A GENERALIZED FORMULA OF HARDY

A GENERALIZED FORMULA OF HARDY

GEOFFREYB. CAMPBELL Instituteof Advanced Studies Schoolof Mathematical Sciences TheAustralianNationalUniversity GPO Box4,Canberra, Australia 2601

(ReceivedNovember 16,

1992)

ABSTRACT. We give new formulae applicable to the theory of partitions. Recent work suggests they also relate to quasi-crystal structure and self-similarity. Other recent work has givencontinued fractionsfor the type offunctionsherein. Hardy originally gavesuchformulaeas oursin earlyworkongap power series which led to his and Littlewood’s "High Indices" Theorem.

Over a decade ago, Mahler and then others proved ^results on irrationality of decimal fractions applicabletotypes offunctionsweconsider.

KEY WORDS AND PHRASES. Combinatorial identities,

Farey

sequences; Analytictheory of partitions, Combinatorialinequalities,Fractals,Partitionsof integers.

1991 AMS SUBJECTCLASSIFICATION CODES. 05A19, 11B57, 11P82, 05A20,^28A80.

1. INTRODUCTION.

Hardyin hisclassicalbookonRamanujan

([12],

p

39),

gives theformula,essentially

(for Rex>0),

o 2kit

-x

log2

-Zr (1.1)

with

’

^{over all non-zero}integers k. The oscillatory part on theright sideeluded Ramanujan, perhapsbecause theEuler Maclaurinformula

([1],

p806,

23.1.30)

yields

1/(x

log

2)

exactly. This assumption led Ramanujanastrayin his theoryof distribution of primenumbers.

The series

,exp(2kx)

in various forms has been considered by ^numerous authors with varying perspectives during the past century. Hardy considered it

hilst

examining gappower series in thecontext of the so called converse

of

Abel’s Theorem. He gave formulae such asin

[9]: If

^{aisan integer}greater than 1, and

Re

x

>

0,

The

’

summed part hereoscillatesbetween finitelimits, ^or "wobbles." These wobblesare self-similarand clearly thefunction inquestionhere satisfiesthefunctionalequation

F(x) F(ax). (1.3)

As mentioned recently in Ninham

[25],

^{and also in his work with Frankel,} Glasser and Highes

[8],

^these wobbling functions scale at every level, ^{but are} not fractal in the sense of Mandelbrot, beingeverywheredifferentiable. Bothtypes offunctions arecontinuous, althoughit appears that the functions we consider here may have physical applications to at least quasi- crystal spectra

(see [25]),

and furthermore, since they arise from Fourier transforms ofperiodic deltafunctionsthey may beofgreatfundamental significancein other ways.

G.B. CAMPBELL

Mahler IO] consideredcases of the function

f(z)-f(-l+i)- e-" (0< < 1)

(1.4)

k--1

whereit happensexceptionally that

limt

+0

f(z)

exists,and tends toa definite finitevalue.

Central to the considerations ofHardy and Littlewood

[8,

11, 13,

16]

^was the idea that as z 0 in

(1.4)

^{the series does not} tend to alimit consistent with a simple converse of Abel’s Theorem

(see

also P61ya

[34]). In [20],

Mahler showed that if

Ig(1/2t)log

(1.5)

with

[z]

^{asusual the}greatest integerⁱⁿ z,and both lira log2

,-,+0

log(1/tj ^f(e-’+

^:=

and

tim 1

N-I

^{e2’’ 4,,}

N-

Y

_n--0 ^g.()

^(1.7)

then,

for 0

<

p

<_

q-1,

(p, q)

1,with qodd,

91(27r

^p/q)

92(2r

p/q)

! _exp(2,r

_ip/q)

n=l

(1.8)

Here r

o(q),

the Euler totient function of q. The sumof

(1.8)

is a Gaussianperiod from the theoryofcyclotomy

(see

^Kummer

[15]).

Mahler

[20]

^showedthat if g

>

2is afixedinteger, the decimal fractiongiven by

0.(1)(9)(92)(z)

where

(g’)

is to mean the number

g’

written in decimal form, ^is irrational. His methods are often simply apphed tothegapseries ofourpresentnote. Several authors havesinceworkedon similarproblems

(see [4,

5, 6, 24,26 to29, 34, 35,

38]).

In recent yearsinterest has arisen in the continued fractionforms of series such as

f(z)

of

(1.4).

^{For anaccount}of such worksee

[4,

23,30, 32, 33,36,

37].

2. SOME NEW RESULTS.

As anundergraduatestudent in 1979 the author gave thefollowingformula

(see [7]),

which hasHardy’sformula

(1.2)

^{asthecasem 0.}

THEOREM 2.1. For non-negativeintegersm, Rez 0, Rea > 1,

--aka

)71 (2.1.1)

’L (m+l

(m +

^)(loga)"+ _d=0 ^j

) FI’"+I-J)(1)(

^lg

xl)

^(2.1.2)

Bm+l (2.1.3)

m+l

(2.1.4)

whereF⁽⁾ is the

3-th

derivative of the gammafinction, B,, the m-th Bernoulli number,

X,’

^is

thesum overnonzerointegersk, and k

....

at k rn O.

Although

[7]

^{was accepted} for publication it never actually appeared in print. There are severalfeaturesof this theorem worth notingat this point:-

(i)

^The derivativesat unity of the gammafunctionasin

(2.1.2)

^areeasily found from the expansion valid for

Iz[

<1,

(see [1,

pp259,6.3.14;and

[3])

r’( + z)

r(1 + z)

^-3’

+ ((2)z ((3)z + ((4)z (2.2)

171.2

¹⁽³⁾

sothat

P’(1)

--),,

P(=)(1) ,,/2 + (1)

-7

2"/r2 ^2((3),

^andsoon.

that

(ii)

The relationship of theintegralin

(2.1.4)

^tothe gamma functionisobviatedby reahsing

log ^dt

-"’" r

²⁾

x x re* ^{x ia} log ^{1 2kit}

x

loga ^(2.3)

j=O

Indeed,theversionofTheorem 2.1 given in

[7]

^{has thestunof}

(2.3)

inplaceofthe integral. Taking

(2.3)

into account,thesummation

(2.1.4)

isseento convergebilaterally, owing^to the fact

[1,

pp 257,

6.1.45]

that for fixedm and1

+ ^{< Ret} < Rea(*

>

0),

( ^2kir (

^{1 2kr}

)

^2krt

^-’

P^(’’) 1

+

log/] ^exp

" ^{og-i E}

³⁼⁰

^A(m,

^j)

^{ot (2.4)}

where

(7) ^r( l+J)(

¹

A(m,

j)

(2)

r()

(iii)

Hardy’smethodofproving

(1.2)

^{reliedonasimple}Mellin inversionapplied tothe left side series. Sucha straightforward application designedto arrive at Theorem 2.1would involve the rather non-trivial evaluation of residuesleading to

1

/

^-+’

{’a ^{} fi ( )}

2i

F(-u)x" S) ^a

^j!

(1 )+

^k=l

(-)" -

+

^{the addition of}

(2.1.2), (2.1.3)

^and

(2.1.4),

in which

S}

are Stirling numbersof the second kind.

(iv)

^Itcanbeseenfrom theabove,that

(2.1.4)

^isasumofm

+

oscillating functions,each multiplied by apower oflogx. Eachoscillating function is of order o(1) and oscillates between finite limits. In Hardy’s original paper

[9]

estimates weregiven for these limitsin terms ofa. In Mahler

[201

^anelementarymethodforobtaining such estimateswasalso given.

However,

ifa 2 in

(1.2)

it is easy to usea small calculator to establish that the oscillatory term onlyenters in afterabout thefourthdecimalplace,sofor a 2,

o(0.001)

^seemsaconservative estimate forthe oscillatory function under

3. PROOFS OF THEOREM 2.1.

Thefirstproofof Theorem2.1 appearedin Campbell

[7].

It dependedon Lemma3.1 below, and comparing partial derivatives with respect to the different variables, ^then comparison to obtain a constant ofintegration

(2.1.3).

^{This was somewhatofa} departurefrom the approach taken by Hardy, since the method in

[7]

^required

distinct toaas anintegergreaterthan unity.

(1.1)

^asit appears in Hardy’sbookonRamanujan

[12]

actually has a misprint indicating the author ^was not thinking of a continuous variable a.

The landmarkpaperof Hardy and Littlewood

[13]

^contains theoremsofsufficient generality to justifyall of the differentiations oftheseriesof kind

(2.1.1)

^astheyoccurin

[7]

^{and in thepresent}

note.

(1.1)

^{is a caseof the}following.

Lemma

8.1. I/’Rez

>

0, Rea

>

1,Ren

>

0,

}=-

a

k’e

z"

loga r(n)- F

n

+ iggt

^{z o’,.}

^(3.1)

PROOF. This comeseasily fromboth straightforward application ofthe r ue

theorem,

and by summingontheMellin inversionformula,sotheintegral

2ri

r(-=)*" (x a==)-I

^d=

is given the twoexpressions

and

(3.2)

E ^{a’’ e-a (3.3)}

k=O

a_kn

.-a-z ₊

R.H.S.

of (3.1)

k--1

PROOF OF THEOREM 2.1. Weconsider the behaviour of the termsin the bilateral seriesonthe left side of

(3.1)

^{asn 0.}

In

the directionwith positivek terms theseries clearly converges, howeverfor negative k terms theseries

E ^{e-- (3.4)}

diverges, each term approaching unity successively. We maycompensateforthisbysubtracting the function

a_kn ^a 1 a k--1

fromboth sides of

(3.1)

^sothat

(3.4)

^becomestheconvergent

E ^{(e-- 1)}

k--1

whenn 0, and^weareleft withHardy’sformula

(1.2),

^aftercalculating^thecorrespondinglimit forthe rightsideofthecompensatedversionof

(3.1). Next,

^{weseethat if}

(3.1)

be written inthe form

+ ^a- (e "-1)

k--0 k=l

__r() x

kr ^{+ ig-g/ (3.)}

z" loga 1 a

"

^loga

wecanformy expandeach side into power series inn and equate coefficients to

ve

at the theorem. This is justifiedsince

(3.5)

^{holds foranyn}

>

0 intheneighbourhoodofn 0.

SKETCH OF

THE EARLIER UNPUBLISHED

PROOF.Whilsttheproofin

Camp-

bell

[7]

^is longer andmore

laboured,

it has themerit of showingan interplay between theinde- pendent variablesz,a, andn. Itstartswith ourLemma3.1 andusesthe fact that

(log ) ak

k=-o

k"ta

ke -az

}

-0

_o _{+ ),.(} _,

(:]X

k=0 k=l

Integrationwithrespecttozgives themajor terms ofTheorem2.1,withthe constantterm

(2.1.3)

obtainedfrom thefunctional equation

/" (’ =/"1 E

^"=

^{f (’ "1}

clearly satisfied bythe series

(2.1.1).

^{Ofcourse the} crucial part ofall this is Lemma 3.1, and neitherof the twoproofscould standwithout it.

4. CONNECTIONS

WITH

PARTITION THEORY.

Theorem 2.1 and indeed Lemma3.1 itself may be used to derive identities for generating functionsof partitions ofvarious sortsintoa-th powers. To begin justwith

(1.2),

^{wecan easily}

show that

THEOREM 4.1.

H

Rea

>

1, andRe

>

0 with z e

-t,

II II ^l+z"-

=o =1 2

(4.1)

(o)o

^{,o. 1 ,o}

{

^-1

^{:2ki’) (} 2ki’)(-,,,,)

¹

xexp

1-ogaE’F\l-ga, l/lg

a 1-2-a-

(log)

Wenotethat as acorollary,ifa 2wehavethe elementaryresult

orsimply

+

^a:’2"

=0 2

Io

(1

^q_

^a:2-, )

^1-x

II

2 log

k=l

(4.2)

IIOOF

^{OFTHEOREM4.1.}

^(4.1)may

^be

^known,

^{sinceit issoeasilyaccessibleby}

(1.2).

Theproofinvolvesapplication of the operation

Z ^{(--I)a+I f(a, zj)j-1}

3--1

to

(1.2),

lettingeither side of

(1.2)

be

f(a, ).

This, for the left side gives thelogarithmofthe left side of

(4.1).

^Forthe right side thesame procedure leads to the required result ifwe know that

Z ^{(-1)+a log_k}

k 2

^{! (log2)}

^-’rlog2

^(4.3)

k=l

(-1)

^+ak-(l+i")

(1 + in)(1

²

-’’)

k=l

forrealn

:/:

^0.

(4.3)

^isgiveninHardy’snoteonVacca’sseries

[10].

^Alittlesimplification thereafter gives the theorem.

JustasTheorem 2.1generalizes Hardy’s^formula

(1.2),

^{thefollowingformula,}which appears to be new,generalizes Theorem4.1.

THEOREM 4.2. /fm,a,z, are asinTheorem 2.1,

)

^(-k)

H

¹

⁺

^e-a" 2

^{exp(G1 (*)+} Z’ ^{G.()) (4.4)}

k=O k=l

where

’

^{isovernon-zero}-1 ^{integersk,and k 1 at k m O,}

Gl(m)

_m

+

¹

^B,+I

^log2

(m;

^m,m,.

ma)r(’)(1)r(’)(1)

log

G2(z)

¹

"+

(_l)r(log(t/rz)),

r(l+2kiw/log^a) _e

(4.5) (4.6)

where thesumin

(4.5)

^{isover m}

m +

^m2

+

ma ⁱⁿnon-negative integers and

m; ml,m2,

ms) m

m!m2!ma!

and also

r/(,)(1 Z(_I),+ ^_I

^log-

Aswith Theorem2.!,Theorem4.2hasseveral notable features:-

(4.7)

(i)

The function

lI ⁺ - ^{P,,() -" (4.8)}

k=0 k=0

clearly generates the number ofpartitionsintono morethank termsof kind a

’,

accordingto themethodsin Andrews

[2].

(ii)

^{Estimates asz} 0 for the function

H

2

(4.9)

k=l

areeasily obtained,together withappropriate inequalities, enablingus,ifwechoose,tofocuson

/4./.

(iii) ^Some

modification of Mahler’s method

[20]

^{may be applicable to Theorem}

4.2,

^thus giving elementaryestimatesfor thefunctionsconcerned.

(iv)

^If

r/(n) E(-1)}+:

_k

k=l

then for theterminology of in

(4.5), a()

(m;ml,m2,rrt)rt

^2kin

r(,,,)

¹

+

^{log- z}

R.H.S. of

(4.6)

The clue to this istheidentity

(1

-t-2-t-

3) E ^(m;

^{rrtl, m2,}

^{m3)? rt, rta (4.10)}

ofdegreem, andthat

(v)

^Itis clear from Theorem 4.2 that

ax (x)

^{is apolynomialin}log

Ft"(1)

^and

/"0(1)

^canbe evaluatedasfinite combinationsofRiemannzetafunctions, andthe constants

(7 70) (see [3])

+1

(.11)

k=l

whichoccurnaturallyintheexpansion

[1,

pp807,

23.2.5],

(.)=

¹

+

O’

+ (-1) ^(s

_!

¹⁾

⁷

^(4.12)

k=l

Hardy

[10]

applied

(4.11);

and recentlyin

[8]

^{it is}explained that

(4.12)

is necessaryfor obtaining precise results for thermodynamicfunctions near the so-called "Hagedorn

temperature"

which occursinparticle physics

(hadron physics),

related to prime andintegergases.

(vi) As

acuriosity, ifz is, say,equalto log 10, the decimal fraction obtained from

(4.8)

^is

trivially calculated, and the corresponding

(.9)

^{typefunctioncanbe}found easily usingasmall calculator. The oscillating functions arenot so easily calculated, but themajor terms of are,as arethe orderofeach of thern

+

¹oscillatingfunctions in

G2(z).

PROOF OF THEOREM 4.2. Thisproof^followsananalogous^linetoourproofofTheo-

rem2.1. Lemma3.1 easily transforms

(under

^thesame

conditions)

^to

( =) ( )-(-/:)}

1

’r

ⁿ

⁺ 10g-a

^{( n}

+ + l)ga

^z

log

^a

whicheasily becomes

(after

dividing B.S. bythecasewith 2zfor

z)

k=0 k=l

r(n)r/(n + 1)

^{log2 1}

E’F

ⁿ

+ l)g

exp z’ loga a 1

log

^a

when thecompensatingfactor

gl (;)= ^{() "-’}

^exp

^{{ }_a}

^log12

is multipedtoech side. Theorem4.2 comeseasily fromequating coecients ninheloghmof ech side of 4.14.

.

CLOSING REMARKS.

The results ofthis paper evidently have consequences in number theory

[12],

^{and also in}

physics ^as shown from

[8]

and

[25].

^Related topics have held the interest of mathematicians such asMahler forover50 years.

(See [17

to

22],

especially

[22],

where Mahler reminisces about the subject.) The approaches taken by Hardy, Littlewood andlater P61ya suggest that series of the kinds representedin this paper have been consideredimportant forvarious reasons.

In [25]

the self-similarity property of theoscillating "wobble" functionwasappliedin aningenious fashion toPenrose tilings and quasi-crystalline structureshavingfivefold symmetries connected with equi-angular spirals andFibonacci sequences. Usingmethods ofour current work,results canbe obtainedfor seriesinvolvingFibonaccinumberssuchas

k=0 k=l

which satisfiesthe functionalequation

ACKNOWLEDGEMENTS.

I-i-

f(az) + f(a )=/(m)

The author is indebted to Professors Baxter and Ninham and also to Dr M.F. Newman atthe Australian National University, for their hospitality, and especially to the latter and the MathematicsResearch Section office fortypingthis paper. ThanksarealsoduetoProfessorG.E.

Andrews.

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of Values of Certain

Gap

Series, Kexue Tongbao, 3_0_, 293-297.

Special Issue on

Intelligent Computational Methods for Financial Engineering

Call for Papers

As a multidisciplinary field, financial engineering is becom- ing increasingly important in today’s economic and financial world, especially in areas such as portfolio management, as- set valuation and prediction, fraud detection, and credit risk management. For example, in a credit risk context, the re- cently approved Basel II guidelines advise financial institu- tions to build comprehensible credit risk models in order to optimize their capital allocation policy. Computational methods are being intensively studied and applied to im- prove the quality of the financial decisions that need to be made. Until now, computational methods and models are central to the analysis of economic and financial decisions.

However, more and more researchers have found that the financial environment is not ruled by mathematical distribu- tions or statistical models. In such situations, some attempts have also been made to develop financial engineering mod- els using intelligent computing approaches. For example, an artificial neural network (ANN) is a nonparametric estima- tion technique which does not make any distributional as- sumptions regarding the underlying asset. Instead, ANN ap- proach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting pa- rameters to obtain the desired results. The main aim of this special issue is not to merely illustrate the superior perfor- mance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making. In this sense, the submissions should es- pecially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelli- gent, easy-to-use, and/or comprehensible computational sys- tems (e.g., decision support systems, agent-based system, and web-based systems)

This special issue will include (but not be limited to) the following topics:

• Computational methods: artificial intelligence, neu- ral networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learn- ing, multiagent learning

• Application fields: asset valuation and prediction, as- set allocation and portfolio selection, bankruptcy pre- diction, fraud detection, credit risk management

• Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, imple- mentation

Authors should follow the Journal of Applied Mathemat- ics and Decision Sciences manuscript format described at the journal site http://www.hindawi.com/journals/jamds/.

Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Track- ing System athttp://mts.hindawi.com/, according to the fol- lowing timetable:

Manuscript Due December 1, 2008 First Round of Reviews March 1, 2009 Publication Date June 1, 2009

Guest Editors

Lean Yu,Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China;

Department of Management Sciences, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong;

[email protected]

Shouyang Wang,Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China; [email protected]

K. K. Lai,Department of Management Sciences, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong; [email protected]

Hindawi Publishing Corporation http://www.hindawi.com