<s>
In	O
mathematics	O
,	O
a	O
Carleman	B-Algorithm
matrix	I-Algorithm
is	O
a	O
matrix	O
used	O
to	O
convert	O
function	B-Application
composition	I-Application
into	O
matrix	O
multiplication	O
.	O
</s>
<s>
It	O
is	O
often	O
used	O
in	O
iteration	O
theory	O
to	O
find	O
the	O
continuous	O
iteration	B-Algorithm
of	I-Algorithm
functions	I-Algorithm
which	O
cannot	O
be	O
iterated	O
by	O
pattern	O
recognition	O
alone	O
.	O
</s>
<s>
The	O
Carleman	B-Algorithm
matrix	I-Algorithm
of	O
an	O
infinitely	O
differentiable	O
function	O
is	O
defined	O
as	O
:	O
</s>
<s>
A	O
generalization	O
of	O
the	O
Carleman	B-Algorithm
matrix	I-Algorithm
of	O
a	O
function	O
can	O
be	O
defined	O
around	O
any	O
point	O
,	O
such	O
as	O
:	O
</s>
<s>
This	O
allows	O
the	O
matrix	B-Architecture
power	I-Architecture
to	O
be	O
related	O
as	O
:	O
</s>
<s>
which	O
makes	O
the	O
Carleman	B-Algorithm
matrix	I-Algorithm
M	O
a	O
(	O
direct	O
)	O
representation	O
of	O
.	O
</s>
<s>
,	O
where	O
is	O
the	O
inverse	O
function	O
(	O
if	O
the	O
Carleman	B-Algorithm
matrix	I-Algorithm
is	O
invertible	O
)	O
.	O
</s>
<s>
The	O
Carleman	B-Algorithm
matrix	I-Algorithm
of	O
a	O
constant	O
is	O
:	O
</s>
<s>
The	O
Carleman	B-Algorithm
matrix	I-Algorithm
of	O
the	O
identity	O
function	O
is	O
:	O
</s>
<s>
The	O
Carleman	B-Algorithm
matrix	I-Algorithm
of	O
a	O
constant	O
addition	O
is	O
:	O
</s>
<s>
The	O
Carleman	B-Algorithm
matrix	I-Algorithm
of	O
the	O
successor	O
function	O
is	O
equivalent	O
to	O
the	O
Binomial	O
coefficient	O
:	O
</s>
<s>
The	O
Carleman	B-Algorithm
matrix	I-Algorithm
of	O
the	O
logarithm	O
is	O
related	O
to	O
the	O
(	O
signed	O
)	O
Stirling	O
numbers	O
of	O
the	O
first	O
kind	O
scaled	O
by	O
factorials	O
:	O
</s>
<s>
The	O
Carleman	B-Algorithm
matrix	I-Algorithm
of	O
the	O
logarithm	O
is	O
related	O
to	O
the	O
(	O
unsigned	O
)	O
Stirling	O
numbers	O
of	O
the	O
first	O
kind	O
scaled	O
by	O
factorials	O
:	O
</s>
<s>
The	O
Carleman	B-Algorithm
matrix	I-Algorithm
of	O
the	O
exponential	O
function	O
is	O
related	O
to	O
the	O
Stirling	O
numbers	O
of	O
the	O
second	O
kind	O
scaled	O
by	O
factorials	O
:	O
</s>
<s>
The	O
Carleman	B-Algorithm
matrix	I-Algorithm
of	O
exponential	O
functions	O
is	O
:	O
</s>
<s>
The	O
Carleman	B-Algorithm
matrix	I-Algorithm
of	O
a	O
constant	O
multiple	O
is	O
:	O
</s>
<s>
The	O
Carleman	B-Algorithm
matrix	I-Algorithm
of	O
a	O
linear	O
function	O
is	O
:	O
</s>
<s>
The	O
Carleman	B-Algorithm
matrix	I-Algorithm
of	O
a	O
function	O
is	O
:	O
</s>
<s>
The	O
Carleman	B-Algorithm
matrix	I-Algorithm
of	O
a	O
function	O
is	O
:	O
</s>
