Also, let II r il = r(i,n) and I| rl| = 1 I|| r = Y Y r(i,n). Then, the Taylor's series
ieS neS' RE S'iES
expansion of-I P- 1I = I P II I Pm I can be expressed as
-IP,-II = P-II -IP,-1| Eq. 9.9
= C(m, r) x H H (P(i In)- 1/2L)(,
r n S'ie S
where
C(mr) = H n r(i, n)! P(,n) r! Eq. 9.10
neS'ieS a=l
(IP--IP- = (IP-II-IP -i ) I==
H H r(i,n)!W") P(i I) r!
nE S'ieS a=1
is the coefficient of the order II r|| monomial term uniquely identified by the nonnegative
integer array r. In these expressions, the symbol r! denotes the operation
r!= n [r(i,n)!].
ne S'ie S
Expressing the value of C(m, r) thus reduces to evaluating the indicated mixed partial
derivative of-I PE,- | = P-I -I I P- at a = I divided by r!.
The coefficient of the order I| r| = |1 Ol = 0 term is
C(m,0)=(|P-I| -P-il ) = = { P-1I -|Pj-II| ,=1
and its value is the constant term of the series, provided by Eq. 9.8. The coefficient of the
first order monomial term which results from setting
=[r(i, n)] where r(i, n)(1 i =in =n
n0 otherwise
is given by
aP(il I n) a=1
= (|P-I -|P,-1) -