FUNDAMENTAL POWER SERIES AND THEIR DIFFERENTIAL EQUATIONS. 35 
“ 
THe GENERAL HypeRGEOMETRIC SERIES. 
Let ¢ denote the differential operator 
B+ By2"D! + Bert eD4 0... +BaD) (28) 
and ~y denote the operator 
Ee | C2D+C,c!D +... + CaD" | (29) 
nas 
ae Dit 
B,, being 2 -1) moe +(n—1)) : : . (30) 
Cn Sy, ae Tere IIB +(n—7)) (31) 
Then if y denote a series of the form 2Ax™ 
that is 
Avan! aie Ava”) + ee anes Sh [01 38 Sia 
and we operate on y with ¢— we obtain 
oot 
{p-Y}y= DUATL(a + (m))2™— DY AX TL(B + (m) — (1) D2) 
The lowest index present in the series on the ie: is (m,)—(1), and the term in which 
it occurs is 
a, nn (e + (m,) = (1))acdm-) 
1 
Choose m, so as to make this term vanish, the possible values of m, are zero and 
(1)—8. Now, from the manner in which (m,)(m,).... are formed, we cannot have 
any such relation as (m,)=(m,,,)—(1) unless the elements p, p.p..... are equal to 
one another, so we choose 
(m,) =0 
Gn) =O) =p, 
(mg) = (2) =p, + py 
Gay Bie 
and 
A, - (B+ (ins) — (1) = AIL, (a + (7n,)) 
Then we have ; 
11,(a) Mo)Iifo-+ (1) 
F=1 Bee kl rates MOS : : 3 
‘MO *? @MeMe+e-O) a 
the general term being 
II,(a)I1,(a + (1) es i os cee II,(a + aa? x”) 
PG MARIE (2)=()).. 2... (B+ (m)-(1))” ok 
and a differential equation 
