We begin with the following well-known results.
Lemma 2.1 (see [47]).
Let
Then,
where
is the Lidstone interpolating polynomial of degree 
and
is the residue term
here
Recursively, it follows that
(
is the Bernoulli polynomial of degree
and
is the
th Bernoulli number
,
;
,
,
,
,
,
,
,
,
).
Lemma 2.2 (see [47]).
The following hold:
(
is the Euler polynomial of degree
and
is the
th Euler number
,
;
,
,
,
)
Theorem 2.3.
Let
Then,
where
is the complementary Lidstone interpolating polynomial of degree 
and
is the residue term
here
Remark 2.4.
From (2.4) and (2.15) it is clear that
;
,
; 
,
;
,
;
,
;
, 
Proof.
In (2.1), we let
and integrate both sides from
to
to obtain
Now, since
and, similarly
it follows that
Next since
for
from (2.7), we get
and similarly, for
we have
Finally, since (2.12) is exact for any polynomial of degree up to
we find
and hence, for
it follows that
Combining (2.23) and (2.25), we obtain
Theorem 2.5.
Let
Then, inequalities (1.5) hold with
.
Proof.
From (2.14) and (2.8) it follows that
Now, from (2.11) and (2.13), we find
However, from (2.9), we have
Thus, from
,
,
and
we obtain
Using the above estimate in (2.29), the inequality (1.5) for
follows.
Next, from (2.11), (2.13) and (2.14), we have
and hence in view of (2.5) and (2.9) it follows that
and similarly, by (2.5) and (2.10), we get
Remark 2.6.
From (2.13), (2.28), and the above considerations it is clear that
Remark 2.7.
Inequality (1.5) with the constants
given in (2.27) is the best possible, as equalities hold for the function
(polynomial of degree
) whose complementary Lidstone interpolating polynomial
and only for this function up to a constant factor.
Remark 2.8.
From the identity (see [47, equation (1.2.21)])
we have
and hence
We also have the estimate (see [47, equation (1.2.41)])
Thus, from (2.27), (2.38), and (2.39), we obtain
Therefore, it follows that
Combining (1.5) and (2.41), we get
Hence, if
for a fixed
as
,
converges absolutely and uniformly to
in
provided that there exists a constant
and an integer
such that
for all
, 
In particular, the function
,
satisfies the above conditions. Thus, for each fixed
expansions
converge absolutely and uniformly in
provided
For
(2.43) and (2.44), respectively, reduce to absurdities,
and
Thus, the condition
is the best possible.
Remark 2.9.
If
then
Thus, in view of
,
we have
Now, since
,
,
from (2.6), we find
and hence by (2.15) it follows that
Using these relations in (2.47), we obtain an approximate quadrature formula
It is to be remarked that (2.50) is different from the Euler-MacLaurin formula, but the same as in [49] obtained by using different arguments. To find the error
in (2.50), from (2.28) and (2.46) we have
Thus, it immediately follows that
From (2.52) it is clear that (2.50) is exact for any polynomial of degree at most
Further, in (2.52) equality holds for the function
and only for this function up to a constant factor.
We will now present two examples to illustrate the importance of (2.50) and (2.52).
Example 2.10.
Consider integrating
over
Here,
,
and
The exact value of the integral is
In Table 1, we list the approximates of the integral using (2.50) with different values of
the actual errors incurred, and the error bounds deduced from (2.52).
Note that
hence the error
when
or
Although the errors for other values of
are large, ultimately the approximates tend to the exact value as 
Example 2.11.
Consider integrating
over
Here,
,
and
The exact value of the integral is
In Table 2, we list the approximates of the integral using (2.50) with different values of
the actual errors incurred, and the error bounds deduced from (2.52).
Unlike Example 2.10, here the error decreases as
increases. In both examples, the approximates tend to the exact value as
Of course, for increasing accuracy, instead of taking large values of
one must use composite form of formula (2.50).