Inverses of lower triangular Toeplitz matrices William F. Trench Define an D 0 if n < 0 and assume that a0 ¤ 0. Let TN D .ar
N s /r;sD1 ;
N D 1; 2; : : : ;
be the family of lower triangular Toeplitz matrices generated by the formal power series f .´/ D
1 X
an ´n :
(1)
nD0
The family of inverses is given by TN 1 D .br
N s /r;sD1 ;
where g.´/ D
N D 1; 2; : : : ;
1 X
bn x n
(2)
nD0
is the formal reciprocal of f .´/; that is, bn D 0 if n < 0, b0 D 1=a0 and n X
ar bn
r
D 0;
n > 0:
r D0
This formula can be used to compute fbn g1 nD1 recursively: bn D
n 1 X ar bn b0 r D1
r:
This seems to me to be the most practical way to proceed. I can think of two “explicit" expressions for bn ; however, although they may be of some theoretical interest, they don’t seem very useful. F ORMULA I. For every n 1, 2 3 2 3 b0 1 6 b1 7 6 0 7 6 7 6 7 TnC1 6 : 7 D 6 : 7 : 4 :: 5 4 :: 5 bn
Applying Cramer’s rule here yields ˇ ˇ a1 ˇ ˇ a2 . 1/n ˇˇ : bn D nC1 ˇ :: ˇ a0 ˇ an 1 ˇ ˇ an
0
a0 a1 :: : an an 1
0 a0 :: : 2 1
an an
:: : 3 2
ˇ ˇ ˇ ˇ ˇ ˇ ˇ: ˇ a0 ˇˇ a1 ˇ 0 0 :: :
F ORMULA II. Temporarily assume that (1) has a positive radius of convergence. Then so does (2). Moreover, f .n/ .0/ g.n/ .0/ an D and bn D : (3) nŠ nŠ According to the formula of Faa di Bruno (Ch.-J de La Vallee Poussin, Cours d’analsye infinitesimale, Vol. 1, 12th Ed., Libraire Universitaire Louvain, Gauthier-Villars, Paris, 1959) for the derivatives of a composite function, 0 n X X nŠ f .x/ k1 f 00 .x/ k2 f .n/ .x/ dn .k/ h.f .x// D h .f .x// n dx k1 Š kn Š 1Š 2Š nŠ kD1
k
!k n
P where k is over all partitions of k as a sum of nonnegative integers k1 ; k2 : : : ; kn such that k1 C k2 C C kn D k
and
k1 C 2k2 C nkn D n:
We’re interested in g.x/ D 1=f .x/; therefore, we take h.u/ D 1=u in obtain 0 n X dn . 1/k kŠ X nŠ f .x/ k1 f 00 .x/ k2 f .n/ .x/ g.x/ D dx n k1 Š kn Š 1Š 2Š nŠ .f .x//kC1 k
kD1
!k n
Setting x D 0 here and recalling (3) yields bn D
n X . 1/k kŠ X
kD1
a0kC1
k
1 ak1 ak2 ankn : k1 Š kn Š 1 2
Although we derived this formula assuming that (1) has a positive radius of convergence, it is a purely algebraic result that doesn’t require this assumption.
