Factors of alternating sums of powers of q-Narayana numbers

The q-Narayana numbers Nq(n,k) and q-Catalan numbers Cn(q) are respectively defined byNq(n,k)=1−q1−qn[nk][nk−1]andCn(q)=1−q1−qn+1[2nn], where [nk]=∏i=1k1−qn−i+11−qi. We prove that, for any positive integers n and r, there holds∑k=−nn(−1)kqjk2+(k2)Nq(2n+1,n+k+1)r≡0(modCn(q)), where 0⩽j⩽2r−1. We also propose several related conjectures.