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).