: 11)—a. i ~^iwg—W 
383] PROBLEMS AND SOLUTIONS. 
575 
duced 
(2) The above formula expressed as 
(Ike) 2 n 2 1 n 2 {n 2 -l 2 ) 1 n 2 (n 2 -1 2 ) (n 2 - 2 2 ) 
i)~ l’x + 1.2 ‘x(x + 1) 1.2.3 
T (x — w) T (x + n) 
x(x-\-l){x+ 2) 
+ &c. 
and show that this equation is subject to the sole restriction that when n is not 
integral x must not be negative. 
Solution by Professor Cayley; and X. U. J. 
Let n be a positive integer, and suppose that 0] № denotes as usual the factorial 
x{x — 1).... (x — n + 1); then we have 
[x 4- lc] n = (1 + A) fc [x\ n — f 1 4- &A + ^ ^ — A 2 + (Ssc-'j [oc\ n 
№+y w»->++&c. ; 
or putting k — — n we have 
n 2 
\x — n\ n — [#] w — y 
n 2 (n 2 — l 2 ) 
1.2 
\x\ l ~ 2 — &c. 
Writing herein (x + n — 1) for x, and dividing by [oc+n— l] n , we have 
[æ-l] n n 2 1 n 2 (n 2 -l 2 ) 1 
[tc+n- l] n ~ ~ï'x + ÏT2 * xJx + V} ~ &C ' ’ 
or, what is the same thing, 
(Yx) 2 
_ n 2 1 n 2 (n 2 — l 2 ) 1 
T (x — n) T (x + n) 1 ' x^ 1.2 ' x(x+l) 
&c., 
which is the formula (2). The foregoing demonstration applies to the case of n a 
positive integer; but as the two sides are respectively unaltered when n is changed 
into —n, it is clear that the formula holds good also for n a negative integer. The 
right hand side is the hypergeometric series F(n, —n, x, 1) and the formula therefore is 
(IV) 2 
T (x — n) T (x + ri) 
a particular case of the known formula 
r (7) r (7 ~ « ~ /3) 
P (7 “ a ) r (y ~ /3) 
— F(n, —n, x, 1), 
= F(cc, /3, y, 1), 
Avhich when a or /3 is a positive integer is a mere identity, true therefore for all 
values of y; but if neither a nor /3 is a positive integer, then the right hand side 
is an infinite series which is only convergent for y > a + ¡3. In the particular case we