Euler Maclaurin summation and Schlömilch series

Transcription

1 Euler Maclaurin summation and Schlömilch series I Thompson and C M Linton Department of Mathematical Sciences, Loughborough University, Loughborough, Leics. U i.thompson@lboro.ac.uk Abstract A method for analysing a class of divergent series is developed from the Euler Maclaurin summation formula. The conditions that the summand must satisfy are explored, and a significant simplification is obtained for cases the summation ranges over all integers. As an example, we consider the Ewald representation for Schlömilch series, and show that this includes Twersky s dual series formula as a limiting case. 1 Introduction This article concerns the behaviour of certain series that do not converge uniformly, in limits they become divergent. In particular, we seek to generalise the formula ] lim (s + x) n e (s+x)δ s n e sδ ds = B n+1(x) δ 0 + (n + 1), (1) s=0 B n ( ) is the Bernoulli polynomial of order n. This seems to have originated in 1, p. 53], it is derived from the Euler Maclaurin summation formula, 3, 4]. It can be used as a means of expanding the series on the left-hand side in negative powers of δ, because the integral can be evaluated exactly in terms of the gamma function 5, 6.1.1]. We will consider a class of series in which e (s+x)δ is replaced by another function, E(s, δ), say, which ensures convergence for δ > 0. The essential idea is simple the Euler Maclaurin formula is used to obtain a closed form expression for the difference between the sum and the integral in the limit δ 0. This is highly desirable because it is usually much easier to integrate a function over a given range than it is to sum a discrete set of its values. The plan for the first half of the paper is as follows. The basic formulae and conditions for the use of Euler Maclaurin summation are set out in, and a generalised version of (1) is obtained from these in 3. Careful attention is paid to the conditions that the function E(s, δ) must satisfy in order that the procedure is valid. Finally, in 4, we obtain a formula that can be used in cases the summation ranges over all integers, which turns out to be much simpler. In the second half of the paper, we provide an illustrative example of the use of the generalised limit formulae, the difficulties that this can entail, and some means of overcoming these. Thus, it turns out that series such as that appearing in (1) (with various forms for E(s, δ)), occur in several places in the theory of the Schlömilch series 0 σ n (X, ) = e ijx + ( 1) n e ijx] H (1) n (j). () Here, X and are real parameters, and H (1) n ( ) is the nth order Hankel function of the first kind. The right-hand side of () is known as a representation, and arises naturally

2 April 30, 009 in the study of wave interactions with periodic structures 6, 7, 8]. Provided that X ± + qπ, q Z, the series in () converges, but it does so rather slowly, and therefore spatial representations cannot be used to efficiently evaluate Schlömilch series. A second representation for σ 0, which is more useful for practical purposes, is well-known 9, 8.54] and easy to derive 10]. This is usually referred to as the dual series formula, or spectral representation. Equivalent results at higher orders are much more difficult to obtain. If a procedure similar to that in 10, 3.3] is followed, one soon encounters a limit involving complicated infinite series that do not converge uniformly. This problem was solved by Twersky in ]. By a sequence of ingenious manipulations, the infinite series were expressed in such a form as to facilitate evaluation of the limit by means of (1), and this lead Twersky directly to the dual series formulae for n > 0. A third representation for the Schlömilch series () appeared in ] (see also 13]). This was derived using Ewald summation, a technique that had been successfully applied some years earlier to analogous problems for spherical wave functions 14]. The Ewald representation contains a positive parameter, here denoted by η, which affects the computational efficiency of the formula, but not its actual value. Taking the limit η 0 retrieves the spatial representation () 15]. It was also noted in 15] that for the case n = 0, the Mellin transform can be used to retrieve the dual series representation in the limit η, but no means of applying this procedure to higher order Schlömilch series has been found. Interestingly, the main obstacle is again a limit involving series with nonuniform convergence. This is the problem that we are concerned with here. The plan for the second half of the paper is as follows. In 5 we set up our notation and give the dual series and Ewald representations for (). Some terms in the dual series formula can be retrieved directly; the others require the use of the generalised limit formulae derived in 3 4, and this procedure is carried out in 6 7. Some concluding remarks are made in 8. Euler Maclaurin summation We will apply Euler Maclaurin summation in the form given in ], which is convenient for our purposes (4, ch. 14] uses a similar formulation, but with an unusual definition of the Bernoulli polynomials). Thus, for a function f and a natural number r, we have r f(s) = s=1 r 1 f(s) ds + f(1) + f(r) + B j f (j 1) (r) f (j 1) (1) ] (j)! + r 1 P m+1 (s) (m + 1)! f (m+1) (s) ds, (3) provided that all of the required derivatives are continuous on the interval 1, r]. Here, B j is the jth Bernoulli number, and P j ( ) is the periodic extension of the Bernoulli polynomial of order j from 0, 1) to R, that is P j (s) = B j (s s]), s] represents the integer part of s. Equation (3) holds for all non-negative integers m (the finite series on the right-hand side disappears if m = 0). Provided that the

3 April 30, resulting series and integrals converge, we can take the limit r to obtain f(s) = s=1 1 f(s) ds + f(1) B j (j)! f (j 1) P m+1 (s) (1) + 1 (m + 1)! f (m+1) (s) ds. (4) Note that the existence of the series on the left-hand side of this expression implies that f(s) 0 as s. Finally, in place of the arbitrary function f(s), we write f(s 1+λ), λ Z, and this recasts (4) into the form f(s) = s=λ λ f(s) ds + B m f; λ] + m f; λ], (5) B m f; λ] = f(λ) B j (j)! f (j 1) (λ), (6) and m f; λ] = λ P m+1 (s) (m + 1)! f (m+1) (s) ds. (7) As outlined in 1, it is our intention to use (5 7) to evaluate limits of certain series that are not uniformly convergent. The finite series in (6) poses no problems in this respect, but the periodic Bernoulli functions impede the evaluation of the integral in (7), and the fact that this is an improper integral can cause difficulties when taking limits. However, if there exists A λ such that f (m+1) (s) is of fixed sign for s > A, then we have A f; λ] f (m+1) (s) ] ds + f (m+1) (s) ds (π) m+1. λ Here, we have used the upper bound for periodic Bernoulli functions of odd order given in 16]. The second integral on the right-hand side can now be evaluated, and, since f(s) 0 as s, we find that m f; λ] (π) m+1 3 Generalised limit formula A A f (m+1) (s) ds + f (m) (A) ]. (8) λ Equation (1) can be obtained by applying Euler Maclaurin summation to the function f(s, δ) = (s + x) n e (s+x)δ, with the variable s retaining the role it played in. The crucial observation regarding this is that if we choose m > n/, then the right-hand side of (8) will disappear in the limit δ 0. Motivated by this, we seek a generalised version of (1) by considering functions of the form f(s, δ) = (s + x) n E(s, δ), with E(s, 0) = E 0. (9)

4 April 30, Here, x and E 0 are constants and n 0 is an integer. We must assume that there exists δ 0 > 0 such that the series and integrals obtained when (9) is substituted into (4) converge for δ (0, δ 0 ). Euler Maclaurin summation will then be applied on the interval λ, ), λ Z, and to this end we also assume that the function E is n 0 times continuously differentiable with respect to s on this interval for all δ 0, δ 0 ), { n + : n odd, n 0 = (10) n + 3 : n even. Later we will see that this condition can be relaxed. Finally, we require the existence of A > 0 such that 0 δ < δ 0 f (n 0 ) (s, δ) ds = f (n0) (s, δ) ds, A the superscript denotes derivatives with respect to s. This last condition is necessary only in that it facilitates the use of (8). A more sophisticated analysis of f; λ] than that in could be used to extend the theory to cases the derivatives of E are of an oscillatory nature. Note that we could consider functions of the form f(s, δ) = p(s)e(s, δ), for an arbitrary polynomial p, but the particular case (9) leads to a number of elegant simplifications, and we can always write p(s) as a finite Taylor series about s = x. Now, from the mean value theorem, we can write E(s + s, δ) E(s, δ) s A = E (c, δ), for some s c s + s. We can then set δ = 0 and subsequently take the limit s 0 to show that E (s, 0) = 0, in view of (9). Consequently, by induction, d j E(s, 0) = 0, (11) dsj for j N such that the derivative exists, and all s λ, ). It now follows immediately from (9) that for j > n we have ] d j lim f(s, δ) = 0, δ 0 dsj because expansion of the bracket using Leibniz product rule leads to an expression each term involves a derivative of E. Hence, taking m > n/, we see that lim mf; λ] = 0; (1) δ 0 the limit δ 0 can be commuted with the finite integral on the right-hand side of (8). According to our assumption regarding the differentiability of E, we now have only one choice for m, namely that m = n/] + 1. (13) Note that any terms with j > n + 1 which appear in the series in (6) if m > (n + 1)/ vanish in the limit δ 0 due to (11). One such term occurs when n is even if we use

5 April 30, (13); also choosing a larger value for m in cases more continuous derivatives exist simply introduces more o(1) terms to B m. Substituting (9) into (6) we find that lim B n/]+1f; λ] = E n/]+1 0 δ 0 (λ + B j x)n lim δ 0 (j)! = E 0 (λ + n+1 B j n!(λ + x) n j+1 E 0 x)n j!(n j 1)! j= ) = E n+1 0 n + 1 having used (11), and also the fact that B j 1 = B j ( n + 1 j d j 1 (s + x) n E(s, δ) ] ds j 1 s=λ (λ + x) n j+1, { 1/ : j = 1, 0 : j > 1. Collecting together our results, it is now clear that inserting a function of the form (9) into (5) yields (s + x) n E(s, δ) = s=λ λ (s + x) n E(s, δ) ds E 0 n + 1 n+1 ( ) n + 1 B j (λ + x) n j+1 + o(1), j (14) provided that m is chosen according to (13). Since B 0 = 1, and the Bernoulli polynomial B n (x) is given by 9, 9.60] we rewrite (14) in the form s=λ B n (x) = n ( ) n B j x n j, j (s + x) n E(s, δ) = I n (x, δ) E 0 n + 1 B n+1(λ + x) + o(1), (15) I n (x, δ) = E 0 n + 1 (λ 0 + x) n+1 + (s + x) n E(s, δ) ds. (16) λ 0 Here, we are free to set the parameter λ 0 to any finite value; it need not equal λ because E 0 n + 1 (λ + x) n+1 (λ 0 + x) n+1] λ0 + (s + x) n E(s, δ) ds 0 as δ 0. (17) λ This greatly increases the possibility of finding suitable standard integrals to facilitate the explicit evaluation of (16). Equation (1) can now be obtained immediately by taking E(s, δ) = e (s+x)δ, λ = 0 and λ 0 = x. Finally, we demonstrate that the conditions on the differentiability of E can in fact be relaxed. Thus, suppose that a sufficient number of derivatives for the application of

6 April 30, (15) exist on Λ, ) for some Λ Z, but not on λ, ). In this case, we simply split the series under consideration by writing Λ 1 (s + x) n E(s, δ) = (s + x) n E(s, δ) + s=λ s=λ Now a repeated application of 9, 9.63()] shows that lim δ 0 (s + x) n E(s, δ). (18) s=λ j 1 B n+1 (y + j) B n+1 (y) = (n + 1) (s + y) n, j N, and with this we obtain Λ 1 ] Λ λ 1 (s + x) n E(s, δ) = E 0 (s + x + λ) n s=λ s=0 s=0 = E 0 Bn+1 (x + Λ) B n+1 (x + λ) ]. (19) n + 1 Thus, after applying (15) to the semi-infinite sum in (18), and using (17) and (19) we actually return to (15), in a situation no assumptions have been made regarding the differentiability of E on λ, Λ) (the conditions remain in force on Λ, )). This is in spite of the fact that the Euler Maclaurin formula, from which (15) was derived is no longer valid on λ, Λ]. 4 Sums over Z The material in the previous section is geared toward the treatment of infinite series with a finite lower limit using Euler Maclaurin summation. We can also consider the case the summation ranges over all integers, again provided that the relevant series and integrals are convergent. In this case, we write and observe that f(s) = f(0) + ( ) f(s) + f( s), (0) s=0 B m f(s); 0] + B m f( s); 0] = f(0), for any m, because the finite series in (6) contains only odd derivatives. Consequently, if we apply equation (5) with λ = 0 to the sum on the right-hand side of (0), we obtain f(s) = f(s) ds + m f(s); 0] + m f( s); 0]. (1) Now the periodic extension of the Bernoulli polynomial that appears in the definition of f; λ] (7) is almost every differentiable, and the derivative exists, we have 9, 9.63(3)] P m(x) = mp m 1 (x). Hence, repeated integration by parts in (7) shows that the right-hand side of (1) is independent of m. Nevertheless, the freedom to choose a value for this parameter leads

7 April 30, to an interesting consequence if f is of the form (9). As an example, consider the case f(s, δ) = e (sδ). Jacobi s imaginary transformation for the theta functions can be used to show that 17, p. 476] π e (sδ) = e (sπ/δ), δ and hence f(s, δ) f(s, δ) ds = π δ s 0 e (sπ/δ). Thus in this case, the difference between the sum of f(s, δ) at integer values of s and the integral of the same function over all s is beyond all asymptotic orders in δ. This is consistent with (1), since (8) shows that m f(s, δ); 0] + m f( s, δ); 0] 4 f (m) (0, δ), (π) m+1 and we can take m to be arbitrarily large, thereby introducing any required power of δ to the upper bound on the right-hand side. 5 Ewald representation of Schlömilch series It is convenient to work in terms of scattering angles, defined via and cos ψ s = X + sπ, s R, () sin ψ s = iγ s = iγ(cos ψ s ), (3) γ(z) = (z 1) 1/ ; γ(0) = i. (4) The branch cuts for this function are placed on the line sections z = ±(1 + iu), so that sin ψ s is either positive real or positive imaginary. Consequently z γ(z) 0 as z ±, meaning that e ±iψs = cos ψ s ± i sin ψ s 0 as s ± ; (5) this is important later. With these definitions, Twersky s dual series representation for Schlömilch series 11] with n > 0 is given by σ n (X, ) = in e inψ 0 ( e inψ s ) ] + + einψ s + µ n (X, ), (6) sin ψ 0 sin ψ s sin ψ s s=1 µ m 1 (X, ) = 1 m 1 ( ) ( 1) j j+1 (m + j 1)! 4π B j+1 π (m j 1)!(j + 1)! µ m (X, ) = i ( ) j j (m + j 1)! 4π ( 1) B j π (m j)!(j)! ( ) X. (7) π ( ) X. (8) π

8 April 30, and B j ( ) is the Bernoulli polynomial of order j. On the other hand, the Ewald representation (again for n > 0) is 10, 3.6] in which and τ n (1) (X, ) = in τ () n (X, ) = n+1 iπ s=1 σ n (X, ) = τ (1) n (X, ) + τ () n (X, ), (9) cos n ψ s sin ψ s n/] ( 1) j Γ( 1 j) ( e isx + ( 1) n e isx) ( s ( ) ( ) n 1 tan j ψ s Γ j j, γs, (30) 4η ) n e /(4u ) e s u u n 1 du. (31) Note that η in our notation is equivalent to ηa in 10]. The symbol Γ(, ) represents the (upper) incomplete Gamma function 5, 6.5.3]; that is Γ(µ, z) = z η t µ 1 e t dt, π < argt] < π, (3) and for x < 0 we define Γ(µ, x) = lim Γ(µ, x iɛ). (33) ɛ 0 + The series in (30) and (31) converge exponentially for η (0, ). It is not difficult to see that τ n (1) 0 as η 0, whilst the integral in (31) becomes a representation of the Hankel function 9, 3.471(9)]; and so () is retrieved. Similarly, τ n () 0 as η, but we cannot commute this limit with the series in (30). Instead, we will take the limit using Euler Maclaurin summation as in 3 4. Before we can do this, the series for τ n (1) must be recast in the appropriate form. Note that () shows that integer powers of cos ψ s are actually polynomials in s of the correct type for use in (9). It is only the presence of sin ψ s in the denominator of (30) that currently stands in our path. To remove this obstacle, we first use the decomposition Γ( 1 j, δ γ s) Γ( 1 j) = erfc(δγ s ) e (δγs) j p=1 (δγ s ) 1 p Γ( 3, (34) p) we have written δ = /(η), and the finite series disappears when j = 0. This formula can be obtained by repeated integration by parts in (3), followed by a substitution, yielding the series and the complementary error function, respectively. In this process, the square roots must be taken in accordance with (3) and (33). Note that for γ s real and positive, (34) can be obtained directly from 9, 8.356(5) & 8.359(3)]. Equation (30) now becomes τ n (1) = S n + R n, (35) S n = in R n = in+1 cos n n/] ψ s erfc(δγ s ) sin ψ s n/] ( ) j n ( 1) j j p=1 ( 1) j ( n j ( 1) p δ 1 p Γ( 3 p) ) tan j ψ s, (36) f j p n j (s, δ) (37)

9 April 30, and f v u(s, δ) = cos u ψ s sin v ψ s e (δγs). (38) The summand f v u has the form (9) because the power of sin ψ s is always even (see equation ()). The behaviour of R n in the limit δ 0 is determined in 6. For S n, we observe that the finite series in (36) can be expressed in closed form; thus Now S n = in = in cos n ψ s sin ψ s erfc(δγ s ) (1 + i tan ψ s ) n + (1 i tan ψ s ) n] cos(nψ s ) sin ψ s erfc(δγ s ). (39) sin(nψ s ) sin ψ s = U n 1 (cos ψ s ), U n ( ) is a Chebyshev polynomial of the second kind in cos ψ s 18, p. 631]; therefore we make the further decomposition S c n = in e inψ 0 ( e inψ s erfc(δγ 0 ) + sin ψ 0 s=1 this series being uniformly convergent, and S d n = in+1 sin nψ 0 sin ψ 0 erfc(δγ s ) + S n = S c n + S d n, (40) sin ψ s erfc(δγ s ) + e inψ s sin ψ s erfc(δγ s ) ( sin nψs s=0 sin ψ s ) ], (41) erfc(δγ s ) sin nψ ) ] s erfc(δγ s ). (4) sin ψ s Here it is convenient to include the term for which s = 0 in the expression for S d n, even though this makes no contribution. Taking the limit δ 0 in S c n retrieves the infinite series component of the dual representation for σ n (6); we need not perform any further manipulations on this expression. Finally, from 18, eqn ], we have S d n = n i n+1 (n 1)/] ( 1) j j ( ) n j 1 j ] g + n j 1 (0, δ) + g + n j 1 (s, δ) g n j 1 (s, δ), (43) s=0 g ± v (s, δ) = cos v ψ ±s erfc(δγ ±s ). (44) Equation (43) expresses S d n in a form suitable for the application of (15); this is undertaken in 7.

10 April 30, The series R n in the limit δ 0 Applying (1) to the fully infinite series that appears in (37), we obtain f v u(s, δ) = = v q=0 ( v q cos u ψ s ( 1 cos ψ s ) v e (δγ s) )( 1) q cos u+q ψ s e (δγs) ds + o(1). Since the effect of the generalised limit formula is to replace the problem of evaluating an infinite series with that of evaluating an integral of the same function, a closed form expression for the finite sum is known automatically at this stage, as this is no longer the case after we evaluate the integral. This is an important general point: once the Euler Maclaurin summation procedure has done its work, we can reverse any decomposition that we have used by taking finite sums etc. under the integral sign and evaluating them explicitly. In this case, we find that f v u(s, δ) = sin v ψ s cos u ψ s e (δγs) ds + o(1). Evidently, the cost of this is the appearance of more complicated integrals; nothing is gained by continuing the process to a point these cause serious difficulties. We therefore evaluate the integral now, and to this end, we write c = cos ψ s, which yields f v u(s, δ) = π ( 1 c ) v c u e δ (c 1) dc + o(1). (45) Clearly this integral is zero if u is odd, which is always the case if n is odd in (37); hence On the other hand, if u = w, then we have lim R m 1 = 0, m N. (46) δ 0 f v w(s, δ) = ( 1)v π 1 t v (1 + t) w 1/ e δt dt + o(1), and we can use (A1) to obtain f v w(s, δ) = ( 1)w δ 1 v w Γ( 1 v w) M ( 1 w, 1 v w, δ) + o(1). When this is substituted into (37), we find that R m = i( 1)m πδ m ( ) j m Γ( 1 + m p) j Γ( 3 p) M ( 1 m + j, 1 m + p, δ) + o(1); (47) p=1 here we have used the reflection formula 5, ]. This expression is reduced to a single finite series in appendix B; the final result is given by (B5).

11 April 30, The series S d n in the limit δ 0 Next consider S d n, given by equation (43). Making use of (), we write g ± v (s, δ) = cos v ψ ±s erfc(δγ ±s ) = ( s + x ±) v E ± v (s, δ), and the constants x + and x are given by E ± v (s, δ) = (±π/) v erfc δγ(cos ψ ±s )], x ± = ±X/(π). We now apply (15) and (16) with λ = 0, λ 0 = x (this choice is made purely for algebraic convenience) and the role of E 0 taken by (±π/) v. Note that the functions g v ± are not differentiable at points γ ±s = 0 (see 3). We find that g v ± (s, δ) = s=0 = x x ( ) s + x ± v E ± n (s, δ) ds (±π/)v ( B ) v+1 x ± + o(1) v + 1 cos v ψ ±s erfc(δγ ±s ) ds (π/)v v + 1 B v+1 ( ) X + H( 1) cos v ψ 0 + o(1), π having applied the symmetry relation B v ( x) = ( 1) v B v (x) + vx v 1 ] 5, 3.1.9] to write all of the Bernoulli polynomials in terms of x +. This leads to the appearance of the extra term, in which H( ) is the Heaviside unit function (note that cos ψ 0 = X/). If we now make the substitution c = cos ψ ±s and then perform an integration by parts, we find that g + v (0, δ) + s=0 ( g + v (s, δ) g v (s, δ) ) = 1 ( 1)v ] v + 1 δ π 3/ 0 e δ (c 1) (π/)v v + 1 B v+1 (c 1) 1/ cv+ dc ( ) X + o(1). (48) π Here, we are once again free to adjust the lower limit of integration, since the contribution from any finite line section disappears as δ 0; it is convenient to use the value 1. Moreover, the integral vanishes completely in cases v is even, which is always the case if n is odd in (43). Thus, setting n = m 1 and then writing m 1 j in place of j, equation (43) becomes S d m 1 = 1 π m 1 ( 1) j (m + j 1)! (m j 1)!(j + 1)! ( ) j+1 ( ) 4π X B j+1 + o(1). (49) π Similarly, if we write n = m, and replace j with m j in (43), we find that, for m > 0, S d m = V m + i π j (m + j 1)! ( 1) (m j)!(j)! ( ) j ( ) 4π X B j + o(1), (50) π in which V m = iδ π 3/ j (m + j 1)! ( 4) (m j)!(j)! 0 t 1/ (t + 1) j e δt dt. (51)

12 April 30, Here, the series for S d m and V m have been extended to begin at 0; the net effect of this is zero, because B 0 (x) 1, and the integral evaluates to π/δ if j = 0. Next, we use (A1), (D) and (D4) to obtain V m = i π (m + j 1)! ( 1) j δ j M ( j, 1 (m j)!j! j, δ). (5) This is reduced to a single finite series in appendix C. The final result is given by (C1), and if we compare this with (B5), we immediately see that V m + R m 0 as δ 0. Equations (49) and (50) therefore give the finite series component of Twersky s dual series formulae (7 8), and we need only add Sn c (41) to construct the complete result, in view of (46). As a final note, we observe that taking the finite series in (51) under the integral sign and using 18, p. 631] yields V m = i( 1)m δ π 3/ m 0 T m ( 1 + t ) t 1/ e δt dt, T n ( ) is the nth order Chebyshev polynomial of the first kind. This is to be expected, because (43) contains a Chebyshev polynomial of the second kind (in cos ψ s ), and we have subsequently performed an integration by parts. Since (C1) gives the exact value of V m (as opposed to an o(1) approximation), we now have 0 T m ( 1 + t ) t 1/ e δt dt = π δ + m π r=1 (m + r 1)! (m r)!r! δ r 1, (53) a result that we have not found in any standard tables. Here, we have separated the term in which r = 0 from the remainder of the series; in this form (53) is valid for all m 0, although we have only derived it for cases m > 0. 8 Concluding remarks Using the method of Euler Maclaurin summation, we have derived a formula that enables us to determine the behaviour of a class of infinite series in a limit they become divergent. Essentially, the formula replaces the problem of summing a set of a function s values with the much simpler problem of integrating the same function. The formula is remarkably robust, in that it continues to work in cases the function to be summed possesses an insufficient number of derivatives for the application of Euler Maclaurin summation. Furthermore, if the required integral is expressed as a sum of proper and improper integrals, the former always disappear in the relevant limit. This allows us to adjust the range of integration without affecting the final result, and greatly enhances the possibility of finding that facilitate an explicit evaluation in standard tables. As an example of the use of the method, we have shown that the Ewald representation for Schlömilch series includes Twersky s dual series formula as a limiting case (η, or equivalently δ 0). The Ewald representation (9) was decomposed into components that converge in the limit δ 0, and components that diverge. Overall the representation is always convergent; thus the problem essentially relates to the order in which certain manipulations are carried out, and the limit δ 0 must be the last of all. The methods

13 April 30, developed in 3 and 4 enable us to express the divergent series as expansions in negative powers of δ, plus constant and (unknown) o(1) terms. When the results are combined, the spurious negative powers of δ disappear, and the limit δ 0 can then be taken easily. Interestingly, the mechanisms by which this cancellation occurs in the cases of odd and even ordered Schlömilch series turned out to be rather different to one another. For odd orders, the singular terms (45) and (48) (which are contained in the integrals) vanish independently. On the other hand, for even orders we found it necessary to combine together all of the expansions in negative powers of δ to achieve the cancellation. It may be that alternative means of casting (30) in a form amenable to the use of (15) and (1) exist and circumvent this complication, but we have been unable to find any. Notwithstanding this, it is possible to deal head-on with the problem, as in appendices B and C, using methods found in 19]. Whilst this is somewhat involved algebraically, it is essentially straightforward. Acknowledgements The authors would like to thank Prof P A Martin for useful discussions regarding certain aspects of this work. A A useful integral Suppose that b > a > 0, with b / N, and let { 1 : a N t 0 = 0 : b a N. Then, provided that t 0 is defined, and Rez] > 0, we have t 0 e zt t a 1 (t + 1) b a 1 dt = πz 1 b M(1 + a b, b, z), (A1) Γ( b) sin(πb) M(,, ) is ummer s confluent hypergeometric function. To obtain this result, use 5, eqn ] to write the relationship between the hypergeometric functions U and M 5, eqn ] in the form U(a, b, z) sin (π(b a)) Γ(b a) sin(πb)γ(b) M(a, b, z) = Equation (A1) now follows directly from 5, eqns & 13..5]. B Series expansion for R m πz 1 b M(1 + a b, b, z). Γ( b)γ(a) sin(πb) Here, we determine an explicit expression for the terms in R m, m > 0, that do not vanish in the limit δ 0. This is achieved by evaluating the finite sums over j and p in (47). It turns out that elementary methods do not suffice, and therefore we will appeal to the Chu Vandermonde theorem 19,..3], which states that n ( n) j (a) j (c) j j! = (c a) n (c) n, (B1)

14 April 30, ( ) n is Pochhammer s symbol (see appendix D). Now, introduce the Maclaurin expansion for ummer s hypergeometric function 5, eqn ] into equation (47) and then apply (D3) and (D) to the Pochhammer symbol in the denominator to obtain R m = i( 1)m πδ m r=0 ( 1) r δr r! ( ) m ( 1 j m + j) r j p=1 Γ( 1 p + m r) Γ( 3 p) + o(1). (B) It is evident that the term with r = m, which we denote R 0 m, is rather different to those in which r < m, and we denote the sum of these by R m. In the first instance, we have m Rm 0 = i( 1)m m πm! (m 1)!! ( ) j m ( 1) j j 1 p ; here we have used (D4) to remove the remaining Pochhammer symbol. Interchanging the order of the summations, we obtain R 0 m = i( 1)m m π (m 1)!! m p=1 ( 1) p 1 p m p p=1 ( 1) j (m j p)!(j + p)!, and the inner series can now be written in the form (B1) using the method described in 19,.7]. Thus, the first term is 1/(p!(m p)!), and the ratio of successive terms is (j + p m)/(j + p + 1), meaning that R 0 m = i( 1)m m m!π (m 1)!! m p=1 ( 1) p 1 p ( ) m p m (p m) j. p (p + 1) j Since (1) j = j!, the Chu Vandermonde theorem (B1), with c = p + 1 and a = 1, yields R 0 m = m i( 1)m m m!mπ (m 1)!! ( ) ( 1) p p m. 1 p p Exactly the same method can now be used to evaluate the sum over p; we find that ( Rm 0 = i( 1)m m 1 m m!π (m 1)!! 1 ) (1 m) p p p! ( ), 3 and after applying (B1) this reduces to p=1 p=0 p R 0 m = i( 1)m mπ. (B3) Next consider R m. Since we are concerned with the terms in (B) with r < m, equation (D) can be applied to the ratio of gamma functions, and this leads us to R m = i( 1)m πδ m m 1 r=0 ( 1) r m r δ r r! ( ) m ( 1 j m + j) r ( 1 ) ( ] 1 j), m r m r

15 April 30, we have used (D5) to evaluate the inner series. Note that the sum over j has been extended to begin at 0. This clearly has no effect on the value of Rm, but it now turns out that the second term in the square bracket makes no contribution. To see this, use (D3) to obtain ( 1 m + j) ( 1 j) = ( r m r ( 1)r 1 j), (B4) m and then (D4) yields ( ) m ( 1 j j) m = (m)! 4 m m! = 0. ( ) m ( 1) j j For the remaining term, we use (B4) again, and then write m r in place of r to obtain R m = i( 1)m π r=1 δ r ( 1 r(m r)! ) r ( ) m ( 1 j) m j ( 1 j). r Once again, the inner sum can be evaluated using the Chu Vandermonde theorem. Thus, after determining the first term, and the ratio of successive terms, we rewrite Rm in the form R m = i( 1)m π r=1 δ r ( 1 r(m r)! ) m ( m) j ( 1 r) j ( 1 )j j!. Applying (B1), and then adding the constant term given by (B3), we arrive at the final result R m = i( 1)m (m + r 1)! δ r + o(1). (B5) π r!(m r)! C Series expansion for V m r=0 It is relatively easy to reduce V m (5) to a single finite series. Again using 5, eqn ], and also (D3) we find that V m = i π (m + j 1)! ( 1) j δ j (m j)!j! j r=0 (1 + j r) r r!( 1 + j r) δ r ; r note that the Maclaurin series terminates due to the Pochhammer symbol in the numerator, which can now be written as a ratio of factorials. This done, we write j r in place of r, and interchange the summations, to obtain V m = i π r=0 δ r r! ( 1) j (m + j 1)! (m j)!(j r)!( 1 + r). j r j=r The inner series can be evaluated using the Chu Vandermonde theorem as in appendix B. After some manipulation, we find that V m = i π r=0 (m + r 1)! (m r)!r! m r ( 1) r δ r (r m) j (r + m) j j!( 1 + r), j

16 April 30, and (B1) along with (D3) now yields V m = i( 1)m π r=0 (m + r 1)! r!(m r)! δ r. (C1) D Pochhammer s symbol For non-negative integers n, Pochhammer s symbol is defined as (z) 0 = 1, (z) n = z(z + 1)... (z + n 1), n N. (D1) or, equivalently note the important reflection formula (z) n = Γ(z + n) ; (D) Γ(z) (z) n = ( 1) n (1 z n) n, (D3) which follows directly from (D1). Pochhammer symbols containing half integer values can be written in terms of factorials. For integers j satisfying 0 j n, we have ( 1 j) = ( 1 n n ) ( 1) j (j 1)!!(n j 1)!! = ( 1 ) n ( 1) j = ( 1 4 )n ( 1) (j)!(n j)! (j)!!(n j)!! j (j)! (n j)!. (D4) j! (n j)! An analogous expression for the case j > n is easy to obtain, but this is not required here. One more result that we will need is ( 3 j) n = 1 n + 1 this is easily verified by induction on m, using the fact that References ( 1 ) n+1 ( 1 m) n+1], (D5) (n + 1) ( 1 m) n + ( 1 m) n+1 = ( 1 m) n+1. 1] N. E. Nørlund. Vorlesungen über Differenzenrechnung. Springer Verlag, Berlin, ] T. M. Apostol. An elementary view of Euler s summation formula. Amer. Math. Monthly, 106(5): , ] G. H. Hardy. Divergent Series. AMS Chelsea Publishing, New York, Originally published by Oxford University Press in Available online at

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