<s>
Certain	O
sequences	O
of	O
approximants	O
lying	O
within	O
a	O
Padé	B-Algorithm
table	I-Algorithm
can	O
often	O
be	O
shown	O
to	O
correspond	O
with	O
successive	O
convergents	O
of	O
a	O
continued	O
fraction	O
representation	O
of	O
a	O
holomorphic	O
or	O
meromorphic	O
function	O
.	O
</s>
<s>
Modern	O
interest	O
in	O
Padé	B-Algorithm
tables	I-Algorithm
was	O
revived	O
by	O
H	O
.	O
S	O
.	O
Wall	O
and	O
Oskar	O
Perron	O
,	O
who	O
were	O
primarily	O
interested	O
in	O
the	O
connections	O
between	O
the	O
tables	O
and	O
certain	O
classes	O
of	O
continued	O
fractions	O
.	O
</s>
<s>
However	O
,	O
it	O
can	O
be	O
shown	O
that	O
,	O
due	O
to	O
cancellation	O
,	O
the	O
generated	O
rational	O
functions	O
Rm	O
,	O
n	O
are	O
all	O
the	O
same	O
,	O
so	O
that	O
the	O
(	O
m	O
,	O
n	O
)	O
th	O
entry	O
in	O
the	O
Padé	B-Algorithm
table	I-Algorithm
is	O
unique	O
.	O
</s>
<s>
Although	O
the	O
entries	O
in	O
the	O
Padé	B-Algorithm
table	I-Algorithm
can	O
always	O
be	O
generated	O
by	O
solving	O
this	O
system	O
of	O
equations	O
,	O
that	O
approach	O
is	O
computationally	O
expensive	O
.	O
</s>
<s>
Usage	O
of	O
the	O
Padé	B-Algorithm
table	I-Algorithm
has	O
been	O
extended	O
to	O
meromorphic	O
functions	O
by	O
newer	O
,	O
timesaving	O
methods	O
such	O
as	O
the	O
epsilon	O
algorithm	O
.	O
</s>
<s>
cells	O
in	O
the	O
Padé	B-Algorithm
table	I-Algorithm
,	O
from	O
position	O
(	O
m	O
,	O
n	O
)	O
through	O
position	O
(	O
m+r	O
,	O
n+r	O
)	O
,	O
inclusive	O
.	O
</s>
<s>
If	O
a	O
particular	O
rational	O
function	O
occurs	O
exactly	O
once	O
in	O
the	O
Padé	B-Algorithm
table	I-Algorithm
,	O
it	O
is	O
called	O
a	O
normal	O
approximant	O
to	O
f(z )	O
.	O
</s>
<s>
If	O
every	O
entry	O
in	O
the	O
complete	O
Padé	B-Algorithm
table	I-Algorithm
is	O
normal	O
,	O
the	O
table	O
itself	O
is	O
said	O
to	O
be	O
normal	O
.	O
</s>
<s>
the	O
Padé	B-Algorithm
table	I-Algorithm
is	O
normal	O
if	O
and	O
only	O
if	O
none	O
of	O
the	O
determinants	O
Dm	O
,	O
n	O
are	O
equal	O
to	O
zero	O
(	O
note	O
in	O
particular	O
that	O
this	O
means	O
none	O
of	O
the	O
coefficients	O
ck	O
in	O
the	O
series	O
representation	O
of	O
f(z )	O
can	O
be	O
zero	O
)	O
.	O
</s>
<s>
There	O
is	O
an	O
intimate	O
connection	O
between	O
regular	O
continued	O
fractions	O
and	O
Padé	B-Algorithm
tables	I-Algorithm
with	O
normal	O
approximants	O
along	O
the	O
main	O
diagonal	O
:	O
the	O
"	O
stairstep	O
"	O
sequence	O
of	O
Padé	O
approximants	O
R0	O
,	O
0	O
,	O
R1	O
,	O
0	O
,	O
R1	O
,	O
1	O
,	O
R2	O
,	O
1	O
,	O
R2	O
,	O
2	O
,	O
is	O
normal	O
if	O
and	O
only	O
if	O
that	O
sequence	O
coincides	O
with	O
the	O
successive	O
convergents	O
of	O
a	O
regular	O
continued	O
fraction	O
.	O
</s>
<s>
In	O
other	O
words	O
,	O
if	O
the	O
Padé	B-Algorithm
table	I-Algorithm
is	O
normal	O
along	O
the	O
main	O
diagonal	O
,	O
it	O
can	O
be	O
used	O
to	O
construct	O
a	O
regular	O
continued	O
fraction	O
,	O
and	O
if	O
a	O
regular	O
continued	O
fraction	O
representation	O
for	O
the	O
function	O
f(z )	O
exists	O
,	O
then	O
the	O
main	O
diagonal	O
of	O
the	O
Padé	B-Algorithm
table	I-Algorithm
representing	O
f(z )	O
is	O
normal	O
.	O
</s>
<s>
Here	O
is	O
an	O
example	O
of	O
a	O
Padé	B-Algorithm
table	I-Algorithm
,	O
for	O
the	O
exponential	O
function	O
.	O
</s>
<s>
This	O
fraction	O
's	O
successive	O
convergents	O
also	O
appear	O
in	O
the	O
Padé	B-Algorithm
table	I-Algorithm
,	O
and	O
form	O
the	O
sequence	O
R0	O
,	O
0	O
,	O
R0	O
,	O
1	O
,	O
R1	O
,	O
1	O
,	O
R1	O
,	O
2	O
,	O
R2	O
,	O
2	O
,	O
</s>
<s>
Mathematicians	O
have	O
also	O
constructed	O
two-point	O
Padé	B-Algorithm
tables	I-Algorithm
by	O
considering	O
two	O
series	O
,	O
one	O
in	O
powers	O
of	O
z	O
,	O
the	O
other	O
in	O
powers	O
of	O
1/z	O
,	O
which	O
alternately	O
represent	O
the	O
function	O
f(z )	O
in	O
a	O
neighborhood	O
of	O
zero	O
and	O
in	O
a	O
neighborhood	O
of	O
infinity	O
.	O
</s>
