Boersma, J.

Doelder, de, P.J.

English

Pure OAI Repository

 Solution to Problem 92-11* : On alternating multiple sumsCitation for published version (APA):Boersma, J., & Doelder, de, P. J. (1993). Solution to Problem 92-11* : On alternating multiple sums. SIAMReview, 35(3), 497-500. https://doi.org/10.1137/1035101DOI:10.1137/1035101Document status and date:Published: 01/01/1993Document Version:Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers)Please check the document version of this publication:• A submitted manuscript is the version of the article upon submission and before peer-review. There can beimportant differences between the submitted version and the official published version of record. Peopleinterested in the research are advised to contact the author for the final version of the publication, or visit theDOI to the publisher's website.• The final author version and the galley proof are versions of the publication after peer review.• The final published version features the final layout of the paper including the volume, issue and pagenumbers.Link to publicationGeneral rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.            • Users may download and print one copy of any publication from the public portal for the purpose of private study or research.            • You may not further distribute the material or use it for any profit-making activity or commercial gain            • You may freely distribute the URL identifying the publication in the public portal.If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, pleasefollow below link for the End User Agreement:www.tue.nl/taverneTake down policyIf you believe that this document breaches copyright please contact us at:openaccess@tue.nlproviding details and we will investigate your claim.Download date: 16. Nov. 2023PROBLEMS AND SOLUTIONS 497theorem" by the authors and yet, if 92-1" were solved, I doubt that it could be made use of inorder to prove the validity of (5), The underlying reason is, in my opinion, the pronouncedasymmetry manifesting itself in the above problem when b c, in contrast to G(t) being ofeven parity in the theorem of 1, 3]. Hence, I wonder whether the latter can really be calleda general sampling theorem. I think that proving (5) on the basis of a theorem as in 1, 3],requires two generalizations of the latter.(1) The entire function G(t) should be deprived of the requirement of having parity orbeing deduced from an entire function possessing parity by way of a linear transformation, asit is for instance the case with (4) when b c:b-)(G()))b=c sinrF(b-))F(la b-,k)a zr a(2) The theory should be extended to other types of boundary-value problems, not involv-ing a differential equation but for instance a difference equation. This can be illustrated bymeans of the problem proposed in 92-1". (x) is known not to satisfy a differential equationof finite order, but ratherr(x + 1)Hence, 1/F (x) is an entire function ofAy(x)=xy(x) y(x) O Vx 6 - - \{0}x--0where Ay(x) := y(x + 1) y(x). The zeros of y(x) are given by Z\N. Similarly, (4) is nota solution of a differential equation, but satisfiesG( + a) G(,k).) +a-cin contrast, (4) with b c is a solution of the regular boundary-value problem2(G"()))b=ca2(G()))b=c, G(b) O.In my opinion, no solution of this kind to 92-1" can be found.REFERENCE[1] A, ZAYED, G. HINSEN, AND P, BUTZER, SIAM J. Applied Math., 50 (1990), pp. 893-909.On Alternating Multiple SumsProblem 92-11", by MALTE HENKEL (Universit6 de Gen6ve, Switzerland) and R. A. WESTON(University of Durham, Durham, UK).Consider the functions(1) S(v) (--1)qx+qYq -1 sin(2vq),qx=l qy:O(2) C(v) Z (--1)qx+qYq-1COS(2Vq),qx=l qy=O498 PROBLEMS AND SOLUTIONS/where q /q2 + q2. Numerical studies have led to the conjectures(3) S(v)=-v/2 if <v <.v/,C(v)=O ifv=4-5/4.Prove or disprove (3). Find the general expressions for C(v) and S(v) for v arbitrary.These sums arose in finite-size scaling studies of the three-dimensional spherical model.Solution by J. BOERSMA and R J. DE DOELDER (Eindhoven University of Technology, Eind-hoven, the Netherlands).Since S(v) and C(v) are odd and even functions of v, respectively, we may restrict theanalysis to the case v > 0. Consider the function f(v; x, y), with v > 0, defined byexp [2ivv/x2 + y2]f(v; x, y) dX2 + y2(x, y) 7 (0, 0),2iv, (x, y) (0, 0).Its two-dimensional Fourier transform is found to beeiex+iYdx dy.T’{f(v; x, y)]v/X2 + y2/ 2rr[i(4v2 2 r/2)-1/2 (2 + r2)-1/2], 2 + t?2 < 4v2,/ 27r[(2 nt r/2 4v2)-1/2 (2 nt r/2)-1/2], 2 -t- r/2 > 4v2,by means of some Hankel transforms from [1, form. 8.2(1), (32), (42)]. Next, by inversion of0r{f} we obtainf(v; x, y) 2+2<4v] ] [(2+ 2__ 4V2)-1/2- (2+ 2)-l/2]e-ix-iyodd"+:+,>4This representation is used in the double seriesT(v)-- (-1)m+"f(v;m,n)-- Z (-1)m+nm oo n cx m n cxz /m2 + n2yieldingT(v) (-1)m+n [i(4v2 2 2)-/2m=-n=- 2+02<402_(2 + 02)-l/e]e-im-inoddoexp[2iv/m2 q--n2] --1+ff [(2 q- r}2 4V2)-1/22+r/2>4V2(2 _ff r12)-l/2]e-im-inoddo !./PROBLEMS AND SOLUTIONS 499After a formal interchange of the summation and integrations we employ Poisson’s summationformula to rewriteZ (--1)me-im 2rr Z( --krr),m=- k oddZ (--1)ne-ino 2rr Z 3(rl err).x: oddAs a result we findk,g odd 2+02<4v2q-r}2 >4V28 [i(4v2/rr2 k2 2)-1/2 (k2 if_ e2)-1/2]k,e odd >0h2+e <4v2/r+8 [(k2 _}_ 2 4vZ/rr2)-l/2 (k2 %_ 2)-1/2].k,e odd >0+e2 >4v2/rOn the other hand, the double series T (v) can be expressed asc (2v/m 2 n:T(v)=U(v;O,O)+4ZZ(-1)m+ncs + )-l+isin(2v/m2+n2)m=l n=0 //m2 + n2c c (__ 1)m+n2iV--4mZn=0 /m2 q- n2 q- 4C(v) q- 4is(v).The latter double series is evaluated by means of a result of Glasser and Zucker [2], viz.cxz cx (__ 1)m+n x (_l)m (__ 1)m+m=l /mg + n2 rn /m2m=l m=l n=l-log2 + lim[(1 2-2s)(2s) (1 2-)fl(s)(s)]s-log 2 + log 2 + (-1 + 21/2)fl Q.Finally, by identifying the real parts and the imaginary parts of the two results for T (v), wefind the general expressionss(v)2C(v) -2--v+2k,e odd <0+2 >4v ]rt2y (k2 _}_ 2)-1/2 _+_ 2k,e odd >0k2+e <4v2/rr(4/)2/rr2 k2 2)-1/2,odd >0k2+e2 >4v2/r[(k2 + 2 4v2/rr2)-l/2-(k2 q- e2)-l/2] q.- Q,valid for v > 0.500 PROBLEMS AND SOLUTIONSLet 0 < v < zr/V, so that 4v2/Tr2 < 2; thenS(v) -v/2, C(v) 2 Z [(k2 nt- 2 4V2/yr2)-l/2 (k2 + 2)-1/2] Or Q.k,(odd>0Thus the conjecture for S(v) is correct. By expansion of (k2 -+- 2 4v2/Tr)-l/2 and by useof another result of Glasser [3], C(v) can be expressed in terms of a single series, viz.1/2 lP ( q- )(1--2-n-1/2)fl /7 q- q-C(v)= " nn=O n! Yr 2 J 132 < 7t’2/2,where (s) is the well-known Riemann zeta function andfl(s) Z(-1)e(2g + 1) -s,-0S>0.We disprove the second conjecture by computing, from this expansion that the smallestzero of C (v) is approximately 1.252129830.REFERENCESA. ERDgLYI, W. MAGIUS, E OBERHETTINGER, AND E G. TRICOMI, Tables oflntegral Transforms, Vol. II, McGraw-Hill. New York, 1954.[2] M.L. GLASSER AND I. J. ZUCKER, in Theoretical Chemistry: Advances and Perspectives, Vol. 5, D. Hendersonand H. Eyring, Academic Press, New York, 1980.[3] M.L. GLASSER, The evaluation of lattice sums. I. Analytic procedures, J. Math. Phys. 14 (1973) pp. 409-413.Also solved by SCOTT ALLEN and RAJ PATHRIA (University ofWaterloo), DAVID BORWEIN(University of Western Ontario) and JONATHAN BORWEIN (University of Waterloo), C. C.GROSJEAN (University of Ghent, Belgium), NORBERT ORTNER and PETER WAGNER (Univer-sity of Innsbruck).Editorial Note. Since the solutions for this problem were of uniformly high quality,although lengthy, the editors regret that space limitations preclude publishing all of them.The Borweins decided to submit their solution in the form of a paper, On some trigonometricand exponential lattice sums, to the SIAM Journal on Mathematical Analysis. Grosjeanpresented accurate, detailed, numerical procedures, which included two finite, high precision,r ’. Ortner and Wagner gave a carefulapproximations to compute C(v) in < v <discussion of the distributional setting and convergence questions. Generalizations to Besselfunctions were mentioned by Grosjean, as well as Allen and Pathria who also considered sumsin arbitrary dimensions and refer to their paper, Analytical evaluation of a class of phase-modulated lattice sums, J. Math. Phys. (in press). Grosjean proved the surprising result, forall real x"Z (-1)m+nJo(xV/m2-l-rt2) =0"Circular Binary SequencesProblem 92-12, by MARK STAMP (Worcester Polytechnic Insitute).The binomial coefficient () can be interpreted as the number of distinct binary sequencesof length n with exactly k ones. Let { be the number of binary sequences of length n with k

Solution to Problem 92-11* : On alternating multiple sums

Abstract

Similar works

Full text

Available Versions

Pure OAI Repository

NARCIS

Pure OAI Repository