(For R-R, special-case a:=q) (c44) 'PRODUCT(MATRIX([0,A],[Q^K,1]),K,0,INF) = MATRIX([0,SUM(A^(N+1)*Q^N^2/QPOCH(Q,Q,N),N,0,INF)],[0,SUM(A^N*Q^(N^2-N)/QPOCH(Q,Q,N),N,0,INF)]); [ inf 2 ] [ ==== n n ] [ \ a q ] [ 0 a > -------------- ] inf [ / qpoch(q, q, n) ] /===\ [ ==== ] | | [ 0 a ] [ n = 0 ] (d44) | | [ ] = [ ] | | [ k ] [ inf 2 ] k = 0 [ q 1 ] [ ==== n n - n ] [ \ a q ] [ 0 > -------------- ] [ / qpoch(q, q, n) ] [ ==== ] [ n = 0 ] (c45) TAYLOR(PRUD(PART(%,1,1),K,0,7) = MAKEPROD(RHS(%)),Q,0,6); [ 2 2 2 2 3 3 2 4 3 2 5 3 2 6 ] [ 0 + . . . a + a q + a q + a q + (a + a ) q + (a + a ) q + (2 a + a ) q + . . . ] (d45)/T/ [ ] = [ 2 2 2 3 2 4 2 5 3 2 6 ] [ 0 + . . . a + 1 + a q + (a + a) q + (a + a) q + (2 a + a) q + (2 a + a) q + (a + 3 a + a) q + . . . ] [ 2 2 2 2 3 3 2 4 3 2 5 3 2 6 ] [ 0 + . . . a + a q + a q + a q + (a + a ) q + (a + a ) q + (2 a + a ) q + . . . ] [ ] [ 2 2 2 3 2 4 2 5 3 2 6 ] [ 0 + . . . a + 1 + a q + (a + a) q + (a + a) q + (2 a + a) q + (2 a + a) q + (a + 3 a + a) q + . . . ] This came from the eensy path-invariant 3x3s {{k, {{0, q^n, 0}, {q^(k + n), q^n, 1 - q^n}, {0, 0, 1}}}, {n, {{q^(k + 2 n)/(1 - q^(1 + n)), 0, q^n}, {0, q^(k + 2 n)/(1 - q^(1 + n)), 1}, {0, 0, 1}}}}. --rwg with the usual big help from Corey&Julian