PART II. 
HYPERGEOMETRIC FAMILIES. 
A. DEFINING PROPERTIES. 
§ 1. REGULAR SINGULAR POINTS FOR BINARY FAMILIES. EXPONENTS. 
In Part I, A, § 4, we have shown that if the hole a is a point, a basis {y a ',y a ") nas 
i properly chosen circle about a, the Laurent's developments 
v zr -f co 
C a 
yj = (x - a) a ' 2 9 J (* - «)", 
(22) l=+l 
provided « is finite; if a is the point infinity, we need only substitute - for (x-a) in 
(22), and this remark will apply throughout the subsequent work where we explicitly 
consider only the case where a is finite 
In the above-mentioned section, wf 
have called a a regular singular point wl 
developments in (22) contain only a finite number of terms with negative values of v\ 
this is readily seen to be equivalent to the definition : A finite regular singular point a 
of a Unary family is an isolated singular point such thai, if (y', y") is some basis of the 
family, there exists a constant a for which the equations 
lim (x — a) a y' - 0, 
x = a 
lim (x — a) a y" = 0, 
x = a 
are always satisfied. These equations will then be true for every basis. We now 
examine in detail the fundamental bases of binary families for a regular singular 
point a. 
If a is ordinary, a fundamental basis for that point always has the developments 
-f-co 
(23) 
V« =(*- <*Y 2 9 J (* - ay, g > * 0, 
v = 
-{-co 
y a " = ( x ~ «)«» 2 g v " {x - a)*, g " * 0. 
i>=0 
