1.3. TERMINOLOG Y USED THROUGHOU T (EXCEP T AS MODIFIED IN CHAPTE R 12) 3

of 1Zm into IZm such that

(P + QY = P' + Q' and {PQ)' = P'Q + P Q'', for any P , Q in ftm.

Henceforth, we let ' denote the unique derivation for 1Zm that satisfies

(w[j)y = w{?+l\ for 1 i m and j 0;

e.g., see [5, page 139] or [6, page A.V.130]. We set: Wi = w\ \ for 1 i ra;

WQ = 1; and

P^1'

= P' , for any polynomial P in 7£m. A polynomial P in 7£m is a

constant when P ' = 0. Thus, the constants of 7£m are the elements of Q. With the

derivation ', 7£m is an ordinary differential ring; e.g., see [38, page 58]. Infinitely

many variables w\ are considered so that each P in 7£m has a corresponding P

in 7£m. This generality is used in Appendix E. However, most of our constructions

involve polynomials over Q in the w\ having 1 i ra and 0 j m — i.

1.3. Terminology used throughout (except as modified in Chapter 12)

We assume that the coefficients c\(z), ..., cm(z) of the rath order homogeneous

linear differential equation

m

(1.1) y^\z) +

J2ci(z)y{m-l)(z)

= Q

2 = 1

are meromorphic functions of a complex variable z on a region ft of the complex

plane. Then, for any meromorphic function p(z) ^ 0 on J?, there are unique

meromorphic functions c\(z), ..., c^(z) on 4? such that the change

(1.2) y(z)=p(z)v(z)

of the dependent variable from y to v transforms (1.1) into

m

(1.3)

v^\z) +

Yl

c

iW

v(m"°

(*)

=

°'

2 = 1

Proposition A.l on page 134 and Remark A.5 on page 138 give details. Alternative

hypotheses for (1.1) and (1.2) are considered in Remark 1.7 on page 8. For any

univalent analytic function £ = g(z) on Q whose inverse is z — /(£) on J?** = g(f2),

there are unique meromorphic functions c^*((), ..., c^(Q on J?** such that the

change

(1-4) z = /(C)

of the independent variable from z to ( transforms (1.1) into

m

(i.5)

u(m)(o+£cno«('n-i)(o

= o)

2 = 1

where n(£) = (y o /)(£). Theorem A.3 on page 136 and Remark A.5 on page 138

provide details.

For any P in 7£m, we let P(z) denote the function on 4? obtained by replacing

each w^ in P with the corresponding cj?'(z) from (1.1); we let P*(z) denote the

function on i? obtained by replacing each w^ in P with the corresponding c^\z)

from (1.3); and we let P**(C) denote the function on 4?** obtained by replacing

each w^ in P with the corresponding c£* (C) from (1.5).