is unpublished stuff in my drawer.)

I hope that I will find time to have a look at your examples.

Best wishes,  Roland Bacher
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