[seqfan] Re: Peculiar Continued Fractions

[Joerg Arndt]

* Paul D Hanna <pauldhanna at juno.com> [Mar 25. 2010 18:01]:
> 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,

sort of:

Define Z  as
        n
                                                      \~~   n ___k
     Z :=z  + z  z  + z  z  z  + z  z  z  z  + ... =   >      | |   z 
      n   1    1  2    1  2  3    1  2  3  4          /__ k=1 | |i=1 i

Then Z   has the continued fraction K(a,b) where
      oo

    a=[0,  1,  z  + 1,  z  + 1,  z  + 1,  z  + 1,  z  + 1, ...]        
                2        3        4        5        6
    b=[1,  z ,   -z ,    -z ,     -z ,     -z ,      -z , ...]
            1      2       3        4        5         6



? Z(k)=z^(k-1)  \\ I start form one, so offset -1
? zprod(k)=prod(j=1,k,Z(j))
? zsumprod(k)=sum(j=1,k,zprod(j))
? n=10
10
? zsumprod(n)  \\ here is your theta-oid
z^45 + z^36 + z^28 + z^21 + z^15 + z^10 + z^6 + z^3 + z + 1
? afunc(j)=if(j<=0,0,if(j==1,1,1+Z(j)))
? bfunc(j)=if(j<=0,0,if(j<=1,Z(j),-Z(j)))
? a=vector(n,j,afunc(j-1))
[0, 1, z + 1, z^2 + 1, z^3 + 1, z^4 + 1, z^5 + 1, z^6 + 1, z^7 + 1, z^8 + 1]
? b=vector(n,j,bfunc(j-1))
[0, 1, -z, -z^2, -z^3, -z^4, -z^5, -z^6, -z^7, -z^8]

(this (likely) is your simple continued fraction)




> and is it useful in evaluating special values of theta2? 

For special values exp( a*Pi ) just ask R.W.Gosper  ;-)

>     Paul 
> [...]