Proceedings of the Edinburgh Mathematical Society
- Proceedings of the Edinburgh Mathematical Society / Volume 43 / February 1924, pp 103-105
- Copyright © Edinburgh Mathematical Society 1924
- DOI: http://dx.doi.org/10.1017/S0013091500036373 (About DOI), Published online: 20 January 2009
Research Article
Note on Binet's Inverse Factorial Series for μ(x)
E. T. Copson
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.
(Received March 06 1924)
Notes
* Journ. de l' Ecole Polyt. 16 (1839), 123. [OpenURL Query Data] [Google Scholar]
† 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
* Landau: Munich Sitzungsberichte (1906) 36, 151–221. [OpenURL Query Data] [Google Scholar]
