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Published online by Cambridge University Press: 20 January 2009
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
then
where 
denotes the coefficient of xr in the expansion of (1 + x)m.
* Journ. de l' Ecole Polyt. 16 (1839), 123.Google Scholar
† The usual proof depends on expressing μ(x) as an integral. See Nielsen, : Handbuch der Theorie der Gammafunktionen; p. 284 et seq.Google Scholar
‡ This is merely the theorem that F(a, b; c; 1) can be expressed as
* Landau, : Munich Sitzungsberichte (1906) 36, 151–221.Google Scholar
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