or 



A«=^ p . . (6) 



n(4- c 3)n (4-0 



and 



On the Partial Fraction Problem. 109 



Therefore [~— 1 



A K c K <f>(ic K ) = i (~-l) a m n 2a .A a+1 



a=0 



2 (-l)X- 2 «-i<? 



a=0 



K— 1 p 



n ($-# n 



y=l y^e + i 



m 



B«=;^^- -r- .... (7) 



n(4-# n (<*-<£) 



Hence 

 2 m„«»— p ( 2 (-l) <, m, s _ 2 ,,_ 1 cj> + 2 (-1)V„_ 2 ^ 



«=0 ^ . «=Q a=Q 



P ** r 2 -4-c 2 



J8 = l 



II. To separate 



X m a x 



into Partial Fractions, 



F W = VV „, , where n<2p, 



a W»- p ! ^ + 



) / ~S)(^ 2 + c2 F" ,c ' " " ' — 



\x 2 + 6 



or « p-i 



Sm/-=S(A K HB K )(f+c 2 ) x . . . (2) 



a=0 K =0 



Equating the odd powers of both sides of this identity, 

 and also the even powers, we obtain 



fir] 



I mn-^x^Jtk^ + tfy, ... (3) 



a=0 k=0 



and r«-i 



U-^4b b (/ + c 2 )«. ... (4) 



