Just eyeballing your msg: Joerg> * Bill Gosper <billgosper@gmail.com <http://gosper.org/webmail/src/compose.php?send_to=billgosper%40gmail.com>> [Feb 23. 2012 07:41]:> [...] Still fails, could be the limits, the order of the matrices in the products, or... N=166; \\ infinity for summation oo=N; \\ infinity for limits q='q+O('q^N); \\ as power series \\ q=0.5; \\ numerically \\ Lambert: \\ for later reference \\Lam=sum(n=1,N,q^(n^2)*(1+q^n)/(1-q^n)) \\ == q + 2*q^2 + 2*q^3 + 3*q^4 + 2*q^5 + 4*q^6 + 2*q^7 + 4*q^8 + 3*q^9 + ... \\ == [1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 6, 2, 4, 4, 5, ...] (series) \\ == 1.60669515241... A065442 (numerically)

This is, of course, just a specialization of Rogers-Fine.> Here are the two matrices:> {{k, {{q^(1 + 2*k + n), -((q*(-1 + q^(2*k + n)))/((-1 + q^k)*(-1 + q^(k +> n))))},> {0, 1}}},> {n, {{q^k, q/(1 - q^(k + n))}, {0, 1}}}}

Km(k,n)={ [ q^(n+2*k+1), (1-q^(2*k+n))/( (1-q^(k)) * (1-q^(k+n)) ) ; 0, 1 ]; } Looks like you lost a factor of q from UR. You should have a handy invariance checker Nm(k,n) Km(k,n+1) = Km(k,n) Nm(k+1,n), and use it frequently. Nm(k,n)={ [ q^(k), q/(1-q^(k+n)); 0, 1 ]; } Alternatively, you could drop the UR q from Nm.

In[1511]:= Assuming[0 < q < 1,> MProd[mats[n] /. k -> 1, {n, 0, Infinity}] .> MProd[Limit[mats[k], n -> Infinity], {k, 1, Infinity}] ==> MProd[mats[k] /. n -> 0, {k, 1, Infinity}] .> MProd[Limit[mats[n], k -> Infinity], {n, 0, Infinity}]]> > [...]

Ln = prod(n=0,N, Nm(1,n) ) \\ == [0, Lam; 0, 1] Lk = prod(k=1,N, Km(k,oo) ) \\ == [0, 2; 0, 1]] L = Ln * Lk \\ == [0, Lam; 0, 1] \\ OK Rn = prod(k=1,N, Nm(k,0) ) \\ == [0, XXX; 0, 1] Eeek, no. Nm bumps n, not k. Rk = prod(n=0,N, Km(oo,n) ) \\ == [0, 1; 0, 1] No, Km bumps k! R = Rn * Rk \\ == [0, XXX; 0, 1] \\ NOT OK That's for sure. Should be Rk Rn. Matrix multiply doesn't commute. Mind my MProds. Ease up on the beer. Hope this helps. --rwg \\ where XXX == \\ == q + 2*q^2 + q^3 + 3*q^4 + q^5 + 2*q^6 + 3*q^7 + 2*q^8 + q^9 + 3*q^10 +... \\ == [1, 2, 1, 3, 1, 2, 3, 2, 1, 3, 3, 2, 2, 2, 2, 5, ...] (series) \\ == 1.41361501351851... (numerically) d = L - R \\ This SHOULD be zero, but is \\ == [0, DD; 0, 0] where \\ DD == q^3 + q^5 + 2*q^6 - q^7 + 2*q^8 + 2*q^9 + q^10 - q^11 +- ... (series) \\ DD == 0.19308... (numerically)

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