From the fragment below, it's not hard to guess the q upper index negation rule (a la GKP (5.14)): QBinomial[n, k, q] == (-1)^k q^(-(1/2) k (-1 + k - 2 n)) QBinomial[-1 + k - n, k, q] (checked out on http://mathworld.wolfram.com/CauchyBinomialTheorem.html). Here's <http://gosper.org/gea.pdf> a clarifying pdf I typesot for George Andrews. (But it *does* work to simply delete GMail's stinking linebreaks one at a time.) I remain convinced that this is all very old hat. --rwg BtW, http://mathworld.wolfram.com/CauchyBinomialTheorem.html works for complex n if you simply run the sum to ∞, and interpret the QPochhammer carefully in terms of q-factorials.
On Tue, Jul 25, 2017 at 8:27 PM, Bill Gosper <billgosper@gmail.com> wrote:
This must be well-known, but not by Wikipedia nor Mathworld. (I've misplaced my BHS again! This all might fall out of the q-Beta function, but not glaringly, due to the reciprocal.)
Here is a neighborhood of the apex. Table[qbin[n, k], {n, -3, 6}, {k, -2, 3}] // TableForm
2 2 2 2 2 3 4 1 + q + q (1 + q ) (1 + q + q ) (1 + q ) (1 + q + q + q + q ) -(----------) --------------------- -(-------------------------------) 3 7 12 0 0 1 q q q
2 2 1 + q 1 + q + q (1 + q) (1 + q ) -(-----) ---------- -(----------------) 2 5 9 0 0 1 q q q
1 1 1 -- -(--) -(-) 3 6 0 0 1 q q q
0 0 1 0 0 0
0 0 1 1 0 0
0 0 1 1 + q 1 0
2 2 0 0 1 1 + q + q 1 + q + q 1
2 2 2 2 0 0 1 (1 + q) (1 + q ) (1 + q ) (1 + q + q ) (1 + q) (1 + q )
2 3 4 2 2 3 4 2 2 3 4 0 0 1 1 + q + q + q + q (1 + q ) (1 + q + q + q + q ) (1 + q ) (1 + q + q + q + q )
2 2 2 2 2 3 4 2 2 2 3 4 0 0 1 (1 + q) (1 - q + q ) (1 + q + q ) (1 - q + q ) (1 + q + q ) (1 + q + q + q + q ) (1 + q) (1 + q ) (1 - q + q ) (1 + q + q + q + q )
qbin[n_, k_Integer] := Factor[qpochgunch[FunctionExpand@QBinomial[x, k, q], x - k + 1, k] /. x -> n]
qpochgunch[xp_, old_, by_] := xp /. QPochhammer[q_^old, q_, n___] :> Product[1 - q^¢, {¢, old, old + by - 1}] QPochhammer[q^(old + by), q, n]
(*Preserves the value of QPochhammers if q is atomic.*) (*NB: I have
In[461]:= Product[f@k, {k, 0, -2}]
Out[461]= 1/f[-1] *) Note q-Pascal's *automatic* asymmetry for negative n.
These values are completely consistent with the two recurrences
{QBinomial[n, m, q] == QBinomial[n - 1, m - 1, q] + q^m QBinomial[n - 1, m, q], QBinomial[n, m, q] == q^(n - m) QBinomial[n - 1, m - 1, q] + QBinomial[n - 1, m, q]}
(From which one can In[445]:= Eliminate[%%, %%[[1, 2, 1]]] , getting the two term recurrence
Out[445]= (-q^m + q^n) QBinomial[n, m, q] == q^m (-1 + q^n) QBinomial[-1 + n, m, q]
and another one.) (As before, imposing "Bermuda" symmetry breaks the recurrences. I think it would be genuinely interesting to find the actual objection that led to this fiasco.)
Mathematica's Limit and FunctionExpand both misbehaved here:
In[458]:= FunctionExpand@QBinomial[-2, 1, q]
Out[458]= Indeterminate
In[459]:= FunctionExpand@QPochhammer[1, q]
Out[459]= QPochhammer[1, q] (* Hello? *)
In[430]:= Limit[QBinomial[n, 3, q], n -> -2]
Out[430]= -(QPochhammer[1/q^4, q]/((-1 + q)^3 (1 + q) (1 + q + q^2) QPochhammer[1/q, q]))
This is 0/0! --rwg