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Note on Binet's Inverse Factorial Series for μ(x)

Published online by Cambridge University Press:  20 January 2009

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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.

Type
Research Article
Copyright
Copyright © Edinburgh Mathematical Society 1924

References

* 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, 151221.Google Scholar