549] NOTE ON THE MAXIMA OF CERTAIN FACTORIAL FUNCTIONS. 
549 
In every case the maxima decrease from Y x which is the greatest, to Y p or F p+1 
which is the least; in particular, n = 2p +1, then 
- i (* - ^rr) - <* -1) 
/i 2p-l L_ 
V 2 2.2p+l ”‘~2.2p + l 
which is 
2 2 ^ +2 . {'Ip +1) 2 ^ 
(2p+iyp ’ 
{r(p + l)-s-Tj} 2 P(p + *) 
4 (2p + l)^ 47t (2p + l) 2p ’ 
we have 
and so 
Suppose p is large; then, as for large values of x, 
Tx = \J{2tt) af~^e~ x , 
r (p +1) = V(27r) (p + hY e ~ p ~ l 
= V(2tr) pP e ph *( 1+ £)*-*>-* = V(2tt) tf-P, 
(2p + lp = (2\p) 2p . e 2pl ° g ( 1+ ^) = 2^p 2 ^ e, 
v _ 27rp 2p e~ 2p _p 2 e~ 2p ~ 1 9 ( 1 
i «4-1 = . ^ —:— — p~ I :r~ 
p+l 4tt2 2p p 2p e 2 2p ~ l 
Also Fi corresponds approximately to 
2e. 
2J3+1 
F, 
2 2p + 1 2n ’ 
113 2n-l 1 
1 j F (w + i) 
— ™n+1 Ì • 2 * 2 •** ( n Ì) 
1 r (2p+ f) 
Now 
and 
so that 
n n+12 Tl 2(2p+l) 2 i ,+1 VO) ' 
T (2p +1) = V(2tt) (2p + f) e- 2 ?-i = V(2tt) (2p) 2 i> V 2p+l)I ° g ^ + e~ 2p ~$ 
= V(2tt) 2 2p+2 p 2p+2 e~ 2p , 
(2p + l) 2 ^ 41 = (2p) 2p+l e [2p+1) ]og ( 1+ *p) = (2p) 2p+l e ; 
1 VIStt) . 2 2p+ ' 2 . p 2p+2 e~ 2p 
Y> = 
2 2p+2 p 2 P +1 e * 
VO) 
= pV(2) 
g2p+l ’ 
so that, p being large, Y l is far larger than Y p+1 .