Solution to Problem 86-5*: Conjectured trigonometric identities

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Solution to Problem 86-5*: Conjectured trigonometric identities

Citation for published version (APA):

Lossers, O. P. (1987). Solution to Problem 86-5*: Conjectured trigonometric identities. SIAM Review, 29(1), 132-135. https://doi.org/10.1137/1029014

DOI:

10.1137/1029014

Document status and date: Published: 01/01/1987

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Also solved by theproposer who uses Whipple’s quadratic

_3F2

transformation as a

generatingfunction andwhodrawsattention tothecase c v aswellasc u.

Conjectured Trigonometrical Identities

Problem 86-5*, by M. HEyEL _{(University of}_{Bonn, W.} _Germany) and J. LACKI (University ofGeneva, Switzerland).

In computing low-temperature series for a Z,-symmetric Hamiltonian [1] (in

statisticalmechanics),weapparentlydiscovered some newtrigonometrical identities. These were verified up to n 100 on a computer. Prove or disprove the following conjecturedidentities for general n.

Inwhatfollows, we usethe notation,

and (1) (2) (3) (4) (6) (7)

(_j)

{

ifx>O, w=exp 0(x)= ifx__<0, n-I n-I

Y

standsfor

Y

stands for

kr= r# kr=1,kr#kl

E E

E

[(

k,)(

wk-k’)(

k3-k2)(

w-k3)l-’

(n

l)(n 2

41

2#13#2 180

}

+

0(k2-

kl)

sin2-- sin2n 45

[

ksink

(n2 1)(22n4 _13n2 -I-15) 945 REFERENCE

M. HENKELANDJ. LACKI,Bonnpreprint, submittedtoJ.Phys.A.

Editorialnote. The proposersalsohadelevenmoresimilarconjecturedidentities with triple, quadrupleand quintuple summations.

Composite solution by S. W. GRAHAM and O. RUEHR (Michigan Technological University), O. P. LOSSERS (Eindhoven University of Technology, Eindhoven, theNetherlands),andM.RENARDY(University ofWisconsin).

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All oftheconjecturedidentities are true. Identities and6follow fromsymmetry arguments alone. The remaining identities reduceto evaluation of thesums

The requiredsums,

n--I _71-k

Sp=

Y

csc2p _p= _{1,2, 3.} k=l n n2 (n2 1)(n2

₊

11)

S1-

$2"-" 3 45 (n2 _1)(2n4

₊

_23n2

₊

₁₉₁₎ 945

may be found in [1] or may be computed using any one of several methods. For

example,

Sp

maybederivedby considering thecontourintegral

cotz

Csc2P

H

whereCconsistsof thevertical lineRez -6andRez mr 6, 0

<

6

<

r,traversed

in opposite directions. Clearly, the integralvanishes, sobyresidue calculus one has z

Sp

=-Res cot z csc

2p-z=0 H

Wenow turn tothe conjecturedidentities.Let

T,

T_,

...,

T7

denote the givensums.

To provethefirstidentity, notethat thesummand changes signif we reversethe

roles of

k

and

k2.

Thus the terms cancel in pairs and the sum vanishes. The same argument proves the sixthidentityaswell.

To prove the second identity, use the trigonometric formula cscxcscy=

csc(x+y)[cot x

+

coty]to writethe summandas

rk,

rk3

r(k2-

kl)

71(k2- k3)

CSC CSC CSC CSC 16 n n n n

011 7rk2

(

7rkl

=7csc2 cot

+

cot

Since

Y--I

cot

zrk/n

O,we have

r(k2-n

kl))(cot

rk--A3

₊

COtn

r(k2n-

k3)).

and

Y

cot

rk=

y

cot 1#2 n 1#2

7r(k2

k)

_-cot

7rk2

n n Thus

7rk3

2

cot=

₂

cot 3#2 n 3#2

7r(k2- k3)

7rk2

-cot

csc2

rk

cot2

rk

T2

k=l

n n (n2 1)(n2 4) 180

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Toprovethethirdidentity,write

7rkl

7rk2

r3

2 2

W(k,,

k:) csd

csd

2 n n where

1{

)] {

if

kiCk2

W(kl

kz)=-+

O(kz k,)

+

O(k,

kz

ifk,

k.

Itfollows that 3 (n2 _1)(4n2 1)

T3

=-

$2

+-

$2 45 Identity4may beestablishedsimilarly.Write

2

"Xkl

-xk2

rk3

T4

Y

2

W(kl, k2,

k3)

c$c

csc2csc

2 123 n n n where W(kl, k2,k3)

_g

Y

+

O(k2- kl)

+

O(k3

k2)’

thesumextending over allpermutationsofk, k,

k.

Itisreadily found that

We

thusfind

W(kl, k2,

k3)=

ifkl,k2,

k3

are distinct,

ifexactlytwoofk, k2,

k3

areequal, if

_kl

_k2

_k3.

5

(//2_

1)(22n4-_13n2

₊

15)

T4

-

S

+

-

S1S

+

-

$3 945

Finally,wemake useof the fact thatsin

rk/n

sinr(n

k)/n

toreduceidentities 5 and 7 to known sums. In particular, toevaluate

Ts,

writethe sum asecond time

withk replaced by n kand addthe two sums.Wethusfind2

_T5

nSl,so

n(n2 _l)

6

Apply the same technique to

TT,

replacing

kl

by n-

k

and

k

by n-

kz.

Thus

2T7

=2nT3, so n(n 1)(4n2 1)

TT-45 REFERENCE

[1] M. E. FISHER,Solution toProblem 69-14",Sums_ofInverse PowersofCosines, byL. A. GARDINER, JR.,this Review, 13 1971),pp. 116-119.

Alsosolvedby C. GEORGHIOU (University ofPatras, Greece), W. B. JORDAN (Scotia, New York), R. RHER6 (Institut ffir Reine und Angewandte Mathematik der

RWTH,

Aachen, W. Germany) andW. VAN ASSCHE (Katholieke

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by D. FOULSER (Saxpy Computer Corporation, Sunnydale, California) and A.A. JAGZRS(Technische Hogeschool

Twente,

Enschede, theNetherlands). Other partial solutions were provided by J. W. S. CASSLS (Cambridge University, England), S. C. DUTTAROY (Indian Institute of Technology,New Delhi, India) andU. EVERLING(Bonn, W.Germany).

Editorial note. A variety of methods was proposed for evaluating

_S.

Different starting points included the Mittag-Lefller expansion of

cscZz

(A. A. Jagers), Chebyshev polynomials(W. van Assche), eigenvaluesof thematrix

2 -1 0

-1 2

0 -1

-1

(D. Foulser) and the formula

71-2

CSC2"/l’X lpt(1 x)

+

ff’(x), 0

<

x

<

1, where

b

denotesthedigamma function(R. Richberg).

LateSolution

Problem 85-24:by S. LJ. DAMJANOVI (TANJUGTelecommunicate

Center,

Belgrade, Yugoslavia).