Here's the "closed form". You may need to temporarily Inactivate the Sums to prevent such automatic cleverness as replacing a sum of two terms by a difference of two infinite series! z! ~ (E^(-EulerGamma z + Sum[(-z)^n Zeta[n]/n, {n, 2, 1 + S}] - Sum[(z Sum[-(-j)^-n z^n/(1 + n), {n, 0, S}])/j, {j, F}]) F!)/Pochhammer[1 + z, F] where S is the number of zeta terms and F is the number of factorial terms. Tune them to meet your price/performance needs. Accuracy drops off smoothly outside |z|≤1/2. This is probably an identity if either S or F blows up. I haven't mapped the region of convergence. Might be everywhere but the poles. —rwg On Mon, Jan 21, 2019 at 2:09 PM Bill Gosper <billgosper@gmail.com> wrote:
For modest |z| and increasing k and n, E^((-z (-z/n)^k LerchPhi[-z/n, 1, 1 + k] + n Log[(n + z)/n])/n)/(1 + z/n) respectably approximates 1. For specific positive integers k, Mathematica finds that
Product[E^((-z (-z/n)^k LerchPhi[-z/n, 1, 1 + k] + n Log[(n + z)/n])/n)/(1 + z/n), {n, m,∞}]
is a multiple of z! E.g., for k=4, m=9, -1/2 < z < 1/2, 1.00000015 > z! E^(EulerGamma z + 1/8 z (-1 + z/16 - z^2/192 + z^3/2048 + (8 Log[1 + z/8])/z) + 1/7 z (-1 + z/14 - z^2/147 + z^3/1372 + (7 Log[1 + z/7])/z) + 1/6 z (-1 + z/12 - z^2/108 + z^3/864 + (6 Log[1 + z/6])/z) + 1/5 z (-1 + z/10 - z^2/75 + z^3/500 + (5 Log[1 + z/5])/z) + 1/4 z (-1 + z/8 - z^2/48 + z^3/256 + (4 Log[1 + z/4])/z) + 1/3 z (-1 + z/6 - z^2/27 + z^3/108 + (3 Log[1 + z/3])/z) + 1/2 z (-1 + z/4 - z^2/12 + z^3/32 + (2 Log[1 + z/2])/z) + z (-1 + z/2 - z^2/3 + z^3/4 + Log[1 + z]/z) - 1/360 z^2 (30 \[Pi]^2 + \[Pi]^4 z^2 - 120 z Zeta[3]))
1 - .00000015. —rwg