[seqfan] Re: Peculiar Continued Fractions

[Paul D Hanna]

SeqFans,
     Should have added that, if we set y=x=q, then the prior conjecture gives a nice continued fraction expansion of the elliptic function: 
Sum_{n>=0} q^(n(n+1)/2) = 1/(1 - [q/(q+1)]/(1 - [q^2/((q+1)*(q^2+1))]/(1 - [q^3/((q^2+1)*(q^3+1))]/(1 - [q^4/((q^3+1)*(q^4+1))]/(1 -...))))). 
and equals  theta2(q^(1/2))/(2*q^(1/8)). 
  
Is this continued fraction expression known, and is it useful in evaluating special values of theta2? 
    Paul 
   
---------- Original Message ----------
From: "Paul D Hanna" <pauldhanna at juno.com>
To: seqfan at list.seqfan.eu
Subject: [seqfan] Re: Peculiar Continued Fractions
Date: Wed, 24 Mar 2010 18:23:53 GMT

SeqFans, 
    Here is a related conjecture I just came up with.  
 
Given the formal power series in x: 
G(x,y) = exp( Sum_{n>=1} x^n/(n*(y^n+1)) ) 
 
then the continued fraction expansion is:  
G(x,y) = 1/(1 - f(1,y)*x/(1 - f(2,y)*x/(1 - f(3,y)*x/(1 - f(4,y)*x/(1 - ...)))). 
 
where 
f(n,y) = y^(n-1)/((y^(n-1)+1)*(y^n+1)) for n>1 with f(1) = 1/(y+1). 

EXAMPLE:  Let y=2, then:  
G(x,2) = exp( x/3 + x^2/10 + x^3/27 + x^4/68 + x^5/165 +...) 

G(x,2) = 1 + 1/3*x + 7/45*x^2 + 31/405*x^3 + 3937/103275*x^4 + 64897/3408075*x^5 +...

G(x,2) = 1/(1 - (1/3)*x/(1 - (2/15)*x/(1 - (4/45)*x/(1 - (8/153)*x/(1 - (16/561)*x/(1 - (32/2145)*x/(1 - ...)))))).
  
Is there a none-formidable formula for [x^n] G(x,y) in this case? 
    Paul 

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