[math-fun] pi / factorial base (versus e)

Well, two natural questions arise: Q1. is there any known series expansion for pi (or something closely related) which, like the usual series for e, converges factorially rather than merely geometrically? If said series arose from an analytic function's Taylor series, then this would necessarily be an "entire" function, ruling out stuff like arcsine, arctan, elliptic functions, elliptic integrals, log, and polylogs, but permitting, e.g, the reciprocal of the Gamma function, and erf and erfc, and new functions. Q2. Of course pi could be defined via sin(pi)=0, or cos(pi/2)=0, or... and such equations are somewhat easier to work with if you are working with factorial-base numbers.