On Wed, Feb 8, 2012 at 12:01 AM, Bill Gosper <billgosper@gmail.com> wrote:
Puzzle: Assuming[0 < q < 1, Limit[QPochhammer[a*c, q]/QPochhammer[b*c, q], c -> \[Infinity]]]
Or maybe Assuming[0 < q < 1, Limit[QPochhammer[a*x, q]* QPochhammer[b*x, q]/QPochhammer[c*x, q]/QPochhammer[a*b*x/c, q], x -> \[Infinity]]]
Wisely, no one touched this. Obviously, for a,b,c >0, the limit doesn't exist. For typical numbers, the plot from x=9^69 to 9^70 resembles cot(sqrt x), i.e. poles at decreasing frequency. For a,b,c <0, things are less clear. For huge x (~9^99) there is a persistent, smooth, tiny ripple of decaying frequency and amazingly slowly decaying amplitude. Mma's plot seems to show the sudden disappearance of this ripple as q increases beyond some threshold, but this is an artifact. At least in 8.04, Plot[QPochhammer[a*x, q]*QPochhammer[b*x, q]/QPochhammer[c*x, q]/QPochhammer[a*b/c*x, q] - 509719561642/10^12 /. {a -> -1/2, b -> -1/2, c -> -1/4, q -> 25/51}, Evaluate[N[{x, 9^96, 97}, 105]]] shows just the tips of these ripples, with the rest of the wave clipped to 0. Assuming the ripples eventually die, converting the limit to a bilaterally infinite product suggests the answer is Assuming[0 < q < 1&&a<0&&b<0&&c<0, Limit[QPochhammer[a*x, q]* QPochhammer[b*x, q]/QPochhammer[c*x, q]/QPochhammer[a*b*x/c, q],x -> Infinity]] == ((QPochhammer[a, q] * QPochhammer[b, q] * QPochhammer[q/a, q] * QPochhammer[q/b, q])/ (QPochhammer[((a * b)/c), q] * QPochhammer[c, q] * QPochhammer[q/c, q] * QPochhammer[((c * q)/(a * b)), q])) Testing, In[56]:= N[% /. {a -> -1/2, b -> -1/2, c -> -1/4, q -> 25/51}] Out[56]= 0.509719561642272 in agreement with the offset in the clipped plot. --rwg As for that QPochhammer generating function, Mma does nothing with FunctionExpand[QHypergeometricPFQ[{x, q}, {0}, q, t] == Sum[QPochhammer[x, q, n]*t^n, {n, 0, Infinity}]] and yet it knows the q-binomial thm: In[55]:= FunctionExpand[QHypergeometricPFQ[{x}, {}, q, t] -> Sum[QPochhammer[x, q, n]/QPochhammer[q, q, n]*t^n, {n, 0, Infinity}]] Out[55]= QPochhammer[t x, q]/QPochhammer[t, q] -> QPochhammer[t x, q]/QPochhammer[t, q] So maybe that simple-looking g.f. is unknown.