# [seqfan] Re: Peculiar Continued Fractions

Paul D Hanna pauldhanna at juno.com
Tue Mar 23 12:58:06 CET 2010

```SeqFans,
Forgive my typo;  obviously I should have stated:

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_{k=0..n} D(k,b,c).

[and not "n!*Product_{n>=0} D(n,b,c)."]

---------- Original Message ----------
From: "Paul D Hanna" <pauldhanna at juno.com>
To: seqfan at list.seqfan.eu
Subject: [seqfan] Re: Peculiar Continued Fractions
Date: Tue, 23 Mar 2010 11:51:34 GMT

Roland,
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)) )
where
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.

Thanks,
Paul

```