284 
POWER SERIES 
For let p, q, r be any three integers. Consider the functions 
F(a/3yx}, + 1, £, y, x) ••• F(cc+p, /3, 7, *), 
FÇa+p, 13 + 1, 7, a), F(a+p, /3 + 2, 7, x) F(u+p, /3 + q, 7,2:), 
We have p + ^ + r + 1 functions, and any 3 consecutive ones 
are contiguous. There are thus p + q + r — 1 linear relations 
between them. We can thus by elimination get a linear relation 
between any three of these functions. 
189. Derivatives. We have 
«•« + !••••« + n—1 • /3 • /3 -t-1 • ••• /3 + n — 1 
& 7,x) = ^n 
^ 1 . 2 • ••• n • 7 • 7 + 1 • ••• 7 + n — 1 
_ yp «•« + !• -••« + w./3-/3 + l- • • • /3 + w 
â 1 • 2 • ••• w+1 • 7 • 7+1 • ••• 7+ n 
— ^ V « + !••••<* + w • /3 + 1 - • • • fi + n x , n 
7 Y 1-2 -••-w+ 1- 7 + 1* •••7 + w 
= ^^(«+1, /3+1, 7 + 1, *). 
7 
Hence 
F" (a, /3, 7, *) = — (« + 1, /3 + 1, 7 + 1, *) 
7 
^C a + 2, /3 + 2, 7 + 2, x) 
and so on for the higher derivatives. We see they are conjugate 
functions. 
190. Differential Equation for F. Since F, F', F" are conju 
gate functions, a linear relation exists between them. It is found 
to be # 
x(x - 1) F" + {(« + /3 + V)x - 7} F' + a/3F — 0. (1 
To prove the relation let us find the coefficient of x n on the left 
side of 1). We set 
a • a + 1 • ••• a + n — 1 • /3 • /3 + 1 • ••• /3 + n — 1 
1 .2 - ... n • 7 • 7 + 1 • ••• 7 + w — 1