Re: [math-fun] tiny 2x2 simultaneously computes both Rogers-Ramanujan sums

10 Oct 2012

      * Joerg Arndt <arndt@jjj.de> [Oct 10. 2012 13:02]:
...
[...]
Now we really like to know the two '?' ...
Writing R1 and R2 for the '?'s and starting
the product at index one, I get:

     inf
    /===\
     | |  [       k ]   [ X  X ]
     | |  [ 0  a*q  ] = [      ]
     | |  [         ]   [ Y  Y ]
    k = 1 [ 1   1   ]

Now we can recognize your matrix product [R1, U; R2, V]:

[0,1;1,1]^(-1) * [ X, X; Y, Y ]
\\ == [  R1,    U;   R2, V ]
\\ == [ -X+Y, -X+Y;  X,  X ]

That is

     inf
    /===\
     | |  [       k ]   [ U  U ]
     | |  [ 0  a*q  ] = [      ]
     | |  [         ]   [ V  V ]
    k = 0 [ 1   1   ]

where (as before)
U(a,q) = a * sum(n=0,N, a^n*q^(n^2+n) / prod(k=1,n, 1-q^k) );
V(a,q) = sum(n=0,N, a^n*q^(n^2) / prod(k=1,n, 1-q^k) );

I'll now put this in.