M.-P.— Vol. I.] WILCZYNSKI— LINEAR SUBSTITUTIONS. 6 1 



( 7 ) ^/=?1/*'. 



Now U^ (yS X ~ ff A = o for X = a, . .m—i, if p>m— I— <r, 



and C^W ( Jk^ _/,) ) = o for X = p, . . w— I, if tr>in—i—p. 

 Therefore 



(8) (i/,W, <7 «)/ =0 



/br p -j- cr > 7/2 — i. 

 So we have 



( ^ k « m - 1 », ^. 1 (i)) = (Z7 ik (-i) s £//>) =... = ( CV-u, U^)=o 



( ^ k <- 2 >, *7/>) = ...= ( ^ ik (m - 2 », Uf**)=o 



It remains to consider those commutators, the sum of whose 

 indices is not greater than m — i. 

 We find 



(9) (u^'\u^)f=o 



if ? < / and / ^ /•. 



Now let i= I, but / < k. Then 



(io) {TjJf\U^f= U^f,(p + <r<f*-i,f>Zk.) 



If i — / and j = k, 



(ii) ( ujp , U^)f = uj^f- U^f 



(p -f <r<m— i). 

 In particular, if «'= X% 



(12) (^, ^)/=0. 



Now let z'^/ but j = k. Then we find in the same 

 manner 



(13) (BT.W, U^>)f=-U^f. 



