OF UNRESTRICTED DEGREE, ORDER, AND ARGUMENT. 
471 
(/* + o. i) 
and 
7TC 
1 
2 sin (n -f on) it II ( — on) 
e~( n+im) *‘ (~Y‘ l F l—n, n+l, 1 -to, — 
1 - ^ 
U+p/ V 
?1+ 1, 1 —TO, 
2 
1 + ft 
7re' 
Q,F(/^ 0,t )~^sin(7H-m) 7 rn(-m){ e< F ( n,W + 1 » 1 m > 2^ 
from these equations we find 
1 1 + /fi y” 
irfi;) F (-».«+1, i-m,^ 
_ e -^Y- — ^Y"‘ «_li i __*v, 1 +/A i 
vr 
e~ hnm Q.,“ (pfOu) - Q, “ (/x - 0 . i) 
1 
77 C 
-( e -0+ m)m — e (»+)«KD /i AM' p / n n ! ^ 1 ^ 
2 sin (n + on) it II (— on) ' '\l + /x/ \ ’ 5 2 /5 
hence we have the relation 
e~ imm Q* (/x + 0 . t) - e’ iiWt Q* - 0 . t) ~ - or e mm P,/« (/x) . . (30), 
where P,/“ (/x) is defined as in the last Art. In the particular case to = 0 , (30) reduces 
to Heine’s relation 
Q« (/x + 0 . t) — Q„ (/X — 0 . i) — — ITT P )4 (/x). 
It is convenient to define Q,/“ (/x) for real values of /x between + 1 and — 1 by 
means of the equation 
e" lnL . Q/ (/x) = i {e-= iiiiU Q„* (/x + 0 . t) + Q„* (/x - 0 . i)} . . (31), 
which gives us 
Q/M = 
7T 
1 f / , \ /1 + /x\ iTO „ / 1 — /X 
cos(n+TO) 7 r. P-) r l—n, ft-fl, 1 —to, 
2 sin (» + m) 7r II ( — m) 
V 1 -/x 
1 — /x\- m -JJ, ( I T 1 1 + /X 
—j F (—«,« + !. 1-m, 
We have also 
(n + on — 1\ 
Q.-W = - 2*-‘ sin =±= » (1 -vf F (5±f±*, Pp 4, /.• 
+ 2 1 cos 
n 
(n + on\ 
»+. n (x) n(_i) 
(n 
2 ■ /«-™-ry (i-^ 2 )>F(~ 
n <—T~ 
m — n +1 m+ii + 2 3 9 
» 3 J 2 ) 
