Bulletin T.CXLV de l Académie serbe des sciences et des arts 2013 Classe des Sciences mathématiques et naturelles Sciences mathématiques, No 38

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1 Bulletin T.CXLV de l Académie serbe des sciences et des arts 213 Classe des Sciences mathématiques et naturelles Sciences mathématiques, No 38 FAMILIES OF EULER-MACLAURIN FORMULAE FOR COMPOSITE GAUSS-LEGENDRE AND LOBATTO QUADRATURES G. V. MILOVANOVIĆ (Presented at the 4th Meeting, held on May 31, 213) A b s t r a c t. Beside a short account on the Euler-Maclaurin formula, using a recent progress in variable-precision arithmetic and symbolic computation and the corresponding Mathematica package Orthogonal- Polynomials Facta Univ. Ser. Math. Inform. 19 (24), 17 36; Math. Balkanica 26 (212), , we obtain extensions of the Euler-Maclaurin formula for the composite (shifted) Gauss-Legendre formula, as well as for its Lobatto modification. Some special cases are also presented. AMS Mathematics Subject Classification (2): 65D3, 65D32, 4C15 Key Words: Gauss-Legendre quadrature, Lobatto quadrature, Euler- Maclaurin formulae, Bernoulli numbers, Bernoulli polynomials, orthogonal polynomials, quadrature sum, remainder term. 1. Introduction The Euler-Maclaurin summation formula plays an important role in the broad area of numerical analysis, analytic number theory, and the theory

2 64 G. V. Milovanović of asymptotic expansions, as well as in many applications in other fields. In connection with the so-called Basel problem (or in modern terminology, with determining ζ(2)), in 1732 Leonhard Euler discovered this formula, n f(k) = k= n + f(x) dx + 1 (f() + f(n)) 2 r j=1 B 2j f (2j 1) (n) f (2j 1) () + E r (f), (1.1) (2j)! which holds for any n, r N and f C 2r, n, where B 2j are Bernoulli numbers (B = 1, B 1 = 1/2, B 2 = 1/6, B 3 =, B 4 = 1/3,...). This formula was also found independently by Maclaurin. While in Euler s case the formula (1.1) was applied for computing slowly converging infinite series, in the second one Maclaurin used it to calculate integrals. A history of this formula was given by Barnes 3, and some details can be found in 19, 1, 11, 12, 5. Bernoulli numbers B n, n =, 1,..., can be expressed as values at zero of the corresponding Bernoulli polynomials, which are defined by the generating function te xt e t 1 = B j (x)t j. j! j= Bernoulli polynomials play a similar role in numerical analysis and approximation theory like orthogonal polynomials. First few polynomials are B (x) = 1, B 1 (x) = x 1 2, B 2(x) = x 2 x + 1 6, B 3(x) = x 3 3x2 2 + x 2, B 4 (x) = x 4 2x 3 + x 2 1 3, B 5(x) = x 5 5x x3 3 x 6, etc. Some interesting properties of these polynomials are B n(x) = nb n 1 (x), B n (1 x) = ( 1) n B n (x), B n (x) dx = (n N). The error term E r (f) in (1.1) can be expressed in the form (cf. 5) E r (f) = ( 1) r + n k=1 e i2πkt + e i2πkt (2πk) 2r f (2r) (x) dx,

3 Families of Euler-Maclaurin formulae 65 or in the form n E r (f) = B 2r (x x ) f (2r) (x) dx, (1.2) (2r)! where x denotes the largest integer that is not greater than x. Supposing f C 2r+1, n, after an integration by parts in (1.2) and recalling that the odd Bernoulli numbers are zero, we get (cf. 14, p. 455) E r (f) = n B 2r+1 (x x ) f (2r+1) (x) dx. (1.3) (2r + 1)! If f C 2r+2, n, using Darboux s formula one can obtain (1.1), with E r (f) = 1 1 ( n 1 ) B 2r+2 B 2r+2 (x) f (2r+2) (k + x) dx (1.4) (2r + 2)! (cf. Whittaker & Watson 26, p. 128). This expression for E r (f) can be also derived from (1.3), writting it in the form E r (f) = = k= ( B 2r+1 (x) n 1 ) f (2r+1) (k + x) dx (2r + 1)! k= B 2r+2 (x) ( n 1 ) f (2r+1) (k + x) dx, (2r + 2)! and then by an integration by parts, the last expression becomes ( B2r+2 (x) n 1 ) 1 f (2r+1) (k + x) (2r + 2)! k= k= B 2r+2 (x) (2r + 2)! ( n 1 k= ) f (2r+2) (k + x) dx. Because of B 2r+2 (1) = B 2r+2 () = B 2r+2, E r (f) can be represented in the form (1.4). Since ( 1) r B 2r+2 B 2r+2 (x) on, 1 and B 2r+2 B 2r+2 (x) dt = B 2r+2, according to the Second Mean Value Theorem for Integrals, there exists η (, 1) such that E r (f) = B ( n 1 2r+2 ) f (2r+2) (k+η) = n B 2r+2 (2r + 2)! (2r + 2)! f (2r+2) (ξ), < ξ < n. k= (1.5)

4 66 G. V. Milovanović Remark 1.1 The Euler-Maclaurin summation formula is implemented in Mathematica as the function NSum with option Method -> Integrate. Practicaly, the Euler-Maclaurin summation formula (1.1) is related with the so-called composite trapezoidal rule, Namely, T n f := T n f I n f = n k= r j=1 f(k) = 1 n 1 2 f() + k=1 f(k) f(n). B 2j f (2j 1) (n) f (2j 1) () + Er T (f), (1.6) (2j)! where I n f := n f(x) dx and the remainder term ET r (f) is given by (1.5) if the function f belongs to C 2r+2, n. Similarly, for a quadrature sum with values of the function f at the points x = k + 1 2, k =, 1,..., n 1, i.e., for the midpoint rule M n f := n 1 k= ( f k + 1 ), 2 there exists the so-called second Euler-Maclaurin summation formula r (2 1 2j 1)B 2j M n f I n f = f (2j 1) (n) f (2j 1) () + Er M (f), (1.7) (2j)! for which j=1 Er M (f) = n (2 1 2r 1)B 2r+2 f (2r+2) (ξ), < ξ < n, (2r + 2)! when f C 2r+2, n (cf. 2, p. 157). The both formulas, (1.6) and (1.7), can be unified as Q n f I n f = r j=1 B 2j (τ) (2j)! f (2j 1) (n) f (2j 1) () + Er Q (f), where τ = for Q n T n and τ = 1/2 for Q n M n. It is true, because of the fact that 24, p. 765 (see also 7) ( 1 ) B j () = B j and B j = (2 1 j 1)B j. 2

5 Families of Euler-Maclaurin formulae 67 If we take a combination of T n f and M n f as Q n f = S n f = 1 3 (T nf + 2M n f), which is, in fact, the well-known classical composite Simpson rule, we obtain S n f I n f = S n f := 1 1 n f() + k=1 r j=2 (4 1 j 1)B 2j 3(2j)! n 1 f(k) + 2 k= ( f k + 1 ) f(n), f (2j 1) (n) f (2j 1) () + Er S (f). (1.8) Notice that the summation on the right hand-side in the previous equality starts with j = 2, because the term for j = 1 vanishes. For f C 2r+2, n it can be proved that there exists ξ (, n), such that Er S (f) = n (4 r 1)B 2r+2 f (2r+2) (ξ). 3(2r + 2)! Some periodic analogues of the Euler-Maclaurin formula with applications to number theory have been developed by Berndt and Schoenfeld 4. In the last section of 4, they showed how the composite Newton-Cotes quadrature formulas (Simpson s parabolic and Simpson s three-eighths rules), as well as various other quadratures (e.g., Weddle s composite rule), can be derived from special cases of their periodic Euler-Maclaurin formula, including explicit formulas for the remainder term. Also, in the papers 8, 23, 25, the authors considered some generalizations of the Euler-Maclaurin formula for some particular Newton-Cotes rules, as well as for 2- and 3-point Gauss-Legendre and Lobatto formulas (see also 2, 9, 16, 17). A recent progress in variable-precision arithmetic and symbolic computation has enabled a development of symbolic/variable-precision software for orthogonal polynomials and quadratures of Gaussian type, as well as for their many generalizations. The corresponding software is available today (Gautschi s package SOPQ in Matlab (cf. 21) and our Mathematica package OrthogonalPolynomials 6, 22). Using this advantage in this paper we give extensions of Euler-Maclaurin formulae by replacing Q n by the composite Gauss-Legendre shifted formula, as well as by its Lobatto modification. Several special cases were obtained by using our Mathematica package OrthogonalPolynomials.

6 68 G. V. Milovanović 2. Euler-Maclaurin formula based on the composite Gauss-Legendre formula Let w ν = wν G and τ ν = τν G, ν = 1,..., m, be weights (Christoffel numbers) and nodes of the Gauss-Legendre quadrature formula on, 1, m f(x) dx = wν G f(τν G ) + Rm(f), G (2.1) where the nodes τ ν are zeros of the shifted (monic) Legendre polynomial π m (x) = ( ) 1 2m P m (2x 1). m Degree of its algebraic precision is d = 2m 1, i.e., Rm(f) G = for all algebraic polynomials of degree 2m 1. The quadrature sum in (2.1) we denote by Q G mf, i.e., m Q G mf = wν G f(τν G ). A characterization of the Gaussian quadrature (2.1) can be done via an eigenvalue problem for the symmetric tridiagonal Jacobi matrix (cf. 18, p. 326), α β1 O β1 α 1 β2 J m =. β2 α.. 2, (2.2) βm 1 O βm 1 α m 1 constructed with the three-term recurrence coefficients, α k = 1/2, β k = , k N. 1 (2k) 2 The nodes τ ν = τν G are the eigenvalues of J m and the weights wν G are given by wν G = β vν,1 2, ν = 1,..., m, where β = µ = dx = 1 and v ν,1 is the first component of the normalized eigenvector v ν = v ν,1 v ν,m T (with v T ν v ν = 1) corresponding to the eigenvalue τ ν, J m v ν = τ ν v ν, ν = 1,..., m.

7 Families of Euler-Maclaurin formulae 69 Golub and Welsch 13 gave an efficient procedure for constructing the Gaussian quadrature rules by simplifying the well-known QR algorithm so that only the first components of the eigenvectors are computed. Such a procedure is implemented in several programming packages including the most known ORTHPOL given by Gautschi 1, as well as in the previous mentioned packages SOPQ (in Matlab) and OrthogonalPolynomials (in Mathematica). The corresponding composite Gauss-Legendre sum for approximating I n f := n f(x) dx can be expressed in the form n 1 G (n) m f = Q G mf(k + ) = k= m wν G k= n 1 f(k + τ G ν ). (2.3) In the sequel we use the following expansion of a function f C s, 1 in Bernoulli polynomias for any x, 1 (see Krylov 15, p. 15) s 1 B j (x) f(x) = f(t) dt+ f (j 1) (1) f (j 1) () 1 f (s) (t)l s (x, t) dt, j! s! j=1 (2.4) where L s (x, t) = Bs (x t) Bs (x) and Bs (x) is a function of period one, defined by B s (x) = B s (x), x < 1, B s (x + 1) = B s (x). (2.5) Notice that B (x) = 1, B 1 (x) is a discontinuous function with a jump of 1 at each integer, and Bs (x), s > 1, is a continuous function. Now, we can prove the following composite formula for the integral I n f = f(t) dt. n Theorem 2.1 For n, m, r N (m r) and f C 2r, n we have G (n) m f I n f = r j=m Q G m(b 2j ) (2j)! f (2j 1) (n) f (2j 1) () + En,m,r(f), G (2.6) where G (n) m f is given by (2.3), and Q G mb 2j denotes the basic Gauss-Legendre quadrature sum applied to the Bernoulli polynomial x B 2j (x), i.e., m Q G m(b 2j ) = wν G B 2j (τν G ) = Rm(B G 2j ), (2.7)

8 7 G. V. Milovanović where R G m(f) is the remainder term in (2.1). If f C 2r+2, n then there exists ξ (, n), such that the error term in (2.6) can be expressed in the form En,m,r(f) G = n QG m(b 2r+2 ) f (2r+2) (ξ). (2.8) (2r + 2)! P r o o f. Suppose that f C 2r, n, where r m. Since the all nodes τ ν = τ G ν, ν = 1,..., m, of the Gaussian rule (2.1) belong to (, 1), using (2.4) with x = τ ν and s = 2r + 1, we have 2r B j (τ ν ) f(τ ν ) = I 1 f + j! j=1 1 (2r + 1)! f (j 1) (1) f (j 1) () f (2r+1) (t)l 2r+1 (τ ν, t) dt, from which, by multiplying by w ν = wν G get and summing in ν from 1 to m, we i.e., m w ν f(τ ν ) = ( m ) w ν f(t) dt 2r ( 1 m + w ν B j (τ ν )) f (j 1) (1) f (j 1) () j! j=1 1 (2r + 1)! ( m ) f (2r+1) (t) w ν L 2r+1 (τ ν, t) dt, Q G mf = Q G m(1) f(t) dt + 2r j=1 Q G m(b j ) j! f (j 1) (1) f (j 1) () + Em,r(f), G where Since Em,r(f) G 1 = (2r + 1)! f (2r+1) (t)q G m (L 2r+1 (, t)) dt. 1, j =, B j (x) dx =, j 1,

9 Families of Euler-Maclaurin formulae 71 and m 1, j =, Q G m(b j ) = w ν B j (τ ν ) =, 1 j 2m 1, because the Gauss-Legendre formula is exact for all algebraic polynomials of degree at most 2m 1, the previous formula becomes Q G mf f(t) dt = 2r j=2m Q G m(b j ) j! f (j 1) (1) f (j 1) () + Em,r(f). G (2.9) Notice that for Gauss-Legendre nodes and weights the following equalities τ ν + τ m ν+1 = 1, w ν = w m ν+1 >, ν = 1,..., m, hold, as well as w ν B j (τ ν ) + w m ν+1 B j (τ m ν+1 ) = w ν B j (τ ν )(1 + ( 1) j ), which is equal to zero for odd j. Also, if m is odd, then τ (m+1)/2 = 1/2 and B j (1/2) = for each odd j. Thus, the quadrature sum for odd j, so that (2.9) becomes Q G mf f(t) dt = m Q G m(b j ) = w ν B j (τ ν ) = r j=m Q G m(b 2j ) (2j)! f (2j 1) (1) f (2j 1) () + Em,r(f). G (2.1) Now, for the (shifted) composite Gauss-Legendre formula we have G (n) m f I n f = = n 1 k= n 1 k= Finally, using (2.1) we obtain n 1 G (n) r m f I n f = k= j=m Q G mf(k + ) k+1 k f(t) dt Q G mf(k + ) f(k + x) dx. Q G m(b 2j ) (2j)! f (2j 1) (k + 1) f (2j 1) (k)

10 72 G. V. Milovanović +Em,r(f(k G + )) = r j=m Q G m(b 2j ) (2j)! f (2j 1) (n) f (2j 1) () + En,m,r(f), G where E G n,m,r(f) is given by En,m,r(f) G 1 = (2r + 1)! ( n 1 k= Since L 2r+1 (x, t) = B2r+1 (x t) B 2r+1 (x) and ) f (2r+1) (k + t) Q G m (L 2r+1 (, t)) dt. (2.11) B 2r+1(τ ν ) = B 2r+1 (τ ν ), B 2r+1(τ ν t) = 1 2r + 2 we have d dt B 2r+2(τ ν t), Q G m (L 2r+1 (, t)) = Q G ( m B 2r+1 ( t) ) Q G ( m B 2r+1 ( ) ) = 1 2r + 2 QG m because Q G m (B 2r+1 ( )) =. Then for (2.11) we get (2r + 2)!E G n,m,r(f) = ( n 1 k= ( ) d dt B 2r+2( t), ) ( ) d f (2r+1) (k + t) Q G m dt B 2r+2( t) dt. By using an integration by aprts, the right-hand side reduces to where RHS = F (t)q G ( m B 2r+2 ( t) ) 1 F (t) = n 1 k= f (2r+1) (k + t). Since B 2r+2 (τ ν 1) = B 2r+2 (τ ν) = B 2r+2 (τ ν ), we have Q G ( m B 2r+2 ( t) ) F (t) dt, F (t)q G ( m B 2r+2 ( t) ) 1 = (F (1) F ())Q G ( m B 2r+2 ( ) ) = Q G m (B 2r+2 ( )) F (t) dt,

11 Families of Euler-Maclaurin formulae 73 so that RHS = Q G m (B 2r+2 ( )) Q G ( m B 2r+2 ( t) ) F (t) dt. Since ( 1) r m Q G m B2r+2 ( ) B2r+2( t) >, < t < 1, (2.12) there exists an η (, 1) such that RHS = F (η) Q G m B2r+2 ( ) B2r+2( t) dt. Because of continuity of f (2r+2) on, n we conclude that there exists also ξ (, n) such that F (η) = nf (2r+2) (ξ). Because of QG m B 2r+2 ( t) dt =, we finally obtain that i.e., (2.8). (2r + 2)!E G n,m,r(f) = nf (2r+2) (ξ) Q G m B 2r+2 ( ) dt, Remark 2.1 Let g G m,r(t) be the left-hand side in (2.12). Typical graphs of functions g G m,r(t) for some selected values of r m 1 are presented in Figure 2.1. Now we consider some special cases of the formula (2.6) for some typical values of m. For a given m, by G (m) we denote the sequence of coefficients which appear in the sum on the right-hand side in (2.6), i.e., G (m) = Q G m(b 2j ) j=m Q = G m(b 2m ), Q G m(b 2m+2 ), Q G m(b 2m+4 ),.... These Gaussian sums we can calculate very easy by using Mathematica Package OrthogonalPolynomials. In the sequel we mention cases when 1 m 7. Case m = 1. Here τ G 1 = 1/2 and wg 1 Q G 1 (B 2j ) = B 2j (1/2) = (2 1 2j 1)B 2j, = 1, so that, according to (2.7), and (2.6) reduces to (1.7). Thus, G (1) = 1 12, 7 24, , , , , , ,....

12 74 G. V. Milovanović Fig Graphs of t g G m,r(t), r = m (solid line), r = m + 1 (dashed line), and r = m + 2 (dotted line), when m = 1, m = 2 (top), and m = 3, m = 4 (bottom) Case m = 2. Here τ1 G = 1 (1 1 ) 3, τ2 G = 1 (1 + 1 ) and w G 1 = w G 2 = 1 2, so that Q G 2 (B 2j ) = 1 2 ( ) B 2j (τ1 G ) + B 2j (τ2 G ) = B 2j (τ1 G ). In this case, the sequence of coefficients is G (2) = 1 18, 1 189, , , , , ,.... so that Case m = 3. Here τ G 1 = 1 1 ( 5 15 ), τ2 G = 1 2, τ 3 G = 1 1 w G 1 = 5 18, wg 2 = 4 9, wg 3 = 5 18, Q G 3 (B 2j ) = 5 9 B 2j(τ G 1 ) B 2j(τ G 2 ) ( 5 + ) 15

13 Families of Euler-Maclaurin formulae 75 and G (3) = 1 28, 49 72, , , , ,.... Case m = 4. The corresponding sequence of coefficients is G (4) = 1 441, , , , ,.... Case m = 5. Here G (5) = , , , ,.... Case m = 6. Here G (6) 1 = , , , ,.... Case m = 7. Here G (7) 1 = , , , , Euler-Maclaurin formula based on the composite Lobatto formula We can also consider the corresponding Gauss-Lobatto quadrature formula f(x) dx = m+1 ν= w L ν f(τ L ν ) + R L m(f), (3.1) with the nodes τ = τ L =, τ m+1 = τm+1 L = 1, and others internal nodes τ ν = τν L, ν = 1,..., m, which are zeros of the shifted (monic) Jacobi polynomial, ( ) 1 2m + 2 π m (x) = P m (1,1) (2x 1), m orthogonal on the interval (, 1) with respect to the weight function x x(1 x). Degree of its algebraic precision is d = 2m + 1, i.e., R L m(f) = for all algebraic polynomials of degree 2m + 1.

14 76 G. V. Milovanović In construction of the Gauss-Lobatto formula Q L m(f) = m+1 ν= w L ν f(τ L ν ), (3.2) we can use parameters of the corresponding Gaussian formula m g(x)x(1 x) dx = ŵν G g( τ ν G ) + R m(g), G which can be calculated from an eigenvalue problem for the symmetric tridiagonal Jacobi matrix (2.2), in this case, with the three-term recurrence coefficients, α k = 1/2, β k = 1 4 k(k + 2) (2k + 1)(2k + 3), k N. The nodes of the Gauss-Lobatto quadrature formula (3.1) are τ L =, τν L = τ ν G, ν = 1,..., m, τm+1 L = 1, and the corresponding weights (cf. 18, pp ) are and w L ν = w L = 1 2 m ŵν G τ ν G (1 τ ν G, ν = 1,..., m, ) ŵ G ν τ G ν The corresponding composite rule is n 1 L (n) m f = Q L mf(k + ) = k=, w L m+1 = 1 2 m m+1 n 1 wν L ν= k= ŵ G ν 1 τ G ν. f(k + τ L ν ), i.e., n L m (n) f = (w L + wm+1) L m n 1 f(k) + w L ν f(k + τν L ). (3.3) k= k= As in the Gauss-Legendere case, there exists a symmetry of nodes and weights, i.e., τ L ν + τ L m+1 ν = 1, w L ν = w L m+1 ν >, ν =, 1,..., m + 1, so that the Gauss-Lobatto quadrature sum Q L m(b j ) = m+1 ν= w L ν B j (τ L ν ) =

15 Families of Euler-Maclaurin formulae 77 for each odd j. Using the same arguments as before, we can state and prove the following result. Theorem 3.1 For n, m, r N (m r) and f C 2r, n we have L (n) m f I n f = r j=m+1 Q L m(b 2j ) (2j)! f (2j 1) (n) f (2j 1) () + En,m,r(f), L (3.4) where L (n) m f is given by (3.3), and Q L mb 2j denotes the basic Gauss-Lobatto quadrature sum (3.2) applied to the Bernoulli polynomial x B 2j (x), i.e., Q L m(b 2j ) = m+1 ν= where R L m(f) is the remainder term in (3.1). w L ν B 2j (τ L ν ) = R L m(b 2j ), If f C 2r+2, n then there exists ξ (, n), such that the error term in (3.4) can be expressed in the form En,m,r(f) L = n QL m(b 2r+2 ) f (2r+2) (ξ). (2r + 2)! Remark 3.1 As in Remark 2.1 we have g L m,r(t) > for < t < 1, where gm,r(t) L := ( 1) r m Q L m B2r+2 ( ) B2r+2( t). Typical graphs of g L m,r(t) for some selected values of r m are presented in Figure 3.1. In the sequel we give the sequence of coefficients L (m) which appear in the sum on the right-hand side in (3.4), i.e., L (m) = Q L m(b 2j ) j=m+1 Q = L m(b 2m+2 ), Q L m(b 2m+4 ), Q L m(b 2m+6 ),..., obtained by using Mathematica Package OrthogonalPolynomials, for some typical values of m. Case m =. This is a case of the standard Euler-Maclaurin formula (1.1), for which τ L = and τ 1 L = 1, with wl = wl 1 = 1/2. The sequence of coefficients is 1 L () = 6, 1 3, 1 42, 1 3, 5 66, , 7 6, , , , ,...,

16 78 G. V. Milovanović Fig Graphs of t g L m,r(t), r = m+1 (solid line), r = m+2 (dashed line), and r = m + 3 (dotted line), when m =, m = 1 (top), and m = 2, m = 3 (bottom) which is, in fact, the sequence of Bernoulli numbers B 2j j=1. Case m = 1. In this case τ L =, τ 1 L = 1/2, and τ 2 = 1, with the corresponding weights w L = 1/6, wl 1 = 2/3, and wl 2 = 1/6, which is, in fact, the Simpson formula (1.8). The sequence of coefficients is 1 L (1) = 12, 5 672, 7 64, , , , , ,.... Case m = 2. Here τ L =, τ L 1 = 1 1 (5 5), τ L 2 = 1 1 (5 + 5), τ L 3 = 1 and w L = wl 3 = 1/12, wl 1 = wl 2 = 5/12, and the sequence of coefficients is 1 L (2) = 21, , , , , , ,.... Case m = 3. Here τ L =, τ L 1 = 1 14 (7 31), τ L 2 = 1 2, τ L 3 = 1 14 (7 + 31), τ L 4 = 1

17 Families of Euler-Maclaurin formulae 79 and w L = 1 2, wl 1 = 49 18, wl 2 = 16 45, wl 3 = 49 18, wl 4 = 1 2, and the sequence of coefficients is 1 L (3) = 3528, , , , ,.... Case m = 4. The corresponding sequence of coefficients is 1 L (4) = 58212, , , , ,.... Case m = 5. Here L (5) 1 = , , , ,.... Case m = 6. In this case the corresponding sequence of coefficients is L (6) 1 = , , , ,.... Case m = 7. Here L (7) 1 = , , , ,.... REFERENCES 1 T. M. Apostol, An elementary view of Euler s summation formula, Amer. Math. Monthly 16 (1999), C. T. H. Baker, G. S. Hodgson, Asymptotic expansions for integration formulae in one or more dimensions, SIAM J. Numer. Anal. 8 (1971), E. W. Barnes, The Maclaurin sum-formula, Proc. London Math. Soc. 2-3 (195), B. C. Berndt, L. Schoenfeld, Periodic analogues of the Euler-Maclaurin and Poisson summation formulas with applications to number theory, Acta Arithmetica 28 (1975),

18 8 G. V. Milovanović 5 P. L. Butzer, P. J. S. G. Ferreira, G. Schmeisser, R. L. Stens, The summation formulae of Euler-Maclaurin, Abel-Plana, Poisson, and their interconnections with the approximate sampling formula of signal analysis, Results. Math. 59 (211), A. S. Cvetković, G. V. Milovanović, The Mathematica Package OrthogonalPolynomials, Facta Univ. Ser. Math. Inform. 19 (24), Dj. Cvijović, J. Klinkovski, New formulae for the Bernoulli and Euler polynomials at rational arguments, Proc. Amer. Math. Soc. 123 (1995), L. Derr, C. Outlaw, D. Sarafyan, Generalization of Euler-Maclaurin sum formula to the six-point Newton-Cotes quadrature formula, Rend. Mat. (6) 12 (1979), no. 3-4, D. Elliott, The Euler-Maclaurin formula revisited, J. Austral. Math. Soc. Ser. B 4 (1998/99), (E), E27 E76. 1 W. Gautschi, Algorithm 726: ORTHPOL a package of routines for generating orthogonal polynomials and Gauss-type quadrature rules, ACM Trans. Math. Software 2 (1994), W. Gautschi, Leonhard Eulers Umgang mit langsam konvergenten Reihen (Leonhard Euler s handling of slowly convergent series), Elem. Math. 62 (27), W. Gautschi, Leonhard Euler: his life, the man, and his works, SIAM Rev. 5 (28), G. Golub, J. H. Welsch, Calculation of Gauss quadrature rules, Math. Comp. 23 (1969), P. Henrici, Applied and Computational Complex Analysis. Volume 2. Special functions Integral transforms Asymptotics Continued Fractions, Pure and Applied Mathematics, John Wiley & Sons Inc., New York London Sydney, V. I. Krylov, Approximate Calculation of Integrals, Macmillan, New York, J. N. Lyness, An algorithm for Gauss-Romberg integration, BIT 12 (1972), J. N. Lyness, The Euler Maclaurin expansion for the Cauchy principal value integral, Numer. Math. 46 (1985), G. Mastroianni, G. V. Milovanović, Interpolation Processes Basic Theory and Applications, Springer Monographs in Mathematics, Springer Verlag, Berlin Heidelberg New York, S. Mills, The independent derivations by Euler, Leonhard and Maclaurin, Colin of the Euler-Maclaurin summation formula, Arch. Hist. Exact Sci. 33 (1985), G. V. Milovanović, Numerical Analysis, Part II, Naučna knjiga, Beograd, 1988 (Serbian). 21 G. V. Milovanović, Chapter 23: Computer algorithms and software packages, In: Walter Gautschi: Selected Works and Commentaries, Vol. 3 (C. Brezinski, A. Sameh, eds.), Birkhäuser, Basel, G. V. Milovanović, A. S. Cvetković, Special classes of orthogonal polynomials and corresponding quadratures of Gaussian type, Math. Balkanica 26 (212), C. Outlaw, D. Sarafyan, L. Derr, Generalization of the Euler-Maclaurin formula for Gauss, Lobatto and other quadrature formulas, Rend. Mat. (7) 2 (1982), no. 3,

19 Families of Euler-Maclaurin formulae A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Integrals and Series, Vol. 3, Gordon and Breach, New York, D. Sarafyan, L. Derr, C. Outlaw, Generalizations of the Euler-Maclaurin formula, J. Math. Anal. Appl. 67 (1979), no. 2, E. T. Whittaker, G. N. Watson, A Course of Modern Analysis, 4th edition, Series: Cambridge Mathematical Library, Cambridge University Press, Mathematical Institute Serbian Academy of Sciences and Arts Kneza Mihaila Belgrade, Serbia gvm@mi.sanu.ac.rs