The b(k)^n in 'sum(b(k),k,1,n) = 'sum(b(k)^n/'prod(b(k)-b(j),j,1,k-1)/'prod(b(k)-b(j),j,k+1,n),k,1,n) n n ==== ==== n \ \ b (k) > b(k) = > ------------------- / / n ==== ==== /===\' k = 1 k = 1 | | | | (b(k) - b(j)) | | j = 1 looks improbable but works. For what I'm unsure. qpoch(q,q,inf)*(1-qpoch(z,q,inf)) = z*'sum((-1)^k*q^(k*(k+3)/2)*qpoch(-z/q^k,q^2,inf)/(qpoch(q,q,k)*(q^k*z+1)),k,0,inf) k (k + 3) --------- k z 2 2 oo (- 1) (- --; q ) q ==== k oo \ q (q; q) (1 - (z; q) ) = z > ------------------------------ oo oo / k ==== (q; q) (q z + 1) k = 0 k has a somewhat oddball lhs, a naked z, and a rather stubborn infinite product in its summand. Computationally (e.g., for evaluating eta(q)/q^(1/24)), it's generally inferior to the q-Exponential qpoch(z,q,inf) = 'sum((-1)^n*q^((n-1)*n/2)*z^n/qpoch(q,q,n),n,0,inf) (n - 1) n oo --------- ==== n 2 n \ (- 1) q z (z; q) = > -------------------- oo / (q; q) ==== n n = 0 (as opposed to the q-exponential). Both sums (and the equivalent infinite product) bog down as |q|->1, which doesn't matter for eta due to the imaginary transformation, but thwarts Pochhammer numerics in general. A partial remedy it to use (z;q)_oo = (z;q)_k (z q^k;q)_oo, qpoch(z,q,inf) = qpoch(z,q,k)*'sum((-1)^n*q^(n*(n+2*k-1)/2)*z^n/qpoch(q,q,n),n,0,inf) n (n + 2 k - 1) oo --------------- ==== n 2 n \ (- 1) q z (z; q) = (z; q) > -------------------------- oo k / (q; q) ==== n n = 0 (of which the previous sum is the k=0 case), which improves the term ratio to -q^(n+k)*z/(1-q^(n+1)) n + k q z - ----------, n + 1 1 - q for however large k that optimizes the sum vs product tradeoff. --rwg