EXTENSIONS OF THE JACOBI IDENTITY 21

define

Y(vin,vin_1,...,vil;zin,zin_1,...,zil;{zi,it\s t};r;[j,k]) =

f Y(vin,Vin_1,...tviliZin,Zin_l,...,Zil;{zi.it\3 t};r - 1; [r,n]),

if j = k;

'Y(vin,Vin_1,...1vil]zinizin_l,...,zil){zilit\3 t};r; [/', fc - 1])+

+z-}

$(ZiiT-Zi*r\.

L if i *;

(1-61)

where we denote by a the permutation ( u u - i • • • fcj). Note that the right-

hand side of (1.61) in the case j = k is independent of j .

The aim of this definition is the step r = n — 1 (and, consequently,

j = k = n). In this case we shall use the following notation:

Y(vn, v

n

_i, • • • ui; z

n

, 2

n

_

l 5

• • • zx\ {z^i j}) =

= Y(vn, u

n

_i, -••«!; z

n

, zn_!, • • • zx\ {zij\i j}; n - 1; [n, n]). (1.62)

Remark 1.8 In the case n = 3 (respectively, n = 4) the expression (1.62)

equals (1.29) (respectively, (1.52)).

The existence of (1.61) must be verified. We follow the approach which

extends the method of Subsection 1.2 (Lemmas 1.1, 1.3, 1.6 and 1.7).

In the rest of this section, we set

^i'o = *%• 1 i n- (1.63)

Lemm a 1.9 The expressions

n ^-i^^r-^V

ijn \ Z M-1 /

lKj

and n ^ft*^),

ijn \ ^i»*-l /

(1.64)

(1.65)

lKj

where each negative power of Yjk=i

zk,k-i

in (1-65) is to be expanded in non-

negative integral powers of Zkyk-i for each k such that i + 1 k j (cf.

Remark 1.2), exist and are equal.