[seqfan] Re: Peculiar Continued Fractions

Paul D Hanna pauldhanna at juno.com
Tue Mar 23 12:51:34 CET 2010

     Thank you for your insights, providing a place to start in analyzing the behavior of these series. 
Following your lead, one should study the formal power series given by:
   G(x,b,c) = exp( Sum_{n>=1} x^n/(n*D(n,b,c)) ) 
   D(n,b,c) = (b+sqrt(b^2-c))^n + (b-sqrt(b^2-c))^n
for some fixed non-zero integers b and c. 
These are unusual power series expansions, and the denominator the coefficients of x^n in G(x,b,c) are some factor of n!*Product_{n>=0} D(n,b,c).  
These functions will require further study, and perhaps may be understood using your methods of analysis. 
It would be interesting to explore the inverse functions of x*G(x,b,c) and G(x,b,c)-1 as well. 
---------- 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.

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