528 
DR. E. W. HOBSON ON A TYPE OF SPHERICAL HARMONICS 
hence 
IT(0) 
n (0) 
_ U ' ( m ~b _ ir (Po + ™ + h) , 1) 
n (m — -|) n o ? 0 + m + |) n o ? 0 + i) 
_ jn'(o) n-(-i), 
i.n(0) ik-i)!" 1- 
“(t +••• + 
+ • 
L 
Po + 1 
'r + 
• + 
i \ 
m — i 
1 
Po + m + b 
Now use the known theorem II (x — ]) n (x — -g) = \Z’2tt . 2" 2 - r . IT (2x 
taking logarithms and differentiating, and then putting x = we find 
IT(0) ITf-i) _ , , 
n (0) n(-j) oge • 
— 1) ; on 
Taking (2), (3), and (4) together, we now have the expression 
(~ l) m 2 m+1 lo « ( 4e ")• sinl1 * o-.e _( - Po+ra+!)<r F (to + p 0 + f, to +1 p 0 + 2, e" 2 ") 
+ (— 1 )* 2 ?H . 
sink" 
a. e 
— (i’o+^+l) O’ 5=® 
n( - |) n {m - j) s f 
i \ — (^o+j+i T" d - v»i+s-± Vp o+m+3+ i) 
where 
and 
II (p 0 + m + s + |) II (??t + s — Q 2s<7 
n (s) 
u r denotes the series -j- + -p + . . . -f — 
L u T 
v r+ x denotes the series 4~ + "V d~ • • • + 1 - • 
i i r + f 
On changing p a into n — 1, we now have the complete expression for the ring 
function P„_f"(cosh <x), (to integral), 
(cosh <x) 
2 II (n — 1) 
IT (n - m-|)II(-}) 
sinh® <r . e {n m i)a 
(j + m)(n - j - m) 2<r 
1+ 1.71-1 6 + '-- 
(i + m). . . (1 + m + n - 2) Q - m - \). . . (- m + Q 2( „_ 1)<r ‘ 
1.2. ..7i-l. 7i-l. n - 2. ..1 
+ (— i) m 2 m+1 . —fy log (4c 0 ') sinh“ cr. e- ( “ +w+ - )<r F(TO + ^+ 2 j TO + |, n+lj e -2<r ) 
/ v qinli»i<T z>-(«+»2+i)<r 
+ (- 1 ) w 2 ” t - n(-|) II ( m _^) S s 0 - v m+n+s _,) 
n (t?i + n + s — ^) n (??i + s — i) _ 2s<r 
H(s) 
