<s>
In	O
discrete	B-Algorithm
calculus	I-Algorithm
the	O
indefinite	B-Algorithm
sum	I-Algorithm
operator	O
(	O
also	O
known	O
as	O
the	O
antidifference	B-Algorithm
operator	I-Algorithm
)	O
,	O
denoted	O
by	O
or	O
,	O
is	O
the	O
linear	B-Architecture
operator	I-Architecture
,	O
inverse	O
of	O
the	O
forward	B-Algorithm
difference	I-Algorithm
operator	I-Algorithm
.	O
</s>
<s>
It	O
relates	O
to	O
the	O
forward	B-Algorithm
difference	I-Algorithm
operator	I-Algorithm
as	O
the	O
indefinite	B-Algorithm
integral	I-Algorithm
relates	O
to	O
the	O
derivative	B-Algorithm
.	O
</s>
<s>
Therefore	O
,	O
each	O
indefinite	B-Algorithm
sum	I-Algorithm
actually	O
represents	O
a	O
family	O
of	O
functions	O
.	O
</s>
<s>
However	O
,	O
due	O
to	O
the	O
Carlson	O
's	O
theorem	O
,	O
the	O
solution	O
equal	O
to	O
its	O
Newton	O
series	O
expansion	O
is	O
unique	O
up	O
to	O
an	O
additive	O
constant	O
C	O
.	O
This	O
unique	O
solution	O
can	O
be	O
represented	O
by	O
formal	O
power	O
series	O
form	O
of	O
the	O
antidifference	B-Algorithm
operator	I-Algorithm
:	O
.	O
</s>
<s>
Indefinite	B-Algorithm
sums	I-Algorithm
can	O
be	O
used	O
to	O
calculate	O
definite	O
sums	O
with	O
the	O
formula	O
:	O
</s>
<s>
Often	O
the	O
constant	O
C	O
in	O
indefinite	B-Algorithm
sum	I-Algorithm
is	O
fixed	O
from	O
the	O
following	O
condition	O
.	O
</s>
<s>
Indefinite	B-Algorithm
summation	I-Algorithm
by	O
parts	O
:	O
</s>
<s>
Some	O
authors	O
use	O
the	O
phrase	O
"	O
indefinite	B-Algorithm
sum	I-Algorithm
"	O
to	O
describe	O
a	O
sum	O
in	O
which	O
the	O
numerical	O
value	O
of	O
the	O
upper	O
limit	O
is	O
not	O
given	O
:	O
</s>
<s>
It	O
is	O
the	O
inverse	O
of	O
the	O
backward	B-Algorithm
difference	I-Algorithm
operator	O
.	O
</s>
<s>
It	O
is	O
related	O
to	O
the	O
forward	O
antidifference	B-Algorithm
operator	I-Algorithm
using	O
the	O
fundamental	O
theorem	O
of	O
discrete	B-Algorithm
calculus	I-Algorithm
described	O
earlier	O
.	O
</s>
<s>
This	O
is	O
a	O
list	B-Algorithm
of	I-Algorithm
indefinite	I-Algorithm
sums	I-Algorithm
of	O
various	O
functions	O
.	O
</s>
<s>
Not	O
every	O
function	O
has	O
an	O
indefinite	B-Algorithm
sum	I-Algorithm
that	O
can	O
be	O
expressed	O
in	O
terms	O
of	O
elementary	O
functions	O
.	O
</s>
<s>
where	O
is	O
the	O
normalized	B-Algorithm
sinc	I-Algorithm
function	I-Algorithm
.	O
</s>
