6jo 



THE r.AMMA, OR FACTOkTAL, FUNCTION. 



Proposition X. — {n)a = — \z^"~'{i+z)"d2 round the unit circle 



27riJ 



ii 11 > — I and a has any vahie. 



Propositions resulting from the above (proofs not given) on 

 extended Binomial formulae and Fractional differentiation, 

 with applications to Hypergeometric series and fractional 

 spherical harmonics. 



The Propositions, though forming, it is hoped, a continuous 

 whole, are not to any great extent interdependent. Some proofs 

 are old established : some are new. 



The ordinary factorial of a positive integer is fully defined by 

 the relation n\ = n («— i) !, with the special value o ! = i. 



If we had a meaning for .r ! when .r is not a positive integer, 

 many algebraical formulae, especially the binomial coefficient, could 

 be simplified. We proceed to discuss whether there is a simple 

 extension of n ! to the general case. The conclusion to which we 

 shall come is as follows : 



PROPOSITION I. 



If n.v is a function satisfying the relation n.v=.rn(.v— i), and 

 equal to .r ! when .r is a positive integer, then there is an infinite 

 variety of possible forms of n.r, but one special form which may 

 be described as the ' simple ' form. 



The relation Ux=xU{x—i) leads of course to 



Ilx = 



Uix+N) 



{x-\-i){x-\-2)...{x+N) 



where A^ is a positive integer. And if n(.r+A^) can be determined 

 when A^ — > oo the definition of Ux is complete. 

 Now if a: is a positive integer 



U{N+x) = {N-i-x) (A^+.v- 1) . . . (A^ + i)nA^ 



which''' 



<nA^ 



{N-i-xy 



< nN . A^* 



1 + 

 1 + 



I 



A^ 



X 



N 



therefore, when .r is a positive integer, -irnrff]^ ^ i- 



* The notation u < 



is used throughout this paper to mean ' « lies 



between a and 6.' Where possible the upper symbol a is reserved for the 

 greater limit. The notation is then equivalent to a > ii > b. It will be seen that 

 the inequality admits of all the transformations of an equation. 



