Lehmer_binom

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Interesting Series Involving the Central Binomial CoefficientAuthor(s): D. H. LehmerSource: The American Mathematical Monthly, Vol. 92, No. 7 (Aug. - Sep., 1985), pp. 449-457Published by: Mathematical Association of AmericaStable URL: http://www.jstor.org/stable/2322496Accessed: 26/09/2010 05:26

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INTERESTING SERIES INVOLVINGTHE CENTRALBINOMIALCOEFFICIENT

D. H. LEHMERDepartment f Mathematics, niversity f California, erkeley, A 94720

The adjective interesting" s used here n a technical enseexplained s follows. series scalled nteresting n casethere s a simple xplicit ormula or ts nth erm nd at the ame time tssum can be expressedn terms f known onstants. hus

1+ - -++*+-+* = 2,2 4 2

1-2+ 34+ *-+( ) -+* =log2,2 34 n1 1 1 +1 .+i

16 81 256 n4 90are familiar xamples f nteresting eries.

The seriesweplan to discuss re of two ypes:

I O an 2n) and II. E (2n

where he a,1 are very imple unctions f n. We beginwith eries f Type .By the binomial heorem ehave

(1) 2()n 1

This converges f IxI< 1/4. If we put x = 1/8, for xample, eget he nteresting eries{2nA

(2) 1?1?4 1? 2 + A __-n + .4 32 128If x = 1/10, we get

(3) 1+.2+.06+.02+.007++

+(n

_Forx= -1/81 3 5 (-)(n)

(4) 1- + 32- 12 + +832 128 8

Averaging 2) and 4) gives s

E ( 2n 3V2+ V6n=o

64n6

Another tep along his ath s to bring n complex ariables. hususing x instead f x leads to

D. H. Lehmer: hispapermarks he ixtieth nniversaryf the publication f the uthor's irst aper n thisMONTHLY.Theauthor opes o submit anuscriptsf other apers rom ime otime s occasion rises.

449


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450 D. H. LEHMER [August-September

8n4 E4J=3+ +2 +

n= 8 n 3 +or

(8n\, 44 (15 V5 ? 6 V2 +dV)n0 60

In general ne can obtain value for he um(2( an+b))

n= an+bn can+b

by setting x = er/c (v = 0,1,2,... ,a - 1), where E = exp(2q7i/a), in (1) and forming heappropriate inear ombination f these esults, process nown s multisection seeRiordan 4]).Another ath out of 1) is to apply perators. f we ntegrate 1) from to x we obtain

E (2n)ftndt0n=O 0 (1 - 4t /

Dividing oth ides by x weget

(5) E (~2n) x =1( -

The coefficients nof the eries re ntegers nown s Catalannumbers. heir irst en values re1, 1, 2, 5, 14,42,132,429,1430,4862,...and they ccurfrequentlyn combinatorial nalysis seeGould 2]).By using tirling's ormula orn we see that

Cn g( n + 1)( 8n)

From thiswe see that or = 1/4 the eries n 5) converges lowly ikeX(1/n3/2). hus 5) givesus the nteresting eries

{2nA1 + 1+ (+ n+) + = 2.4 8 64 4n(n +1)

We can treat 1) a little ifferently y transposing ts first erm o the right ide, dividing othsidesby x and then ntegrating. his gives s the mproper ntegral

E(1 n )tn dt jx( t(l - 4t)1/2 t)

That s,

(6) , (2)n = 21og( 1

Putting = weget the eries

1 3 5 35 nlog4.2 16 48 512 n4f


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1985] INTERESTING SERIES INVOLVING THE CENTRAL BINOMIAL COEFFICIENT 451

If we alternate he igns f this eries,weget

1 3 5 35 (-))?(n) I V?I- ---+--- ~ + + n + lo2 16' 48 512 n4f =l g2]Adding r subtracting hese wo erieswe can derive nteresting eries or he wo ums

d n2n4n or n )n4nn odd n even

n>0

We can integrate 6) and obtain

(7) x x, n ) n= 2xlog +x +4(log4-1)nY2 n(n +1 x + 2 2xlg41-If we substitute = 1/4, we obtain he nteresting eries

1 1 5 7 (n)-+--~ +~ ++**++**=log4 -1.4 16 192 512 n(n + 1)4

Any number f examples f this ort an be obtained y repeated ntegrations f 1).Another perator f thiskind s differentiation. et 6 be the operator

= xddx

If we apply0 to 1),we get00

(8) ~ n= (2,~l) n (=3 /n=1

n l~4x

If we set x -, we get

1 3 15 35 + n_ + 2-+ +125+ ~~+=4 16 125 512 8 ~ 2Operating gain by , weobtain

n2( 2n xn = 2(1 - 4x)-1/2 2x(2x + 1)n=1 nfl (1 -x5/

Setting = we find

1 3 45 35 1575 n2(2n)-+-+~ ~++ +n+4 8 128 128 8192 8 4

If in (1)we replacex by x2 and then ntegrate oth ides, we get

(o 2n)

n 2n + x2 = x (arcsin2x)n O

and for x = 400 (2n)n

n=O (2n + 1)16's 3We turn now to series of type I which nvolve he central binomial oefficient n the

denominator. hese series are more mysterious nd less well understood. n an informal


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discussion f Apery's roof f the rrationality f '(3),A. J. van der Poorten 5] refers o fourseries of type I. The following heorem as quoted and its proof by Z. A. Melzak [3] wasdescribed s "not quite ppropriate". e give n entirely ifferently roof.

THEOREM.f lxi 1,(9) ~~~~~~~~2arcsin - 2x 2

(9) / 2=E (2)x 2 m=i M(2mn

Proof. We use the familiar regory eries00 (_j)m-1t2m

(10) tarctant Em=1 2m-1

and set t X/l1 - X2, so that rctan = arcsin . Then 10)becomes

x arcsin () m-2m2 acm-E (2m -1)(1- _2)m

m- m-1 Z )m-1 j=O

_0xr L_)-(r-)r=1 m=1 (m- ) (r- m) (2m - 1)

It suffices o showthat he oefficient f X2r is half f that n the right ideof 9).That s,

(11) ~~~~~r() v (r - v - 1) (2v + 1) =22rlTo prove hiswe use Wallis' ntegral

(7/2 (sin 2r- 1dO = 2 46 .. (2r - 2)_Josln, da 1 3 5 ..(2r- 1)

as follows. he left-hand ideof 11) can be writtenr( 2r E (1) (r 1) 1 = r(2 r)l E l(r- y-dy

r____ V+1

- r(2rr)|(1 Y2)) dyr~~~~~r

= rK2r)I / sinG)2r dO

by substituting = cos 0. UsingWallis' ntegral e find

(2rr 22 (r -)r(r - 1) = 22r-rk (2r - 1)

Thisproves 11) and hence he heorem s established.If we substitute = 2, weget the result van der Poorten 5],p. 202)


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1985] INTERESTING SERIES INVOLVING THE CENTRAL BINOMIALCOEFFICIENT 453

(12) - 9m=1 m( m )

as our first nteresting eries of Type II. If we wish to alternate he signs n this series, we merelyput x = i/2 in (9). This gives us

m 1 m( 2 m ( 2 )/

If we divide both members f (9) by 2x and then ntegrate, we obtain

(13) 2(arcsinX)2 = L _(2X)2mm=1 M(2)

from which we get, for x -200 1 T2 1

(14) m = 1

where ?(s) is Riemann's zeta function. Also,00 mlogz

( ( + 1 )2- -2~~~~~~~

If we divide both sides of (13) by 2x and integrate, we obtain

E (2 x)2m 4X (arcsin y)2 dm=1 m3( 2m )

This integral is a "higher transcendent". t is closely connected with Spence's transcendent,Clausen's integral, nd the trigamma unction. or x = 2 we obtain

L 1 = 4f1 /(arcsiny)2dm

=-2f x log(2 sin ) dx

=- 3 -72 +3) (3);

where +(x) is the trigamma function. Van der Poorten [6] rejects this evaluation as beingnon-instructive. owever, he does give the nteresting eries

1 1 1 - 1 (-1)~ +iI- - - + _ + -+ ( 1 + *- =2(3)2 48 540 4480 m3(2 m)

as well as Comtet's [1] remarkable00 1 177T4Z

m3240

m1 m(4)


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There re no known nteresting eries f the form

r mkfor k > 4.

If we apply the operator k to both sides of (9), we get an almost unlimited umber finteresting eries.

To begin with Oweobtain

(15) E (2X)2m_ x2 + x arcsinm=( 2rn) 1-X2 (1-x2)3/2

m

From this we obtain, or xample,

1 1 1 1 1 9 + 2'nV32 6 20 70 (2n) 27

If we replace x by ix in 15) and set x = a/b, we get nother orm f 15)

E ( 1)m1(2a/b)2m ab2[log a+ h ah]m=1 (2m h)m

where h= a2 + b2. For example, f a= 1, b 72, h =/, we get

1 1 1 1 ( 1)n1 1 4J x/S?1- - - + - - - + + ( ( 2) ) + 5_ 25og 26 #20 70 (2n) 5 25 og 2

For a = 23660, b = 23661,we have h = 33461 nd we obtain

00 ~ /23660n

n

( n-1 (23661 23660 23660+ (23661) 57121nEl(1l) (2n) (33461)2 33461 log23661]n

= .811587506....This series s of course onvergent. f one examines he ratio f consecutive erms ne finds hatthe erms ncrease n absolute alueuntil n = 11830when he erms re as large s 117. After hatthey ecrease o zero. t is well that wehave our formula or he um.

If weoperate n (9) by higher owers f 4, the eries we obtain re of the form

(16)00 m -2 (2x )2m ( k > 0)

M=1 {2m2mand the valueof the umdepends, s a function f k, on two equences f polynomials k t) andWk (t) defined ecursivelys follows:

V1(t)=1, w1(t) = O,

(17) Vk+l(t) = {(2k - 2)t + 1}Vk(t) + 2(1 - t)4Vk(t),(18) Wk+,(t) = ((2k - 4)t + 2}WK(t) + 2(1 - t) 4Wk(t) + Vk(t).The first ewpolynomials re

V2= 1, W1= 1,


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1985] INTERESTING SERIESINVOLVING THE CENTRAL BINOMIAL COEFFICIENT 455

V3= 1 + 2t, W3 = 3,V4= 1 + 10t+ 4t2, W4= 7+ 8t,

V5 =1 + 36t + 60t2 + 8t3, W5 = 15 + 70t + 20t2.

The valueof the um 16)can be given xplicitly n terms f Vk and Wk by

k-2(l _ X2)k-1/2 Iarcsinxkk(x)+ l - x2Wk(x2)]

If we operate n this y and collect he oefficients f arcsin x and x 1 - we obtain heformulas 17) and 18). A trigonometric orm f this esult s,with x = sinG,

00 mk-24m (sinG0)2m_ sin2GEm - (2rn) sin k [2GVk(sin2') + sin2OWk(sin?)].

If we replace x by ix weget for he lternating eries(o 1) m-1mk-24m sinhZ)2m sinh2z [2log{sinhz + coshz}

m=2 (2rz) (2cosh2z)k

XVk(-sinh2z) + sinh2zWk( - sinh2z)I.

Thesetwoformulas ith heir woparameters anqd yield widevariety f nteresting eries sexamples.We list only he following. he sums xtend rom to oo.

Zm2 2( m) 2 ( q7Tr + 9)2m 2

(2m) 81

__ m3 = 2 ?375T 114052m

(~1) ml12 4 +E ( m) =125[2a + 15] (a = Z log 2=1.076022352)

(2m) 1251 ]

( - 1)m[ -m3 2

m2(2m3) 8m 625

2mm 7T2(2mn) 2


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==- + 12mA 2mJ(Trn)

m2m(2m)

m2 m 1 32m 77+{2mA 5 {2m 2

m42m~~~~m42 = 113,r + 3552m

=229093376w 719718067.(2m)mJ

In general,?? mk2mM= = aiT b,

1(2m)

where b/a is a close approximation o v.

3 2 q72m2 2m) 9m

m(2m) T3 = v = 3.627598728

3mE 2)=2v+ 3

3m

= lOv+ 18(2mm

m 3~ = 2(43v + 78)2m

E ( J = p73, where = VYlog(2+ 4)= 2.281037989m(2m)

( 1)m-12m= 2 + 3(,2m) 9

mJ

( 1)m-1 2m

(2m) 3


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1985] THE LOGIC OF GRAPH-THEORETIC DUALITY 457

(-1)" m1m22m 1

(2m) 27m

( 1)rnm 2m _1- )m2 = [ p + 15]2rn 81/?]m

(2-I2)tm 3 - 2(2______ - ~~( TV24)

m

(1) m I32m 48 15log2?-+

4m( 2m) 125 16m

2n1 2 - ft3)tm T2 2m2 2m(3)m

References

1. Louis Comtet, AdvancedCombinatorics, reidel, Dordrecht, 974,p. 89.2. H. W. Gould, Bibliography f the Euler-Fuss-Segner-Catalan equence,manuscript ontaining 43 titles.3. Z. A. Melzak,Companion o Concrete Mathematics, ol. 1, Wiley, 973, p. 108.4. John Riordan, Combinatorial dentities,Wiley, 968, pp. 131-135.5. A. J. van der Poorten, A proof hat Euler missed, Math. ntelligencer, ol. 1 (1979) 195-203.6. __, Somewonderful ormulas, ueen's Papers n Pure and Applied Math., 54 (1979) 269-286.

THE LOGICOF GRAPH-THEORETICDUALITY

T. A. McKEEDepartment f Mathematics nd Statistics, Wright tate University, ayton, H 45435

Dualities nd duality rinciples reprizedwherever hey ccur n mathematics. omeoptimistssee them s mechanically oubling henumber f results n a theory. thers also optimists) eethem s halving he number f results, ut packingmore ubstance nto ach.

There are limitations hich eep duality rom eing s powerful n graph heory s it is inmany ther reas,but t s still majorunifying heme. obin Wilson's ntroductory ext 24] sorganized round duality, nd it s also central o many rea of applications. s evidence f thelatter, onsider he mphasis n duality n books uch s Johnson Johnson's raph Theory withEngineering pplications8],Price's perations esearch onographraphs nd Networks 19],andNakanishi's raph heory nd the eynman ntegral 16].

Classicalgraph-theoretic uality enters n the relationship etween ircuits nd cutsets, ithspanning rees lso playing basicrole.Thefirst ection f this aper urveys his uality nd ts

formulation s a syntactical rinciple. key feature f this duality s the role played by a

Terry A. McKee: After eceiving Ph.D. in mathematical ogic from he University f Wisconsin-Madison n1974, I workedwith pplications f ogicto subjects uch as topology nd geometry. have since grown ncreasinglyfond of graph theory nd now nhabit niche whichmay be described s "graph metatheory." y other nterestsinclude ap swimming nd studying hemethodologies videnced n vintage etective iction.

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