22

CRISTIANO HUSU

Proof. By Proposition 8.8.5 in [6], the expression (1.64) can be written

as

n ^^^r-^V (

L66

)

By induction on n (cf. Lemma 1.1 and Lemma 1.6), we assume that (1.66)

exists with n replaced by n — 1. Then, the coefficient of

IKo1-''',

a,€Z,

1= 1

exists by inductive hypothesis and therefore (1.64) and (1.66) exist. The

existence of (1.65) is seen directly by means of the coefficient of

and without the need of induction.

The equality of the two expressions is seen in the same way as in the

proof of Lemma 1.6 (equality of (1.53) and (1.54)). •

L e m m a 1.10 The expressions

Y{Y{. • • Y(Y(Y(vn, znm)Y(vn-U zn_1)TO) • • •

)'"V2,z2i)vuz1y (1.67)

and

Y(Y(- • • Y(Y(Y(vn, znm)Y(vn.u zn.1m) •••

) ^771+ 1,771 )vm ? Zm—l,m—2

)--v2,z21)vuz1)- (1.68)

(n n

Tui'-^f1^))

n ^^{^-^A,

where 0 ra n, ezistf.