Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-24T13:59:49.675Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  20 January 2009

E. T. Copson
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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
Information
Proceedings of the Edinburgh Mathematical Society , Volume 43 , February 1924 , pp. 103 - 105
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, 151–221.Google Scholar