EXTENSIONS OF THE JACOBI IDENTITY
23
Proof. We first show the existence of
Y(Y(- Y(Y(vn, 2
n
, n - l K - l , Zn-l.n-2) ' ' ' ^2, Z2i)v1, Zi)-
n
^4^^ (1^)
lijn \ *Ji«-l /
(the case ra = n 1 in (1.67)). Applying Lemma 1.8 to the product of
^-functions, the expression (1.69) can be written as
Y(Y(- Y(Y(vn, *
n
, n - l ) V n - l , *n-l,n-2 ) ' " V2y Z21)vU ZX)-
n
*;£i*(nr*M-1)
(i-70)
lijn \ Z J-1 /
(expanded as in Lemma 1.8). The coefficient of
n *i£"-\ ^-i€z,
l i j n
in (1.70) equals
Y(Y(. r ( y ( t ; n ^ n , n . l K - l , ^ . l
l
n - 2 ) * ' ' (1-71)
t2,*2i)ti,*i) n ( z ) * M - I
lijn \k=i
By induction on n and by means of the coefficient of
zlnn_x,
/ £ Z, (as in
the proof of Lemma 1.7) we see that (1.71) and, therefore, (1.70) and (1.69)
exist.
The existence of the expression (1.67) in the other cases (0 m n 1)
is now seen, for instance, extracting the coefficient of
71 771
II E[
zijJ f°r a*i G
^' 1 i n m,
i=ra+l j=0
and using the existence of (1.69) (with n replaced by m).
Finally the existence of (1.68) is seen by means of the coefficient of
*m+l,m-l*m+l,m H Zt,m+1 *, 6, Cj £ Z.
t=m+2
D
O j , l - l
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