Expansions of generalized Euler's constants into the series of polynomials in $\pi^{-2}$ and into the formal enveloping series with rational coefficients only

0.55 MB·English

by  Iaroslav V. Blagouchine

Expansions of generalized Euler’s constants into the series of
polynomials in π 2 and into the formal enveloping series
−
with rational coefficients only
IaroslavV.Blagouchine
∗
UniversityofToulon,France.
6
1
0
2

c
e Abstract
D
Inthiswork,twonewseriesexpansionsforgeneralizedEuler’sconstants(Stieltjesconstants)γm are
6
1 obtained. ThefirstexpansioninvolvesStirlingnumbersofthefirstkind,containspolynomialsinπ−2
withrationalcoefficientsandconvergesslightlybetterthanEuler’sseries ∑n−2. Thesecondexpan-
]
T sionisasemi-convergentserieswithrationalcoefficientsonly. Thisexpansionisparticularlysimple
N and involves Bernoulli numbers with a non-linear combination of generalized harmonic numbers.
h. It also permits to derive an interesting estimation for generalized Euler’s constants, which is more
t accuratethanseveralwell-known estimations. Finally, inAppendixA,the readerwillalsofindtwo
a
m simpleintegraldefinitionsfortheStirlingnumbersofthefirstkind,aswellanupperboundforthem.
[
Keywords: GeneralizedEuler’sconstants,Stieltjesconstants,Stirlingnumbers,Factorialcoefficients,

4
Seriesexpansion,Divergentseries,Semi-convergentseries,Formalseries,Envelopingseries,
v
0 Asymptoticexpansions,Approximations,Bernoullinumbers,Harmonicnumbers,Rational
4 coefficients,Inversepi.
7
0
0
. I. Introductionandnotations
1
0
5 I.1. Introduction
1
Theζ-function,whichisusuallyintroducedviaoneofthefollowingseries,
:
v
i ∞
X ∑ 1 , Res>1
r n=1 ns
a ζ(s)=  1 ∑∞ (−1)n−1 , Res>0, s =1 (1)
1 21 s ns 6
 − − n=1
∗Correspondingauthor.Phones:[phone],[phone].
Emailaddress:[email protected], [email protected](Iaroslav
V.Blagouchine)
Note to the readers of the 4th arXiv version: this version is a copy of the journal version of the
article, which has been published in the Journal of Number Theory (Elsevier), vol. 158, pp. 365-396,
2016. DOI 10.1016/J.JNT.2015.06.012 http://www.sciencedirect.com/science/article/pii/S0022314X15002255
Artcile history: submitted 1 January 2015, accepted 29 June 2015, published on-line 18 August 2015.
The layout of the present version and its page numbering differ from the journal version, but the
content, the numbering of equations and the numbering of references are the same. This version also
incorporates some minor corrections to the final journal version, which were published on-line in the
same journal on December 8, 2016 (DOI 10.1016/J.JNT.2016.11.002). For any further reference to the
material published here, please, use the journal version of the paper, which you can always get for free
by writing a kind e-mail to the author.
isoffundamentalandlong-standingimportanceinmodernanalysis,numbertheory,theoryofspecial
functionsandinavarietyotherfields.Itiswellknownthatζ(s)ismeromorphicontheentirecomplex
s-planeandthatithasonesimplepoleats = 1withresidue1. ItsexpansionintheLaurentseriesin
aneighbourhoodofs=1isusuallywrittenthefollowingform
1 ∞ ( 1)m(s 1)m
ζ(s) = + ∑ − − γ , s =1. (2)
m
s 1 m! 6
− m=0
where coefficients γ , appearing in the regular part of expansion (2), are called generalized Euler’s
m
constantsorStieltjesconstants,bothnamesbeinginuse.2,3 Series(2)isthestandarddefinitionforγ .
m
Alternatively,theseconstantsmaybealsodefinedviathefollowinglimit
n lnmk lnm+1n
γ = lim ∑ , m=0,1,2,... (3)
m
n→∞(k=1 k − m+1 )
The equivalence between definitions (2) and (3) was demonstrated by various authors, including
Adolf Pilz [69], Thomas Stieltjes, Charles Hermite [1, vol. I, letter 71 and following], Johan Jensen
[87, 89], Je´roˆme Franel [56], Jørgen P. Gram [69], Godfrey H. Hardy [73], Srinivasa Ramanujan [2],
WilliamE.Briggs,S.Chowla[24]andmanyothers,seee.g.[16,176,84,128].Itiswellknownthatγ =
0
γ Euler’sconstant, seee.g.[128], [19, Eq.(14)]. Higher generalizedEuler’sconstants arenot known
toberelatedtothe“standard”mathematicalconstants,nortothe“classic”functionsofanalysis.
Inourrecentwork[18],weobtainedtwointerestingseriesrepresentationsforthelogarithmofthe
Γ-functioncontainingStirlingnumbersofthefirstkindS (n,k)
1
1 1
lnΓ(z) = z lnz z+ ln2π+
− 2 − 2
(cid:18) (cid:19)
+ 1 ∑∞ 1 ⌊∑21n⌋( 1)l (2l)!·|S1(n,2l+1)| (4)
π n n! − (2πz)2l+1
n=1 · l=0
1 1 1 1
lnΓ(z) = z ln z z+ + ln2π
− 2 − 2 − 2 2 −
(cid:18) (cid:19) (cid:18) (cid:19)
1 ∑∞ 1 ⌊∑21n⌋( 1)l (2l)!·(22l+1−1)·|S1(n,2l+1)| (5)
− π n=1n·n! l=0 − (4π)2l+1· z− 21 2l+1
aswellastheiranalogsforthepolygammafunctions Ψ (z).4 Theprese(cid:0)ntpap(cid:1)erisacontinuationof
k
thispreviouswork,inwhichweshowthattheuseofasimilartechniquepermitstoderivetwonew
2ThedefinitionofStieltjesconstantsaccordingly toformula(2)isduetoGodfreyH.Hardy. Definitions, introducedby
Thomas Stieltjesand Charles Hermite between 1882–1884, did not contain coefficients ( 1)m and m! In fact, use of these
−
factors is not well justified; notwithstanding, Hardy’s form (2) is largely accepted and is more frequently encountered in
modernliterature.Formoredetails,see[1,vol.I,letter71andfollowing],[110,p.562],[19,pp.538–539].
3SomeauthorsusethenamegeneralizedEuler’sconstantsforotherconstants,whichwereconceptuallyintroducedandstud-
iedbyBriggsin1961[23]andLehmerin1975[114].Theyweresubsequentlyrediscoveredinvarious(usuallyslightlydifferent)
formsbyseveralauthors,seee.g.[173,140,190].Furthergeneralizationofboth,generalizedEuler’sconstantsdefinedaccord-
inglyto(2)andgeneralizedEuler’sconstantsintroducedbyBriggsandLehmer,wasdonebyDilcherin[49].
4Bothseriesconvergeinapartoftherighthalf–plane[18,Fig.2]atthesamerateas∑ nlnmn −2,wherem=1forlnΓ(z)
andΨ(z),m=2forΨ1(z)andΨ2(z),m=3forΨ3(z)andΨ4(z),etc. (cid:0) (cid:1)
2
seriesexpansionsforgeneralizedEuler’sconstantsγ ,bothseriesinvolvingStirlingnumbersofthe
m
firstkind. Thefirstseriesisconvergentandcontainspolynomialsinπ 2withrationalcoefficients(the
−
latterinvolvesStirlingnumbersofthefirstkind). Fromthisseries,byaformalprocedure,wededuce
thesecondexpansion, whichissemi-convergentandcontainsrationaltermsonly. Thisexpansionis
particularly simple and involves only Bernoulli numbers and a non-linear combination of general-
izedharmonicnumbers. Convergenceanalysisofdiscoveredseriesshowsthattheformerconverges
slightlybetterthanEuler’sseries∑n 2,inaroughapproximationatthesamerateas
−
∞ lnmlnn
∑ , m =0,1,2,...
n2ln2n
n=3
Thelatterseriesdivergesveryquickly,approximatelyas
∞ lnmn n 2n
∑( 1)n−1 , m =0,1,2,...
− √n πe
n=2
(cid:16) (cid:17)
I.2. Notationsandsomedefinitions
Throughout the manuscript, following abbreviatednotations are used: γ = 0.5772156649...for
Euler’sconstant,γ formthgeneralizedEuler’sconstant(Stieltjesconstant)accordinglytotheirdef-
m
inition(2),5 (k)denotesthebinomialcoefficientCn, B standsforthenthBernoullinumber,6 H and
n k n n
(s)
H denotethenthharmonicnumberandthenthgeneralizedharmonicnumberoforders
n
H ∑n 1, H(s) ∑n 1 ,
n ≡ k n ≡ ks
k=1 k=1
respectively. Writings x stands for the integer part of x, tgz for the tangent of z, ctgz for the
⌊ ⌋
cotangent of z, chz for the hyperbolic cosine of z, shz for the hyperbolic sine of z, thz for the hy-
perbolic tangent of z, cthz for the hyperbolic cotangent of z. In order to avoid any confusion be-
tweencompositional inverseandmultiplicative inverse,inverse trigonometric andhyperbolicfunc-
tions are denoted as arccos, arcsin, arctg,... and not as cos−1, sin−1, tg−1,.... Writings Γ(z) and
ζ(z) denote respectively the gamma and the zeta functions of argument z. The Pochhammer sym-
bol (z) , which is also known as the generalized factorialfunction, is defined as the rising factorial
n
(z) z(z+1)(z+2) (z+n 1) = Γ(z+n)/Γ(z).7,8 For sufficiently large n, not necessarily
n
≡ ··· −
integer,thelattercanbegivenbythisusefulapproximation
(z)n = nn+Γz(−z)21√en2π 1+ 6z2−126nz+1 + 36z4−120z32+881n220z2−36z+1 +O(n−3)
(cid:26) (cid:27)
(6)
nz Γ(n) z(z 1) z(z 1)(z 2)(3z 1)
= Γ·(z) 1+ 2−n + − 24−n2 − +O(n−3)
(cid:26) (cid:27)
5Inparticularγ1=−0.07281584548...,γ2=−0.009690363192...,γ3=+0.002053834420....
6InparticularB0 =+1,B1 =−21,B2 =+61,B3 =0,B4 =−310,B5 =0,B6 =+412,B7 =0,B8 =−310,B9 =0,B10 =+656,
B11 = 0,B12 = −2679310,etc.,see[3,Tab.23.2,p.810],[109,p.5]or[59,p.258]forfurthervalues. Notealsothatsomeauthors
mayuseslightlydifferentdefinitionsfortheBernoullinumbers,seee.g.[72,p.91],[116,pp.32,71],[71,p.19,no138]or[11,
pp.3–6].
7Fornonpositiveandcomplexn,onlythelatterdefinition(z)n≡Γ(z+n)/Γ(z)holds.
8Notethatsomewriters(mostlyGerman-speaking)callsuchafunctionfaculte´ analytique orFaculta¨t, seee.g.[157], [158,
p.186],[159,vol.II,p.12],[72,p.119],[106].Othernamesandnotationsfor(z)narebrieflydiscussedin[92,pp.45–47]andin
[68,pp.47–48].
3
whichfollowsfromtheStirlingformulafortheΓ-function.9 Unsigned(orsignless)andsignedStirling
numbers of the first kind, which arealso known asfactorialcoefficients, aredenoted as S (n,l) and
1
| |
S (n,l) respectively (the latter are related to the former as S (n,l) = ( 1)n l S (n,l) ).10 Because
1 1 ± 1
− | |
in literature various names, notations and definitions were adopted for the Stirling numbers of the
first kind, we specify that we use exactly the same definitions and notation as in [18, Section 2.1],
that is to say S (n,l) and S (n,l) are defined as the coefficients in the expansion of rising/falling
1 1
| |
factorial
n∏−1(z+k) = (z) = Γ(z+n) = ∑n S (n,l) zl = ∑∞ S (n,l) zl
n Γ(z) | 1 |· | 1 |·
k=0 l=1 l=0
n∏−1(z k) = (z n+1) = Γ(z+1) = ∑n S (n,l) zl = ∑∞ S (n,l) zl (7a,b)
− − n Γ(z+1 n) 1 · 1 ·
respectivke=ly0,where z C and n > 1. Note th−atif l / l[=1,1n], where l issul=p0posedtobe nonnegative,
∈ ∈
thenS (n,l)=0,exceptforS (0,0)whichissetto1byconvention. Alternatively,thesamenumbers
1 1
maybeequallydefinedasthecoefficientsinthefollowingMacLaurinseries
lnl(1 z) ∞ S (n,l) ∞ S (n,l)
( 1)l − = ∑| 1 |zn = ∑ | 1 |zn, z <1, l =0,1,2,...
− l! n! n! | |
 n=l n=0
lnl(1+z) = ∑∞ S1(n,l)zn = ∑∞ S1(n,l)zn, z <1, l =0,1,2,... (8a,b)
l! n! n! | |
Signed Stirling numbne=rls of the first kni=n0d, as we defined them above, may be also given via the fol-
lowingexplicitformula
S (n,l) = (2n−l)! n∑−l 1 ∑k (−1)rrn−l+k (9)
1 (l 1)! (n+k)(n l k)!(n l+k)! r!(k r)!
− k=0 − − − r=0 −
l [1,n], which maybe usefulfor the computation of S (n,l) when n isnot verylarge.11 Allthree
1
∈
abovedefinitions agreewiththose adoptedbyJordan[92, Chapt.IV],[90,91],Riordan[147, p.70et
seq.], Mitrinovic´ [125],Abramowitz&Stegun[3,no24.1.3,p.824]andmanyothers(moreover,mod-
ernCAS,such asMapleorMathematica, alsosharethese definitions; inparticularStirling1(n,l)
intheformerandStirlingS1[n,l]inthelattercorrespondtoourS (n,l)).12 Kroneckersymbol(or
1
Kronecker delta) of arguments l and k is denoted by δ (δ = 1 if l = k and δ = 0 if l = k).
l,k l,k l,k 6
RezandImzdenoterespectivelyrealandimaginarypartsofz. Letteriisneverusedasindexandis
√ 1. Thewritingresz=a f(z)standsfortheresidueofthefunction f(z)atthepointz = a. Finally,by
−
therelativeerrorbetweenthequantity Aanditsapproximatedvalue B,wemean(A B)/A. Other
−
notationsarestandard.
9Asimplervariantoftheaboveformulamaybefoundin[177].
10Thereexistmorethan50notationsfortheStirlingnumbers,seee.g.[67],[92,pp.vii–viii,142,168],[101,pp.410–422],[68,
Sect.6.1],andwedonotinsistonourparticularnotation,whichmayseemforcertainnotproperlychosen.
11Fromtheabovedefinitions,itfollowsthat: S1(1,1) = +1,S1(2,1) = −1,S1(2,2) = +1,S1(3,1) = +2,S1(3,2) = −3,
S1(3,3)=+1,...,S1(8,5)=−1960,...,S1(9,3)=+118124,etc. NotethatthereisanerrorinStirling’streatise[172]:inthe
lastlineinthetableonp.11[172]thevalueof|S1(9,3)|=118124andnot105056.ThiserrorhasbeennotedbyJacquesBinet
[17,p.231],CharlesTweedie[179,p.10]andsomeothers(itwasalsocorrectedinsometranslationsof[172]).
12Aquick analysisofseveralalternativenames, notationsanddefinitionsmaybefoundinworksofCharlesJordan[92,
pp.vii–viii,1andChapt.IV],Gould[67,66],andDonaldE.Knuth[68,Sect.6.1],[101,pp.410–422].
4
II. AconvergentseriesrepresentationforgeneralizedEuler’sconstantsγminvolvingStirlingnum-
bersandpolynomialsinπ−2
II.1. Derivationoftheseriesexpansion
In1893JohanJensen[88,89]bycontourintegrationmethodsobtainedanintegralformulaforthe
ζ-function
π/2 ∞
1 1 (cosθ)s 2sinsθ 1 1 sin(sarctgx)
ζ(s)= + +2 − dθ = + +2 dx
s 1 2 ˆ e2πtgθ 1 s 1 2 ˆ (e2πx 1)(x2+1)s/2
− 0 − − 0 −
(10)
∞
1 1 1 (1 ix) s (1+ix) s
= + + − − − − dx, s =1
s 1 2 iˆ e2πx 1 6
− −
0
which extends (1) to the entire complex plane except s = 1. Expanding the above formula into the
Laurentseries about s = 1 and performing the term-by-term comparison of the derived expansion
withtheLaurentseries(2)yieldsthefollowingrepresentationforthemthStieltjesconstant
∞
1 1 dx lnm(1 ix) lnm(1+ix)
γ = δ + − , m=0,1,2,... (11)
m 2 m,0 iˆ e2πx 1 1 ix − 1+ix
− (cid:26) − (cid:27)
0
which is due to the Jensen and Franel.13 Making a change of variable in the latter formula
x = 1 ln(1 u),wehave
−2π −
1
ln(1 u) ln(1 u)
lnm 1 − lnm 1+ −
1 1 − 2πi 2πi du
γm = 2δm,0+ 2πiˆ  1(cid:20) ln(1−u) (cid:21) − 1(cid:20)+ ln(1−u) (cid:21) u (12)
− 2πi 2πi
0
 
where m =0,1,2,...
Now,inwhatfollows,wewilluseanumberofbasicpropertiesofStirlingnumbers,whichcanbe
foundinanamountsufficientforthepresentpurposeinthefollowingliterature: [172,79,106],[112,
Book I, partI], [52, 155, 156, 157], [158, pp. 186–187],[159, vol. II, pp. 23–31], [10, 32, 33, 34, 21, 62],
[29, p. 129], [92, Chapt. IV], [90, 91, 132], [133, pp. 67–78], [134, 179], [68, Sect. 6.1], [101, pp. 410–
422],[40, Chapt.V],[50],[141, Chapt.4, 3,no196–no210],[72, p.60etseq.], [130], [147,p.70etseq.],
§
[169, vol. 1], [15], [36, Chapt. 8], [3, no24.1.3, p. 824], [102, Sect. 21.5-1, p. 824], [13, vol. III, p. 257],
[135,171],[44,pp.91–94],[185,pp.2862–2865],[11,Chapt.2],[125,65,67,66,183,29,31],[137,p.642],
[152,61,189,126,14,188,174,80,27,26,83,4,175,70,117,163,164,154,148,149,76,105,18]. Note
thatmanywritersdiscoveredthesenumbersindependently,withoutrealizingthattheydealwiththe
13Intheexplicitform,thisintegralformulawasgivenbyFranelin1895[56](intheabove,wecorrectedtheoriginalFranel’s
formulawhichwasnotvalidform=0).However,itwasremarkedbyJensen[89]thatitcanbeelementaryderivedfrom(10)
obtainedtwoyearsearlieranditishardtodisagreewithhim. Bytheway,itiscuriousthatinworksofmodernauthors,see
e.g.[42,37],formula(11)isoftenattributedtoAinsworthandHowell,whodiscovereditindependentlymuchlater[8].
5
Stirlingnumbers.Forthisreason,inmanysources,thesenumbersmayappearunderdifferentnames,
differentnotationsandevenslightlydifferentdefinitions.14
ConsiderthegeneratingequationfortheunsignedStirlingnumbersofthefirstkind,formula(8a).
Thispower seriesisuniformly andabsolutelyconvergentinside the disk z < 1. Putting l+m 1
| | −
insteadofl,multiplyingbothsidesby(l) andsummingoverl =[1,∞),weobtainfortheleftside
m
∞ ln(1 z) l+m−1 ∞ ln(1 z) l+m−1
∑(l) − − = ∑ − − =
m
· (l+m 1)! (l 1)!
l=1 (cid:2) −(cid:3) l=1(cid:2) − (cid:3)
= ln(1 z) m ∑∞ −ln(1−z) l−1 = ( 1)m lnm(1−z)
− − · (l 1)! − · 1 z
l=1(cid:2) − (cid:3) −
(cid:2) (cid:3)
e−ln(1−z)
| {z }
whiletherightsideof(8a),invirtueoftheabsoluteconvergence,becomes
∑∞ (l) ∑∞ S1(n,l+m−1) zn = ∑∞ zn n−∑m+1(l) S (n,l+m 1)
m m 1
· n! n! · · −
l=1 n=0(cid:12) (cid:12) n=0 l=1
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12)
∞ S (n+1,m+1)
= m! ∑ 1 zn
· n!
n=0 (cid:12) (cid:12)
(cid:12) (cid:12)
Whence
lnm(1−z) = ( 1)mm! ∑∞ S1(n+1,m+1) zn, m =0,1,2,... (13)
1−z − ·n=0 (cid:12) n! (cid:12) z <1
(cid:12) (cid:12) | |
Writinginthelatter zforz,andthensubtractingonefromanotheryieldsthefollowingseries
−
lnm(1−z) lnm(1+z) = 2( 1)mm! ∑∞ S1(2k+2,m+1) z2k+1 (14)
1 z − 1+z − · (2k+1)!
− k=0 (cid:12) (cid:12)
(cid:12) (cid:12)
m = 0,1,2,..., which is absolutely and uniformly convergent in the unit disk z < 1, and whose
| |
coefficientsgrowlogarithmicallywithk
S1(2k+2,m+1) lnmk , k ∞, m =0,1,2,... (15)
(2k+1)! ∼ m! →
(cid:12) (cid:12)
(cid:12) (cid:12)
invirtueofknownasymptoticsfortheStirlingnumbers,seee.g.[91,p.261],[92,p.161],[3,no24.1.3,
p.824],[188,p.348,Eq.(8)].Usingformulæfrom[40,p.217],[163,p.1395],[104,p.425,Eq.(43)],the
14Actually, only in thebeginningoftheXXth century, thename“Stirling numbers”appeared inmathematicalliterature
(mainly,thankstoThorvaldN.ThieleandNielsNielsen[132,179], [101,p.416]). Othernamesforthesenumbersinclude:
factorialcoefficients,faculty’scoefficients(Faculta¨tencoefficienten,coefficientsdelafaculte´analytique),differencesofzeroandevendiffer-
entialcoefficientsofnothing. Moreover,theStirlingnumbersarealsocloselyconnectedtothegeneralizedBernoullinumbersBn(s),
alsoknownasBernoullinumbersofhigherorder,seee.g.[29,p.129],[65,p.449],[67,p.116];manyoftheirpropertiesmaybe,
therefore,deducedfromthoseofBn(s).
6
lawfortheformationoffirstcoefficientsmaybealsowritteninamoresimpleform
1, m =0

(cid:12)(cid:12)S1(2(k2+k+2,1m)!+1)(cid:12)(cid:12) = H21 2kH+221k,+1−H2(2k+)1 , mm ==21 (16)
For higher m, valuesof this coefficien16t(cid:8)(cid:8)mHa23yk+b1e−si3mHil2ak+rl(cid:9)1yHr2(e2kd+)u1c+ed2Hto2(3ka+)n1(cid:9)on,-linear comm=bin3ation of the
generalized harmonic numbers. Since expansion (14) holds only inside the unit circle, it cannot be
directlyused fortheinsertion intoJensen–Franel’sintegralformula(11). However, ifweputin(14)
z = 1 ln(1 u),weobtainfortherightpart
2πi −
2( 1)mm! ∑∞ S1(2k+2,m+1) ln2k+1(1−u) =
− · (2πi)2k+1 · (2k+1)!
k=0 (cid:12) (cid:12)
(cid:12) (cid:12)
see(8a)
∞ ( 1)k S|(2k+{z2,m+}1) ∞ S (n,2k+1)
= 2i( 1)mm! ∑ − 1 ∑ 1 un (17)
− · (2π)2k+1 · n!
k=0 (cid:12) (cid:12) n=1(cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
= 2i( 1)mm! ∑∞ un ⌊∑12n⌋ (−1)k S1(2k+2,m+1) · S1(n,2k+1)
− · n! · (2π)2k+1
n=1 k=0 (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
Therefore,form =0,1,2,...,wehave
ln(1 u) ln(1 u)
lnm 1 − lnm 1+ −
1 − 2πi 2πi
 (cid:20) (cid:21) (cid:20) (cid:21) =
2πi  1 ln(1−u) − 1+ ln(1−u) 
− 2πi 2πi (18)
 = (−1)mm! ∑∞ un ⌊∑21n⌋ (−1)k S1(2k+2,m+1) · S1(n,2k+1)
π n! · (2π)2k+1
n=1 k=0 (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
which uniformly holds in u < 1 and also is valid for u = 1.15 Substituting (18) into (12) and per-
| |
formingtheterm-by-termintegrationfromu = 0tou = 1yieldsthefollowingseriesrepresentation
formthgeneralizedEuler’sconstant
γ = 1δ + (−1)mm! ∑∞ 1 ⌊∑21n⌋ (−1)k S1(2k+2,m+1) · S1(n,2k+1) (19)
m 2 m,0 π n n! (2π)2k+1
n=1 · k=0 (cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12)
wherem =0,1,2,....Inparticular,forEuler’sconstantandfirstStieltjesconstant,wehavefollowing
15Theunitradiusofconvergenceofthisseriesisconditionedbythesingularitythemostclosesttotheorigin.Suchsingularity
isabranchpointlocatedatu=1.Notealsothatsincetheseriesisconvergentforu=1aswell,invirtueofAbel’stheoremon
powerseries,itisuniformlyconvergenteverywhereonthedisc u 61 ε,wherepositiveparameterεcanbemadeassmall
| | −
asweplease.
7
Figure1: Absolutevaluesofrelativeerrorsoftheseriesexpansionfor γ , γ and γ givenby(19)–(20),loga-
0 1 2
rithmicscale.
seriesexpansions
γ = 1 + 1 ∑∞ 1 ⌊∑12n⌋ (−1)k·(2k+1)!· S1(n,2k+1)
2 π n n! (2π)2k+1
n=1 · k=0 (cid:12) (cid:12)
(cid:12) (cid:12)
1 1 1 1 1 3 3 1 3
= + + + +
2 2π2 8π2 18 π2 − 4π4 96 π2 − 2π4
(cid:18) (cid:19) (cid:18) (cid:19)
1 12 105 15 1 60 675 225
+ + + + +...
600 π2 − 4π4 4π6 4320 π2 − 4π4 4π6
(cid:18) (cid:19) (cid:18) (cid:19)
(20)
γ = 1 ∑∞ 1 ⌊∑21n⌋ (−1)k·(2k+1)!·H2k+1· S1(n,2k+1)
1 −π n n! (2π)2k+1
n=1 · k=0 (cid:12) (cid:12)
(cid:12) (cid:12)
1 1 1 1 11 3 1 11
=
−2π2 − 8π2 − 18 π2 − 8π4 − 96 π2 − 4π4
(cid:18) (cid:19) (cid:18) (cid:19)
1 12 385 137 1 60 2475 2055
+ + ...
−600 π2 − 8π4 16π6 − 4320 π2 − 8π4 16π6 −
(cid:18) (cid:19) (cid:18) (cid:19)
respectively. As one can easily notice, each coefficient of these expansions contains polynomials in
π 2 withrationalcoefficients. The rateof convergenceof this series, depictedinFig. 1, isrelatively
−
slow and depends, at least for the moderate number of terms, on m: the greater the order m, the
slowertheconvergence.Amoreaccuratedescriptionofthisdependence,aswellastheexactvalueof
the rateof convergence, bothrequireadetailedconvergenceanalysisof (19), whichis performedin
thenextsection.
II.2. Convergenceanalysisofthederivedseries
The convergence analysis of series (19) consists in the study of its general term, which is given
by the finite truncated sum over index k. This sum has only odd terms, and hence, by elementary
8
transformations,maybereducedtothatcontainingbothoddandeventerms
⌊∑21n⌋( 1)k S1(2k+2,m+1) · S1(n,2k+1) =
− (2π)2k+1
k=0 (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
=⌊∑12n⌋(−1)21(2k+1)−21 S1(2k+1+1,(m2π+)21k)+1· S1(n,2k+1) (21)
k=0 (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
= 1∑n 1 ( 1)l ( 1)21(l−1) S1(l+1,m+1) · S1(n,l) = ...
2 − − · − · (2π)l
l=1 (cid:12) (cid:12) (cid:12) (cid:12)
(cid:2) (cid:3) (cid:12) (cid:12) (cid:12) (cid:12)
where,inthelastsum,wechangedthesummationindexbyputtingl =2k+1.Now,fromthesecond
integralformulafortheunsignedStirlingnumbersofthefirstkind,see(A.4),itfollowsthat
( 1)21(l−1) S1(l+1,m+1) = (−1)m (l+1)! + i l lnm+1(1−z) dz
− · (2π)l 2π · (m+1)! ‰ 2πz z2
(cid:12) (cid:12) (cid:20) (cid:21)
(cid:12) (cid:12) z=r
| |
( 1)l ( 1)21(l−1) S1(l+1,m+1) = (−1)m (l+1)! i l lnm+1(1−z) dz
− · − · (2π)l 2π · (m+1)! ‰ −2πz z2
(cid:12) (cid:12) (cid:20) (cid:21)
(cid:12) (cid:12) z=r
| |
where 0 < r < 1. Therefore, since (l+1)! = xl+1e xdx taken from 0 to ∞, the last sum in (21)
−
´
reducestothefollowingintegralrepresentation
( 1)m n i l i l lnm+1(1 z)
... = − ∑ S (n,l) (l+1)! − dz
4π(m+1)! l=1 1 · ·‰ "(cid:18)2πz(cid:19) −(cid:18)−2πz(cid:19)# z2
(cid:12) (cid:12) z=r
(cid:12) (cid:12) | |
∞
( 1)m n ix l ix l lnm+1(1 z)
= 4π(−m+1)! ·ˆ l∑=1 S1(n,l) ‰ "(cid:18)2πz(cid:19) −(cid:18)−2πz(cid:19)# z2 − dzxe−x dx (22)
0  (cid:12)(cid:12) |(cid:12)(cid:12)z|=r 
 
∞
( 1)m lnm+1(1 z) ix ix
= − − xe−xdx dz
4π(m+1)! ·‰ z2 ˆ 2πz − −2πz 
z=r 0 (cid:20)(cid:18) (cid:19)n (cid:18) (cid:19)n(cid:21) 
| |
 
Theintegralincurlybracketsisdifficulttoevaluateinaclosed-form,butatlargen,itsasymptotical
valuemaybereadilyobtained.
Function 1/Γ(z) is analytic on the entire complex z-plane, and hence, can be expanded into the
MacLaurinseries
1 γ2 π2 ∞
= z+γz2+ z3+... ∑ zka , z <∞, (23)
Γ(z) 2 − 12 ≡ k | |
(cid:18) (cid:19) k=1
where
1 1 (k) ( 1)k (k)
a = − sinπx Γ(x)
k ≡ k! ·(cid:20)Γ(z)(cid:21)z=0 πk! ·h · ix=1
9
seee.g.[3,p.256,no6.1.34],[188,pp.344&349],[77].UsingStirling’sapproximationforthePochham-
mersymbol(6),wehaveforsufficientlylargen
ix ix n2iπxz Γ(n) n−2iπxz Γ(n)
· · =
(cid:18)2πz(cid:19)n−(cid:18)−2πz(cid:19)n ∼ Γ 2iπxz − Γ −2iπxz
(24)
(cid:16) (cid:17) (cid:16) (cid:17)
ixlnn ∞ ix k ixlnn ∞ ix k
= (n 1)! exp ∑ a exp ∑( 1)ka
k k
− " (cid:18) 2πz (cid:19)k=1 (cid:18)2πz(cid:19) − (cid:18)− 2πz (cid:19)k=1 − (cid:18)2πz(cid:19) #
Substituting this approximation into the integral in curly brackets from (22), performing the term-
by-term integration16 and taking into account that z sΓ(s) = xs 1e zxdx taken over x [0,∞),
− − −
∈
yields ´
∞
ix ix
xe−xdx
ˆ 2πz − −2πz ∼
(cid:20)(cid:18) (cid:19)n (cid:18) (cid:19)n(cid:21)
0
(n 1)! ∑∞ a i k (k+1)! 1+ ilnn −k−2 ( 1)k 1 ilnn −k−2 (25)
k
∼ − k=1 (cid:18)2πz(cid:19)· ((cid:20) 2πz (cid:21) − − (cid:20) − 2πz (cid:21) )
32iπ3z3 4π2z2 3ln2n
(n 1)! − , n ∞,
∼ − · 4π(cid:0)2z2+ln2n 3 (cid:1) →
where, atthe final stage, we re(cid:0)tainedonly th(cid:1)e firstsignificant termcorrespondingtofactor k = 1.17
Now,if z 61 e 1 0.63,thentheprincipalbranchof lnm+1(1 z) 61independentlyofmand
−
| | − ≈ −
argz. Analogously,onecanalwaysfindsuchsufficientlylargen ,thatforanyhoweversmallε>0,
(cid:12) 0 (cid:12)
(cid:12) (cid:12)
32iπ3z3 4π2z2 3ln2n
− <ε, n>n , (26)
(cid:12) 4π(cid:0)2z2+ln2n 3 (cid:1)(cid:12) 0
(cid:12) (cid:12)
(cid:12) (cid:12)
on the circle z = 1 e−1 (cid:12)(cid:12)(for e(cid:0)xample, if ε =(cid:1) 1, th(cid:12)(cid:12)en n0 = 1222; if ε = 0.1, then n0 = 38597;
| | −
if ε = 0.01, then n = 33220487; etc.).18 Combining all these results and taking into account that
0
dz = z dargz,weconcludethat
| | | |
1 ⌊∑12n⌋ (−1)k S1(2k+2,m+1) · S1(n,2k+1) <
n n! (cid:12) (2π)2k+1 (cid:12)
· (cid:12)(cid:12)k=0 (cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12) (27)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) < 1 ε <(cid:12) C , n>n ,
n2 · 2 1 e 1 (m+1)! n2 0
− −
Numericalsimulations,seeFig.2,showthat(cid:0)thissimp(cid:1)leinequality,validforallm,mayprovidemore
orlessaccurateapproximationforthegeneraltermof(22),andthisgreatlydependsonm. Moreover,
16Series(23)–(24)beinguniformlyconvergent.
17Thesecondtermofthissum,correspondingtok=2,is
396iγπ3z3 4π2z2 ln2n lnn 1
(n 1)! − =(n 1)! O , n ∞,
− · 4π(cid:0)2z2+ln2n 4 (cid:1) − · (cid:18)ln5n(cid:19) →
andhence,maybeneglectedatlargen(cid:0). (cid:1)
18Notethatforfixedn,theleft-handsideof(26)reachesitsmaximumwhenzisimaginarypure.
10