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