Re: [math-fun] (further) generalized Lambert series with Theta-convergence (brain explosion)
* Joerg Arndt <arndt@jjj.de <http://gosper.org/webmail/src/compose.php?send_to=arndt%40jjj.de>> [Feb 21. 2012 18:09]:> * Bill Gosper <billgosper@gmail.com <http://gosper.org/webmail/src/compose.php?send_to=billgosper%40gmail.com>> [Feb 11. 2012 15:10]:> > [...]
OK, here is what I get:> [...] And this is Clausen's> [...]
I tried to write that up, but then my brain exploded. (There were just to many changes in variables and notation in what was in this mail). Could define K(k,n) and N(k,n) and show which product gives Clausen's expressions? If possible, please omit any variables that are specialized out of existence later. 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}}}} In[1495]:= picheck[mats[]] Out[1495]= True 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}]] During evaluation of In[1511]:= Sum::div:Sum does not converge. >> During evaluation of In[1511]:= Sum::div:Sum does not converge. >> I took the HoldForms out of MProd so they wouldn't clutter up these InputForms. This was my reward: Out[1511]= {{0, (Log[1 - q] + QPolyGamma[0, 1, q])/Log[q]}, {0, 1}} == {{0, Sum[-((q^(1 + (-1 + k53753)*(1 + k53753))*(-1 + q^(2*k53753)))/(-1 + q^k53753)^2), {k53753, 1, Infinity}]}, {0, 1}} In[1512]:= newindex[%[[1, 1, 2]], n] == newindex[%[[2, 1, 2]], k] Out[1512]= (Log[1 - q] + QPolyGamma[0, 1, q])/Log[q] == Sum[HoldForm[-((q^(1 + (-1 + k)*(1 + k))*(-1 + q^(2*k)))/(-1 + q^k)^2)], {k, 1, Infinity}] This is how it looks with the HoldForms. In[1513]:= zapsum[qform[Simplify[#1]] & , zumorg[Out[1504], 1]] Out[1513]= Sum[HoldForm[q^n/(1 - q^n)], {n, 1, Infinity}] == Sum[HoldForm[(q^k^2*(1 + q^k))/(1 - q^k)], {k, 1, Infinity}] Sum should mean LeaveThisSumTheHellAlone, and only do its present contortions under a new function, say ClosedForm. Or FunctionExpand in a pinch. --rwg
* Bill Gosper <billgosper@gmail.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 ]; } Nm(k,n)={ [ q^(k), q/(1-q^(k+n)); 0, 1 ]; }
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] Rk = prod(n=0,N, Km(oo,n) ) \\ == [0, 1; 0, 1] R = Rn * Rk \\ == [0, XXX; 0, 1] \\ NOT OK \\ 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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Joerg Arndt