GENERATING FUNCTIONS IN THE THEORY OF NUAIBERS. 
155 
whence writing 
we obtain the matrix 
Cpx) —- (1, 
1) 
1 
^21 
^31 
1 
^32 
X,,.2 
1 
1 
1 . 
Kis 
■ 
1 
1 
1 
1 
and we can interpret the coefficient of in the corresponding particular 
redundant product as in Art. 15. 
Again, writing 
which, as far as a-p,p+i is concerned. Art. 59 shows to be legitimate, we have 
l^p,i/ — ^p^p+\ • • • ^'^—1 — 1 — suppose, 
and the matrix 
L 
^21 *^1 
Hi ^12 
Hi ^13 
f^nl ^1, yi— 1 
1 
1 
X 32 Xg 
Hz ^23 
(^n2^2,n-\ 
^31 
H'Sl -^12 
1 
^2 
1 
^ 4.3 ^3 
n-l 
^+1 
^42 
1 
1 
* /^n4 ^^4, n—1 
P'il ^13 
^''42 ^23 
•U 3 
) 
/J'la Aj, u-i 
/J'ti'l ^2, «—1 
'-JiS 
P'nZ ^3, »-l 
^ 1 , n —1 
and the new particular redundant product is :— 
1 
Xj 
+ 
Xgi XqX-^ 
+ 
Hi ^31 
+ 
Hi ^41 
+ • • 
• "h Pki X, 
1 
+ 
X 3 
+ 
X 32 X 3 X 3 
+ 
Hz ^42 
+ • . 
• + P ',12 X., 
^31 
^31 *'2 
+ 
1 
+ 
X 3 
+ 
X 43 X 4 X 3 
+ • • 
• "b H^ X, 
A 4 ] 
+ 
^42 
+ 
1 
+ 
+ • . 
• + Hi X, 
A‘'41 ^32 
Hz ^3 
■ + 
+ 
Am3 
+ 
+ . 
• + YC„ 
/^m X«—1, 2 1,1 n 
X 2 
Y' 
y. 
