On the Partial Fraction Problem. 561 



From (2) 



Qir+i=»Vn — aQiK — ^Qi/c-i 



=m K+1 -flS(-lFS(-iraf7m^. fl 



= a=0 \ H / 



LVJ K-l-20 /a + /9\ 



-6 S (-l)W S (-livf^^.^, (4) 



/3^0 a=0 \ H / 



Let /c be even, then 

 Q, I+1 =™, tl + I (-ll'^Vt-ll'+'a'+'KV^-. 



/3=0 a = \ P / 



K l 



2 K-2^-1 /«4-fl\ 



+ S(-ir^ +1 S (-l)V( a t A, )m c . l . w . a .(5) 



Designating the first double summation by s x and the 

 second by s 2 , then 



2 k-2/3+1 



K+l-2,3-a 



5 K+l-20 /a+rt— 1\ 



= S(-1)W 2 (-l)v( a+ g 



K 



but the last summation is equal to m K+l , since if /S^O, 

 Therefore 



t =^ o (-l)W S q (-ljV( a + | 1 ) OT « +1 _^: a -m« +1 (6) 

 Phil. Mag. S. 6. Vol. 31. No, 186. June 1916. 2 P 



