T 



[ 560 ] 

 LXVII. Note on the Partial Fraction Problem. 



By I. J. SCHWATT *. 



separate 



n 



2 m K x n ~ K 



where 2p>n, and a 2 — 46 <0, into partial fractions, using 

 the division method. 



n-2 



Let 2 Q lK % n ~ 2 ~ K denote the quotient and A # + B the 

 remainder (which will, at most, be linear) obtained by 

 divid 

 then 



dividing 2 m K x n ~ K by x 2 + ax + b 



K=0 



2m K z n -' 2Q 1K ^- 2 - K 



K = = * = Apff + -Dp _. 



O 2 + aa? + />) p 0» + ax + ft)^" 1 + (.z 2 + a# + &)* * l ; 



Clearing o£ fractions and equating like powers of x, we 

 have (Q a £ = 0, for negative values of ft), 



Qi, B + «Qi, K -i + ^Qi, K -2=m K (te = Q, 1, 2, ...,n-2),1 



We then derive 



Qio = ^o; Qn = w2 1 — am ; Q l i 2 = m 2 ~am 1 + a 2 tn — bm ; 

 Q J3 = »?? 3 — awi 2 + a 2 w2i — a 3 m — 6?n x + 2bam ; 

 Qm = ?% — « m 3 + a2}n 2 — a 3 >™i + a 4 w*o ~ ° ( m 2 — 2am x + 3a 2 m ) + b 2 m 

 Q15 = m s — a?n 4 + a 2 m z — a z m 2 + a 4 'm 1 — a 5 m — b(m 3 — 2am 2 + Za 2 ^^ 

 — 4a 3 m ) + i 2 (??i 1 — 3a77i ) . 

 We now asssume 

 pel 

 Q lK =J o (-l)^/(-l)«a a ^ + ^^_ 2 ,_ a , . (3) 



where ^ is the integral part of |, and we shall prove 

 that this form holds for Q : K+1 . 



* Communicated by the Author. 



o > 



