OF UNRESTRICTED DEGREE. ORDER, AND ARGUMENT. 
4G1 
this result is proved by Heine* for the special case in which n and m are both 
integers. We see, therefore, that when n + to is a positive integer, the formula (15) 
is undetermined, the formula (17) must in that case be used. 
When n — to is a positive integer we must use (15), since (17) become in this case 
undetermined. When n + to is a negative integer, Q(p) is infinite, but we can 
take (p) sin (n fi- m) tt as a finite solution of the differential equation. 
When n and to are both real integers, and to is positive and > n, the form (17) is 
finite, but if to5p both the forms (15), (17) are undetermined, and must be modified 
by applying the rule for the determination of undetermined forms 0/0. 
The functions P„ m (p), Q„ m (p) are defined by Olbrecht for the general case, by 
means of equations, in our notation, 
P»* (p) = constant F ( — n,n + 1,1 + to, -— , 
Q„ m (p) = constant — j-'j F n, n + 1, 1 fi- to, 1 —) > 
this definition of Q/ ! (p) is, however, not consistent with the usual definition as in (10), 
in the form of a hypergeo metric series whose fourth element is —. 
Relation betiveen the functions Q„ m , P„ m . 
11. In the formula (15), write — n — 1 for n, we have then 
Q-rW = 
7re mm 
1 
2 sin (m — n — 1) tt II (— m) 
3 ± (r + 1) rrt 
p + iy* t i - p 
' h ( — n, n + 1,1 — to, —— 
p 
- l 
p-iy™ / . _ . i + p 
On eliminating the second hypergeometric series between this equation and (15), 
we find 
Q„ m (p) sin (n + to) tt — (p) sin (n — to) 7r 
(<,— + e ~) F f_ 
2n (— ?«) 1 \p — i 
= Tre mm COS TO7T • Vu l (p), 
TO. 
1 — p 
b v (5)- 
We thus obtain the formula 
* See ‘ Kugelfunctionen,’ vol, 2, pp. 238, 336. 
