566 Prof. I. J. Schwatt on the 



Substituting *-l, t-% £-3, ..., 3, 2, 1 for * in 



Qt, IC + l = Q*-l, K + l — a Q<, K ~~ l>Q.t, K-l? 



and adding the resulting equations, we obtain 

 t t 



Qt, «+l = W K + i — aSQy.K — &SQy, ic-l 



y=l y=l 



-.-l:f(-»'<-«)---.-|,f + r 1 )e w r- 1 ) 



-ji ] ""j*-»>-(-«)-...,- 1 -,.i(' 3 +r 1 )(" + ' J+ '- 1 ). 



j8=0 «=o y=l\ P /\ a J 



Now 



/^ + 7 _l\/ a + ^ + 7 _l\ = / a + ^\/ a + / 3 + 7 -l\ 



therefore 



_/« + /5\i/« + /3 + 7-l\ 

 ~\ /3 ) y t\ « + £ ) 



= ^f) x coefficient of x^ in S (l + «) a+lP+lr - 1 



= ( a ^) x coefficient of ^+^+ 1 in [(1 + *)«+*+*- (l + *) a+ *] 



Hen 



ce 



f«1 



+1 1 T-»«-r(r + f- , X* + &'" 1 )*»* 



If a = in tbe first double summation, / £ 1 = 0. 

 The term corresponding to « = /3 = is m K+ i. The upper 

 limit of j3 can be written — ^— . If /3 = is the second 



double summation, ( J =0. 



