Out[1131]= k q Sum -------------------------------, 1 - 2 k 2 1 + 2 k 2 (1 - q ) (1 - q ) k,-∞,∞ = 1 2 4 (1 + q) QFactorial[-(-), q ] 2 ----------------------------- 2 4 (1 - q ) q-Pochhammers in the sum but not the summand? "q-Mittag Leffler"? It q-retracts to 16 Sum[-----------, {k,-∞,∞}] == 2 2 (1 - 4 k ) 2 2 π Mathematica is dumbstruck. For minutes on end. Are these actually common? Test for q=1/2: In[1132]:= N[Activate[%1131/.q->.5]] Out[1132]= True Term ratio: In[1180]:= DiscreteRatio[%1131[[1,1]],k] Out[1180]= (q (1-q^(1-2 k))^2 (1-q^(1+2 k))^2)/((1-q^(-1-2 k))^2 (1-q^(3+2 k))^2) Nine terms of the sum give 27 term agreement with the product, despite the lack of q^k²: In[1189]:= Series[%1131/.\[Infinity]->9/.-\[Infinity]->-9//Activate,{q,0,29}] Out[1189]= 1+5 q+12 q^2+24 q^3+44 q^4+68 q^5+102 q^6+150 q^7+202 q^8+270 q^9+354 q^10+446 q^11+561 q^12+693 q^13+838 q^14+1002 q^15+1195 q^16+1407 q^17+1638 q^18+1906 q^19+2179 q^20+2491 q^21+2842 q^22+3202 q^23+3610 q^24+4042 q^25+4504 q^26+5008 q^27+5549 q^28+6114 q^29+O[q]^30==1+5 q+12 q^2+24 q^3+44 q^4+68 q^5+102 q^6+150 q^7+202 q^8+270 q^9+354 q^10+446 q^11+561 q^12+693 q^13+838 q^14+1002 q^15+1195 q^16+1407 q^17+1638 q^18+1906 q^19+2179 q^20+2491 q^21+2842 q^22+3202 q^23+3610 q^24+4042 q^25+4504 q^26+5008 q^27+5550 q^28+6114 q^29+O[q]^30 In[1190]:= Subtract@@% Out[1190]= -q^28+O[q]^30 I got it by mildly abusing the (bilateral) "q-Dixon's theorem" in http://math.sun.ac.za/~hproding/pdffiles/Triples-long.pdf which is not the (unilateral) q-Dixon's I get from path invariant matrices. --rwg