Georgian Mathematical Journal Volume 8 (2001), Number 3, 447–453

EULER POLYNOMIALS AND THE RELATED QUADRATURE RULE

GABRIELLA BRETTI AND PAOLO E. RICCI

Dedicated to the Memory of Prof. Niko Muskhelishvili

Abstract. The use of Euler polynomials and Euler numbers allows us to construct a quadrature rule similar to the well-known Euler–MacLaurin quadrature formula, using Euler (instead of Bernoulli) numbers, and even (instead of odd) order derivatives of a given function evaluated at the extrema of the considered interval. An expression of the remainder term and numerical examples are also given.

2000 Mathematics Subject Classification. 33C45, 33F05, 65D32. Key words and phrases. Euler polynomials, Euler numbers, Euler–Mac-Laurin quadrature formula, quadrature rules.

1. Introduction

Bernoulli polynomials B_{n}(x) are usually defined (see, e.g., [1], p. xxix) starting from the generating function

G(x, t) := \frac{te^{xt}}{e^{t} − 1} = ∑_{n=0}^{∞} B_{n}(x) \frac{t^{n}}{n!},  |t| < 2π,  (1.1)

and, consequently, Bernoulli numbers β_{n} := B_{n}(0) can be obtained by the generating function

\frac{t}{e^{t} − 1} = ∑_{n=0}^{∞} β_{n} \frac{t^{n}}{n!}.  (1.2)

Bernoulli numbers (see [2]–[3]) are contained in many mathematical formulas such as the Taylor expansion in a neighborhood of the origin of trigonometric and hyperbolic tangent and cotangent functions, the sums of powers of natural numbers, the residual term of the Euler–MacLaurin quadrature formula.

Bernoulli polynomials, first studied by Euler (see [4]–[5]–[6]), are employed in the integral representation of differentiable periodic functions, and play an important role in the approximation of such functions by means of polynomials. They are also used in the remainder term of the composite Euler–MacLaurin quadrature formula (see [7]).

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Euler polynomials E_{n}(x) are defined by the generating function

\frac{2e^{xt}}{e^{t} + 1} = ∑_{n=0}^{∞} E_{n}(x) \frac{t^{n}}{n!},  |t| < π.  (1.3)

Euler numbers E_{n} can be obtained by the generating function

\frac{2}{e^{t} + e^{−t}} = ∑_{n=0}^{∞} \frac{E_{n}}{n!} t^{n},  (1.4)

and the connection between Euler numbers and Euler polynomials is given by

E_{n} (\frac{1}{2}) = 2^{−n}E_{n}, n = 0, 1, 2, ….

The first Euler numbers are given by

E_{0} = 1, E_{1} = 0, E_{2} = −1, E_{3} = 0, E_{4} = 5, E_{5} = 0, E_{6} = −61, E_{7} = 0, ….

For further values see [8], p. 810.

Euler polynomials are strictly connected with Bernoulli ones, and are used in the Taylor expansion in a neighborhood of the origin of trigonometric and hyperbolic secant functions.

Recursive computation of Bernoulli and Euler polynomials can be obtained by using the following formulas:

∑_{k=0}^{n−1} \binom{n}{k} B_{k}(x) = nx^{n−1},  n = 2, 3, …;  (1.5)

E_{n}(x) + ∑_{k=0}^{n} \binom{n}{k} E_{k}(x) = 2x^{n},  n = 1, 2, ….  (1.6)

Further results can be found in a recent article by M. X. He and P. E. Ricci [9].

In this note, starting from the Euler polynomials, we construct a quadrature rule similar to the well-known Euler–MacLaurin quadrature formula, using Euler (instead of Bernoulli) numbers in the remainder term, and even (instead of odd) order derivatives of a given function evaluated at the extrema of the considered interval.

In a forthcoming paper the same procedure will be applied in a more general framework of Appell polynomials, including both the Euler–MacLaurin quadrature formula and the results of this article.

2. Definition and Simple Properties of Euler Polynomials

Euler polynomials E_{n}(t) can also be defined as polynomials of degree n ≥ 0 satisfying the conditions

(i) Em^{′}(t) = mE_{m−1}(t), m ≥ 1;

(ii) E_{m}(t + 1) + E_{m}(t) = 2t^{m}, m ≥ 1.

Note that a polynomial set E_{m}(t) satisfying (i) and (ii) is such that

E_{m}(t + 1) = ∑_{k=0}^{m} \frac{1}{k!}m(m − 1) ⋯ (m − k + 1)E_{m−k}(t)

= ∑_{k=0}^{m} \binom{m}{k}E_{m−k}(t) = ∑_{s=0}^{m} E_{s}(t) = ∑_{s=0}^{m} \binom{m}{s}E_{s}(t),

and, consequently, from (ii), assuming m ≥ 1,

2t^{m} = E_{m}(t + 1) + E_{m}(t) = ∑_{k=0}^{m} E_{k}(t) + E_{m}(t)

= ∑_{k=0}^{m−1} \binom{m}{k}E_{k}(t) + 2E_{m}(t)

so that

E_{m}(t) = t^{m} − \frac{1}{2} ∑_{k=0}^{m−1} \binom{m}{k}E_{m}(t),  m ≥ 1,  (2.1)

which is equivalent to (1.6).

Then E_{m}(t) are determined if we know E_{0}(t) and E_{1}(t). The latter are obtained by using the following procedure. Put E_{0}(t) = a and E_{1}(t) = at + b; writing condition (ii), for m = 1, yields

2t = E_{1}(t + 1) + E_{1}(t) = a(t + 1) + b + at + b = 2at + a + 2b.

Now, equating the first and the last term in t, we find

a = 1 and b = −\frac{1}{2}

and, consequently,

E_{0}(t) = 1 and E_{1}(t) = t − \frac{1}{2}.  (2.2)

The above considerations show that a polynomial set {E_{m}(t)} satisfying (i) and (ii), where E_{m}(t) is of degree m ≥ 0, is uniquely determined by the recurrence relation (2.1) with the initial conditions E_{0}(t) and E_{1}(t) given by (2.2). The existence of such a set of polynomials is trivial since the set defined by (2.1) and (2.2) satisfies conditions (i) and (ii).

Condition (i), for m = 2, gives

E2^{′}(t) = 2E_{1}(t),

and, after integrating term by term,

E_{2}(t) = t^{2} − t,

and, for m = 3,

E3^{′}(t) = 3E_{2}(t),

and, after integrating,

E_{3}(t) = t^{3} − \frac{3}{2}t^{2} + \frac{1}{4},

and so on.

Euler polynomials satisfy the symmetry relation

E_{m}(1 − t) = (−1)^{m}E_{m}(t).

Since E_{m}(0) = E_{m}(1) = 0, for even m, we put, if m is odd:

e_{m} = E_{m}(0) = −E_{m}(1).

The connection of e_{m} with Euler numbers is given by the formula

e_{m} = −\frac{1}{2^{m}} ∑_{h=0}^{m} \binom{m}{h}E_{m−h}.

3. The Quadrature Formula

The main result of this paper is expressed by the following theorem.

Theorem 3.1. Consider a function f(x) ∈ C^{2m}[a, b] and the corresponding integral over [a, b]. Let x_{i} = a + ih, i = 0, 1, …, n, where h := \frac{b−a}{n}, and f_{i} = f(x_{i}), f_{i}^{(p)} = f^{(p)}(x_{i}), p = 1, 2, …, 2m. Then the following composite trapezoidal rule holds true:

∫_{a}^{b} f(x)dx = h (\frac{1}{2}f(a) + f_{1} + ⋯ + f_{n−1} + \frac{1}{2}f(b))

− ∑_{k=1}^{m−1} \frac{e_{2k+1}}{(2k + 1)!}h^{2k+1} [f^{(2k)}(b) + f^{(2k)}(a)] + R_{E}[f],  (3.1)

where the correction terms are expressed by means of the even derivatives of the given funcion f(x) at the extrema, and the error term is given by

R_{E}[f] = \frac{h^{2m}}{(2m)!} ∑_{i=0}^{n−1} ∫_{x_{i}}^{x_{i+1}}f^{(2m)}(x)E_{2m} (\frac{x − x_{i}}{h} ) dx.  (3.2)

Proof. We consider first the trapezoidal rule extended to the whole interval.

Put f(x) = f(a + ht) = g(t), where h := b − a denotes the length of a given interval, and t = \frac{x−a}{h}. By using successive integration by parts we find

∫_{a}^{b} f(x)dx = h ∫_{0}^{1} g(t)dt = h ∫_{0}^{1} g(t)dE_{1}(t)

= h {[g(t)E_{1}(t)]_{0}^{1} − ∫_{0}^{1} g′(t)E_{1}(t)dt}

EULER POLYNOMIALS AND THE RELATED QUADRATURE RULE 451

= h \{\frac{1}{2} [g(1) + g(0)] − \frac{1}{2} [g′(t)E_{2}(t)]_{0}^{1} + \frac{1}{2} ∫_{0}^{1} g′′(t)E_{2}(t)dt\}

= h \{\frac{1}{2} [g(1) + g(0)] − \frac{1}{6} [g′′(t)E_{3}(t)]_{0}^{1} − \frac{1}{6} ∫_{0}^{1} g′′′(t)E_{3}(t)dt\}

= h \{\frac{1}{2} [g(1) + g(0)] − \frac{e_{3}}{6} [g′′(1) + g′′(0)] − \frac{1}{24} [g′′′(t)E_{4}(t)]_{0}^{1}

+\frac{1}{24} ∫_{0}^{1} g^{(iv)}(t)E_{4}(t)dt\}

= h \{\frac{1}{2} [g(1) + g(0)] − \frac{e_{3}}{6} [g′′(1) + g′′(0)] + \frac{1}{120} [g^{(iv)}(t)E_{5}(t)]_{0}^{1}

−\frac{1}{120} ∫_{0}^{1} g^{(v)}(t)E_{5}(t)dt\} = …

= h \{\frac{1}{2} [g(1) + g(0)] − ∑_{k=1}^{m−1} \frac{e_{2k+1}}{(2k + 1)!} [g^{(2k)}(1) + g^{(2k)}(0)]

+\frac{1}{(2m)!} ∫_{0}^{1} g^{(2m)}(t)E_{2m}(t)dt\} .

Recalling that f(x) = g(t), the preceding formula yields

∫_{a}^{b} f(x)dx = h \{\frac{1}{2} [f(b) + f(a)] − ∑_{k=1}^{m−1} \frac{e_{2k+1}}{(2k + 1)!}h^{2k} [f^{(2k)}(b) + f^{(2k)}(a)]

+\frac{h^{2m−1}}{(2m)!} ∫_{a}^{b} f^{(2m)}(x)E_{2m} (\frac{x − a}{h}) dx\} .  (3.3)

Consider now the partition of the interval [a, b] into n partial intervals by means of equidistant knots x_{i} = a + ih, i = 0, 1, …, n, where h := \frac{b−a}{n}, f_{i} = f(x_{i}), f_{i}^{(p)} = f^{(p)}(x_{i}), p = 1, 2, …, 2m. By (3.3) we have

∫_{a}^{b} f(x)dx = ∑_{i=0}^{n−1} ∫_{x_{i}}^{x_{i+1}}f(x)dx = h (\frac{1}{2}f(a) + f_{1} + ⋯ + f_{n−1} + \frac{1}{2}f(b))

− ∑_{k=1}^{m−1} \frac{e_{2k+1}}{(2k + 1)!}h^{2k+1} [f^{(2k)}(b) + f^{(2k)}(a)]

+\frac{h^{2m}}{(2m)!} ∑_{i=0}^{n−1} ∫_{x_{i}}^{x_{i+1}}f^{(2m)}(x)E_{2m} (\frac{x − x_{i}}{h} ) dx.

Note that the remainder term of the above procedure is expressed by

R_{E}[f] = \frac{h^{2m}}{(2m)!} ∑_{i=0}^{n−1} ∫_{x_{i}}^{x_{i+1}}f^{(2m)}(x)E_{2m} (\frac{x − x_{i}}{h} ) dx.

Remark 3.1. The above quadrature rule is similar to the well known Euler– MacLaurin quadrature formula, but the correction terms of the trapezoidal rule are expressed by the even (instead of odd) derivatives of the function f computed at the extrema of the interval [a, b], and it makes use of Euler (instead of Bernoulli) numbers.

By using equations (3.1)–(3.3), it is possible to write explicitly summation formulas for values of an analytic function (in particular of a polynomial) on equidistant knots similar to those which appear in [10], pp. 158–159, formulas (5.8.11)–(5.8.15).

4. Numerical Examples

We consider here two simple examples of the application of formula (3.1).

Consider the elementary integral

I := ∫_{0}^{1} \frac{1}{1 + x}dx = ln 2 ≃ 0.69314718 ….

By using (3.1) with n = 90 and m = 6 we have found numerical approximation

I ≃ 0.69315476 …

so that an absolute error is less than 10^{−5}, and a relative error is less than 2 × 10^{−5}.

Consider now the integral

J := ∫_{1}^{2} \frac{e^{x}}{x} dx ≃ 3.0591166 ….

By using (3.1) with n = 90 and m = 7 we have found numerical approximation

J ≃ 3.0591352 …

so that an absolute error is less than 2 × 10^{−5}, and a relative error is less than 10^{−5}.

Acknowledgement

This paper was prepared (concluded) in the framework of the Italian National Group for Scientific Computation (G.N.C.S.) whose partial support we wish to acknowledge.

References

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3. L. Saalschuetz, Vorlesungen über dir Bernoullischen Zahlen. Berlin, 1893.

4. L. Euler, Instituziones calculi differentialis. Opera Omnia, Series prima: opera mathematica 10, Teubner, 1980.

5. N. E. Nörlund, Vorlesungen über Differenzenrechnung. Berlin, 1924.

6. H. Bateman and A. Erdélyi, Higher trascendental functions. The Gamma function. The hypergeometric function. Legendre functions. Vol. 1. McGraw-Hill, 1953.

7. J. Stoer, Introduzione all’Analisi Numerica. Zanichelli, Bologna, 1972.

8. M. Abramowitz and I. A. Stegun, Handbook of mathematical functions. Dover, New York, 1965.

9. M. X. He and P. E. Ricci, Differential equation of Bernoulli polynomials via the factorization method. J. Comput. Appl. Math. (accepted, in print).

10. F. B. Hildebrand, Analisi numerica. Ambrosiana, Milano, 1967.

(Received 10.03.2001)

Authors’ address: Dipartimento di Matematica “Guido CASTELNUOVO” Università degli Studi di Roma “La Sapienza” P.le A. Moro, 2 – 00185 Roma Italia E-mail: bretti@mat.uniroma1.it

riccip@uniroma1.it