OP UNRESTRICTED DEGREE, ORDER, AND ARGUMENT. 
521 
or 
Q»* (cosh xjj) 
cos rn 77- 2“ II (rn — A) II ( — A) . , 
= ---— sinh® xfj 
e (n+i)u 
^ (2 cosh u — 2 cosh 4) 
'//l+n 
, du 
( 120 ), 
where n — | is a real integer, and the real parts of m — n, ^ — m are positive. 
If m — 0, we have 
g(» +1) M 
Q« (cosh 0) = I - 
(2 cosh u — 2 cosh i|r) 
du 
( 121 ), 
where n — is a negative real integer. 
For all values of m and n such that n — \ is a real integer, the path may be 
taken to be a circle of radius unity with the origin as centre; we obtain on 
putting h — e^, since h 2 — 2/xh + 1 = he Ln (2 cosh xfj — 2 cos <£), 
Q,~M = e ! “" . 2*. n ( ” f n ( sinh™ rp 
g(m—j _ g‘<#> 
or 
Q** (g) = ie 
(vi—h) ITT 
2.n ( »-i)n(- 1 ), iihJ+ 
g( m4 T> i7r (2 cosh -vp — 2 cos c f>) m+ - 
COS (4 + A) (f) 
d<f), 
7T 
o (2 cosh 'yjr — 2 COS (/)) 
—(122). 
Recurrent Relations for Successive Vcdues of n, m in P,/" (/x), Q/' (/x). 
54. Denote the integral j —— — —cZA, by U (w, m),the integral being taken 
along any closed path, that is, one in which after completion the integrand returns to 
its initial value. 
We find 
— ^ m) = ( 2m + 0 f (i _ + W ).n dh = (2m + 1) U (n, m + 1); 
d fj, — h 2m i /n t t \ f — 1 
dh ’ (1 - 2ydi + A 2 )“ +i = (1 - 2/A/ + + ^ 2m + 1 ' (T- 2/xA + ^)»+t ' 
Hence 
W - 1) 
dU (4, m) 
d\i 
— [ h ll+m+1 
— 2m 
(1 — 2/x/i A A. 2 )'" +i 
+ 
_d_ 
dh 
- h 
(1 - 2 fih + hf 
<il+h 
dh 
= - 2mU (n+ 1, w) - | (1 _ (n + m + 1) cZ7? 
= — 2mU (>? + 1, m) — (n + m 4- 1) {/xIT (n, m) — U (n 4- 1, w)}, 
MDCCOXCVI,— A- 3 X 
