Neil fixed up my Mac with the next-to-latest version of Robert Munafo's ries searcher (www.mrob.com/ries), which led me to (1/4)! == Pi^(3/4)* EllipticTheta[3, 0, Exp[-4*Pi]]/(2^(1/4) + Sqrt[2]) which I had only known as a quotient of nested radicals, having neglected to attack it with Corey's denester. This also gives the simplifications {EllipticTheta[4, 0, E^(-4*Pi)] == (2*2^(1/16)*(1/4)!)/((-1 + Sqrt[2])^(1/4)* Pi^(3/4)), EllipticTheta[2, 0, E^(-4*Pi)] == ((-2^(1/4) + Sqrt[2])*(1/4)!)/Pi^(3/4)} Combining the theta[3] and theta[4], (1/4)! == ((2*Pi)^(3/4)*EllipticTheta[3, 0, E^(-16*Pi)])/ (1 + 2^(1/4) + 2^(13/16)/(-1 + Sqrt[2])^(1/4)) Taking the theta[3] terms just through first order, In[609]:= {(1/4)!, 2^(3/4)*(2/E^(16*Pi) + 1)* Pi^(3/4)/(2^(13/16)/(Sqrt[2] - 1)^(1/4) + 2^(1/4) + 1)} 2 3/4 (1 + ------) (2 Pi) 16 Pi 1 E {(-)!, ----------------------------} 4 13/16 1/4 2 1 + 2 + ----------------- 1/4 (-1 + Sqrt[2]) N[%, 88] {0.9064024770554770779826712889669180007487919207200163668583444998924798108846822804045900, 0.9064024770554770779826712889669180007487919207200163668583444998924798108846822804045892} --rwg Equal@@<the previous pair> tricks Mma into not saying False.