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