22 THE REV. F. H. JACKSON ON 
we have an identity 
m—Tt _ 7 ' , jee ill 
: - = ae pe —1 -p 2 ; 
[x a5 Yn — ae ot 2p { a p = il . p™ = il bg de p™ a 1 (ail henal ils . (56) 
Subject to a proper interpretation of [«],,, this expansion holds for all values of m 
provided the series be convergent, the condition for which is p>1. If m be integral 
1 
Ene es ee ae 
[2] denotes 
The theorem in its generality is discussed in a paper on ‘Series connected with the 
Enumeration of Partitions,” series 2, vol. i, Proc. Lond. Math. Soc. In the following 
work we require only the simple cases in which m is an integer positive or negative. 
For y substituting -2n 
x 2n + 2r 
l 2 
m x (a positive integer) 
we have 
fn rm [an r+ Signer RA [an Br} 2 
Now [27+ 27],_, is 
[20+ 27] [2n4+2r-2].... [27+2s+2] = [m+r][ntr-1]).... [n+s4+1]- a 
and [-2n], = [-2n][-2n-2]..... [ —2n —2s+ 2] 
= (-l)p"**[2n] [2n+2] . . . [2n+2s—-2] 
= (- 1)%p-ns-#+5) 7] [n+ 1] Cent Mi [n+s = 1} ; Cp 
So that we have 
(2),[r]! = [m+r][mt+r-1]... [n+ eee 
2 Re r|! 2), E : 2) ah 
+ Pas Lies wt is]! oy ‘[m4+r].... [wts+ ee? x 
[n][m+1]:.. [n+s- ies 
Dividing throughout by [7 +7]! (2),+,{7']! (2), this reduces 
ee 1 Ft Wn a 1 
[R+7! Ona ~ QT, + S- VP ss] @)a@ets! On * 
ee te ley, aoa OS) 
Dividing throughout again by [7]! (2), this identity becomes 
1 a oe es 1 
[mtr]! [7]! (2)nrr(2)r [r]!(2)n Er)! [r]! (2),(2), z 
S(— ype! 1 (nj [n+1] ... [e+s-1) 
2 ale TCC aC) oS 
1 
* Goa Oa) 69) 
