300 
PROFESSOR SYLVESTER AND MR. J. HAMMOND 
a 
■>/ + -[ 
— b n + 
a H (a n + 1) 
1 . 2 
7, _ J_ n 7, a n( a n +1) (-«» + 1) 
u n + 1 — ~T tl ii u n U-z-~ o- 
1 . 2.0 
C n+ 1 — 
7 . . "P 1) 7 i ®«(®« "P 1) (&» “P 2 ) (3a„ -r 1) 
i “T" ^ ^- o'# “T” - 
1 . 2 
1 . 2 . 3.4 
Now 
(1 — x ) ff « = 1 -j- a„a; + 
C( n (fl n + 1) ^2 , a n( a n P 1) ( a n + 2) 
a : 2 + 
1.2 1 1.2.3 
when multiplied by 
F ?i (x) = a n x n + b n x n + 1 + c„af +a + c/^af + 3 + 
a: 3 + 
gives 
(1 — x) "" F„(x) = a n x 11 + b, L x n +1 + c*&” + 2 + cl H x n + 3 4 . . 
+ a„V + 1 + +2 + a ;i c ;i aP + s 4 . . 
ci n ~(a n - 1 - 1 ) n + 2 , ci n (a n + 1 ) 
+ 
1 . 2 
-s» + s + g 7 b u x^ + • • 
+ + !) (On + 2) jgm + 3 _|_ 
1.2.3 
Comparing this with 
+ • 
f 
F Wfl (ai) = b H x n + l + c n x n + 3 + cl u x n + 3 + . . . 
a, ' a ' 1 + 1 ^ x n +1 4 - CLub n x n + 2 4 a n c n x n + 3 + . . . 
1 . 2 
_p g "( g ” + ^ + ^ ^ + 2 _p a "( a >‘ + 1 ) b nX n + 3 _p 
1.2.3 1.2 
+ 
a n( a >i + 1) {ct n + 2) (3 a„ + 1) 
1 . 2 . 3.4 
we see that the difference of the two expressions is 
x 
.« + 3 
+ • • • 
+ • • • 
(a n - IK ,^ + 1 , fa ~ 1) «»(«»+ 1) +9 («, ; -lK(«. M + l)(a,, + 2) ;; + 3 
^ 1.2.3 X ^ 1.2.3.4 33 + 
which is equal to 
a n x n + - - 9 " x 
Thus 
X n ] (1 — — at t-1 (l — x). 
F ;; + 1 (x) = (1 — x)~ a n F„(.x) — at' -1 (l — x)~ a * + 1 + a:" _1 (l — a-).' 
Multiplying this equation by (1 — a , )v+q where 
s » +1 — a o H~ a ] 4~ a 2 + • • • + a » -1 4" a n, 
* See Note 2, p. 312. 
