Binet shewed that the function
can be expanded as an inverse factorial series. This note furnishes a new and much simpler proof of his result, based on a formula which is an analogue of the Binomial Theorem for factorials.
This formula is that, if we denote by [x]n the ratio
where denotes the coefficient of xr in the expansion of (1 + x)m.
(Received March 06 1924)
† The usual proof depends on expressing μ(x) as an integral. See Nielsen: Handbuch der Theorie der Gammafunktionen; p. 284 et seq.
‡ This is merely the theorem that F(a, b; c; 1) can be expressed as