{ (E^(-(1/9) + (-2 \[Pi]^2 + 3 PolyGamma[1, 1/3])/( 36 Sqrt[3] \[Pi])) Glaisher^(4/3))/3^(25/72) == Hyperfactorial[1/3], (2^(2/3) E^(-(1/9) + (2 \[Pi]^2 - 3 PolyGamma[1, 1/3])/( 36 Sqrt[3] \[Pi])) Glaisher^(4/3))/3^(49/72) == Hyperfactorial[2/3], (E^(-(3/32) + Catalan/(4 \[Pi])) Glaisher^(9/8))/Sqrt[2] == Hyperfactorial[1/4], (3^(3/4) E^(-(3/32) - Catalan/(4 \[Pi])) Glaisher^(9/8))/(2 Sqrt[2]) == Hyperfactorial[3/4], (E^(-((\[Pi] (5 + 2 Sqrt[3] \[Pi]) - 3 Sqrt[3] PolyGamma[1, 1/3])/ (72 \[Pi]))) Glaisher^(5/6))/(2^(11/72) 3^(23/144)) == Hyperfactorial[1/6], (E^(-((\[Pi] (5 - 2 Sqrt[3] \[Pi]) + 3 Sqrt[3] PolyGamma[1, 1/3])/( 72 \[Pi]))) (5 Glaisher)^(5/6))/(2^(59/72) 3^(119/144)) == Hyperfactorial[5/6]} But we have Glaisher == E^(1/12 - Zeta'[-1]) and Zeta'[-1, 1/2] == - Log[2]/24 -1/2 Zeta'[-1] and Log[Hyperfactorial[z]] == z Log[z] - Zeta'[-1] + Zeta'[-1, z] and D[Zeta[s, a_Integer] == 2^s \[Pi]^(-1 + s) (-s)! Sin[1/2 \[Pi] (4 a + s)] Zeta[1 - s],s] and BarnesG[1 + n] n!^-n Hyperfactorial[n] ==1 I betcha Adamchik left WRI all this stuff. —Bill It's weird that we have "closed forms" for Hyperfactorial[fraction] for seven different fractions and Factorial[fraction] for only ½.