Sums of products of Apostol-Bernoulli and Apostol-Euler polynomials

13 Pages·2014·0.29 MB·English

HeandAraciAdvancesinDifferenceEquations2014, 2014:155
http://www.advancesindifferenceequations.com/content/2014/1/155
RESEARCH OpenAccess
Sums of products of Apostol-Bernoulli and
Apostol-Euler polynomials
YuanHe1*andSerkanAraci2
*Correspondence:
[email protected] Abstract
1FacultyofScience,Kunming
Inthispaper,afurtherinvestigationfortheApostol-BernoulliandApostol-Euler
UniversityofScienceand
Technology,Kunming,Yunnan polynomialsandnumbersisperformed.Someclosedformulaeofsumsofproducts
650500,People’sRepublicofChina ofanynumberofApostol-BernoulliandApostol-Eulerpolynomialsandnumbersare
Fulllistofauthorinformationis
establishedbyapplyingthegeneratingfunctionmethodsandsomesummation
availableattheendofthearticle
transformtechniques.Itturnsoutthatsomewell-knownresultsarederivedasspecial
cases.
MSC: 11B68;05A19
Keywords: Bernoullipolynomialsandnumbers;Eulerpolynomialsandnumbers;
Apostol-Bernoullipolynomialsandnumbers;Apostol-Eulerpolynomialsand
numbers;summationformulae
1 Introduction
TheclassicalBernoullipolynomialsB (x)andEulerpolynomialsE (x)areusuallydefined
n n
bymeansofthefollowinggeneratingfunctions:
text (cid:2)∞ tn (cid:3) (cid:4) ext (cid:2)∞ tn (cid:3) (cid:4)
= B (x) |t|<π and = E (x) |t|<π . (.)
et– n n! et+ n n!
n= n=
Inparticular,therationalnumbersB =B ()andintegersE =nE (/)arecalledthe
n n n n
classicalBernoullinumbersandEulernumbers,respectively.
Asiswellknown,theclassicalBernoulliandEulerpolynomialsandnumbersplayim-
portant roles in different areas of mathematics such as number theory, combinatorics,
specialfunctionsandanalysis.Numerousinterestingpropertiesforthemcanbefoundin
manybooks;see,forexample,[–].Hereisthewell-knownEulerformulaontheclassical
Bernoullinumbers:
(cid:5) (cid:6)
(cid:2)n
n
B B =–nB –(n–)B (n≥). (.)
k n–k n– n
k
k=
Thisformulahasbeenextendedindifferentdirectionsbymanyauthors.Forexample,a
directgeneralisationof(.)isthefollowingidentityontheclassicalBernoullipolynomials
duetoNörlund[]:
(cid:5) (cid:6)
(cid:2)n
n
B (x)B (y)=–n(x+y–)B (x+y)–(n–)B (x+y) (n≥), (.)
k n–k n– n
k
k=
©2014HeandAraci;licenseeSpringer.ThisisanOpenAccessarticledistributedunderthetermsoftheCreativeCommons
AttributionLicense(http://creativecommons.org/licenses/by/2.0),whichpermitsunrestricteduse,distribution,andreproduction
inanymedium,providedtheoriginalworkisproperlycited.
HeandAraciAdvancesinDifferenceEquations2014, 2014:155 Page2of13
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whichhasbeengeneralisedtoasymmetricformbyHeandZhang[].Wealsomention
[–]forfurtherdiscoveriesoftheaboveNörlundresultfollowingtheworkofCarlitz
[],AgohandDilcher[,].Ontheotherhand,Eie[],SitaramachandraroandDavis
[]generalised(.)tosumsofproductsofthreeandfourBernoullinumbers.Afterthat,
Sankaranarayanan[]gavetheclosedexpressionofsumsofproductsoffiveBernoulli
numbers, and Zhang [] derived the closed ones of sums of products of less than and
equaltosevenBernoullinumbers.Inparticular,Dilcher[]obtainedsomeclosedformu-
laeofsumsofproductsofanynumberofclassicalBernoulliandEulerpolynomialsand
numbers.PetojevićandSrivastava[–]gotseveralnewformulaeofsumsofproducts
ofanynumberofclassicalBernoullinumbersincludingtheEulertypeandDilchertype
sumsofproductsofBernoullinumbers.Recently,Kim[]developedanewapproachto
givetheclosedformulaofsumsofproductsofanynumberofclassicalBernoullinumbers
byusingtherelationofvaluesatnon-positiveintegersoftheimportantrepresentationof
themultipleHurwitzzetafunctionintermsoftheHurwitzzetafunction.Further,Kim
andHu[]obtainedtheclosedformulaofsumsofproductsofanynumberofApostol-
BernoullinumbersbyexpressingthesumsofproductsoftheApostol-Bernoullipolynomi-
alsintermsofthespecialvaluesofmultipleHurwitz-Lerchzetafunctionsatnon-positive
integers.
WenowturntotheApostol-BernoullipolynomialsB (x;λ)andApostol-Eulerpolyno-
n
mialsE (x;λ),whichareusuallydefinedbymeansofthefollowinggeneratingfunctions
n
(see,e.g.,[,]):
text (cid:2)∞ tn (cid:3) (cid:4)
= B (x;λ) |t|<π ifλ=;|t|<|logλ|otherwise (.)
λet– n n!
n=
and
ext (cid:2)∞ tn (cid:3) (cid:4)
= E (x;λ) |t|<π ifλ=;|t|<|log(–λ)|otherwise . (.)
λet+ n n!
n=
Moreover,B (λ)=B (;λ)andE (λ)=nE (/;λ)arecalledtheApostol-Bernoullinum-
n n n n
bersandApostol-Eulernumbers,respectively.Obviously B (x;λ)and E (x;λ)reduceto
n n
B (x)andE (x)whenλ=.ItisworthnoticingthattheApostol-Bernoullipolynomials
n n
werefirstlyintroducedbyApostol[](seealsoSrivastava[]forasystematicstudy)in
ordertoevaluatethevalueoftheHurwitz-Lerchzetafunction.Forsomeelegantresults
andnicemethodsonthesepolynomialsandnumbers,oneisreferredto[–].
Inthepresentpaper,wewillbeconcernedwithsomeclosedformulaeofsumsofprod-
ucts of any number of Apostol-Bernoulli and Apostol-Euler polynomials and numbers.
The idea stems from the two identities of Xu and Cen [] applying the famous Faà di
BrunoformulatoansweraproblemposedbyGuoandQi[].Weprovethetworesults
duetoXuandCeninabriefwayagain.Asfurtherapplications,weobtainsomeclosed
formulaeofsumsofproductsofanynumberofApostol-BernoulliandApostol-Eulerpoly-
nomialsandnumbersbyapplyingthegeneratingfunctionmethodsandsomesummation
transformtechniques.Itturnsoutthatsomeknownresultsincludingtheonesstatedin
[,]arederivedasspecialcases.
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2 Someauxiliaryresults
WefirstlyrecalltheStirlingnumbersofthefirstkinds(n,k)andtheStirlingnumbersofthe
secondkindS(n,k)whichcanbefoundinthestandardbook[].TheStirlingnumbers
ofthefirstkindarethecoefficientsintheexpansion
(cid:2)n
(x) = s(n,k)xk (n≥), (.)
n
k=
andtheStirlingnumbersofthesecondkindarecharacterisedbytheidentity
(cid:2)n
xn= S(n,k)(x) (n≥). (.)
k
k=
Thenotation(x) appearingin(.)and(.)standsforthefallingfactorial(x) ofordern
n n
definedby(x) =and(x) =x(x–)(x–)···(x–n+)forpositiveintegernandcomplex
 n
numberx.Infact,theStirlingnumbersofthefirstkinds(n,k)andtheStirlingnumbersof
thesecondkindS(n,k)canbedefinedbymeansofthefollowinggeneratingfunctions:
(cid:2)∞ (cid:2)∞
(ln(+t))k tn (et–)k tn
= s(n,k) and = S(n,k) . (.)
k! n! k! n!
n=k n=k
Ifmakinguseof(.)thenonecaneasilyobtainthepairofclassicalinverserelations(see,
e.g.,[,p.]):
(cid:2)n (cid:2)n
f(n)= s(n,k)g(k) ⇐⇒ g(n)= S(n,k)f(k), (.)
k= k=
withnbeingnon-negativeintegerandf(n)andg(n)beingtwosequences.
Motivatedbytwoidentitiesappearingin[],GuoandQi[,]posedthefollowing
problem:fort(cid:6)=andpositiveintegern,determinethenumbersa for≤k≤nsuch
n,k–
that
(cid:5) (cid:6)
 (cid:2)n  (k–)
=+ a . (.)
(–e–t)n n,k– et–
k=
Stimulatedbythisproblem,theauthors[]madeuseofthemathematicalinductionto
establisheightidentitieswhichrevealthefunctions/(–e±t)nandthederivatives(/(e±t–
))(k)canbeexpressedbyeachotherbylinearcombinationswithcoefficientsinvolvingthe
combinatorialnumbersandStirlingnumbersofthesecondkind.Further,XuandCen[]
appliedthefamousFaàdiBrunoformulatounifytheeightidentitiesduetoGuoandQi
[]totwoidentitiesinvolvingtwoparametersα andλ.Wenextstatetheirresultsand
giveabriefproof.
Theorem.([]) Letnbeanon-negativeintegerandα,λtwoparameters.Then
(cid:5) (cid:6)
 (n) (cid:2)n+ (–)n+k–αn(k–)!S(n+,k)
= , (.)
–λeαt (–λeαt)k
k=
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andforpositiveintegern,
(cid:5) (cid:6)
 (cid:2)n (–)n–ks(n,k)  (k–)
= . (.)
(–λeαt)n (n–)!αk– –λeαt
k=
Proof Inviewofthebinomialseries
(cid:5) (cid:6)
(cid:2)∞
β
(+t)β = tn (βcomplexnumber), (.)
n
n=
wediscover
(cid:5) (cid:6) (cid:5) (cid:6)
(cid:2)∞ (cid:3) (cid:4) (cid:2)∞ (cid:3) (cid:4)
 –n n+j–
= –λeαt j= λeαt j. (.)
(–λeαt)n j j
j= j=
Iftakingx=–jin(.)thenweobtain,forpositiveintegern,
(cid:5) (cid:6)
(cid:2)n
 n+j–
s(n,k)(–)n–kjk–= . (.)
(n–)! j
k=
Applying(.)to(.)yields
(cid:2)n (cid:2)∞ (cid:3) (cid:4)
 
= (–)n–ks(n,k) λeαt jjk–. (.)
(–λeαt)n (n–)!
k= j=
Itiseasytoseefrom(.)that
(cid:5) (cid:6)
 (k–) (cid:2)∞ (cid:3) (cid:4) (cid:2)∞ (cid:3) (cid:4)
= λj eαtj (k–)=αk– λeαt jjk–. (.)
–λeαt
j= j=
Thus,combining(.)and(.)givestheformula(.).Itfollowsfrom(.)and(.)
that(.)iscomplete.Thisconcludestheproof. (cid:2)
Itbecomesobviousthatinthecaseα=in(.)arises,forpositiveintegern,
(cid:5) (cid:6)
 (cid:2)n (–)k–s(n,k)  (k–)
= (.)
(λet–)n (n–)! λet–
k=
and
(cid:5) (cid:6)
 (cid:2)n (–)n–ks(n,k)  (k–)
= . (.)
(λet+)n (n–)! λet+
k=
Wenextgiveanotherauxiliaryresultasfollows.
Theorem. Letnbeanon-negativeinteger.Then
(cid:7) (cid:5) (cid:6) (cid:8)
(cid:3) (cid:4) (cid:2)∞ (cid:2)n n tm
ext f(y,t) (n)= (–x)n–kf (x+y) , (.)
m+k
k m!
m= k=
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andforpositiveintegerr,
(cid:7) (cid:5) (cid:6)
(cid:3) (cid:4) (cid:2)∞ (cid:2)n
n f (x+y)
ext g(y,t) (n)= (–x)n–k m+k+r
k (cid:8)m+k+(cid:9)
r
m= k=
(cid:8)
(cid:9)
(–)n+xm+n+  tm
+ tm(–t)nf (x+y–xt)dt , (.)
r–
(r–)! m!

where (cid:8)x(cid:9) denotes the rising factorial (cid:8)x(cid:9) of order n defined by (cid:8)x(cid:9) =  and (cid:8)x(cid:9) =
n n  n
x(x+)(x+)···(x+n–)forpositiveintegernandcomplexnumberx,f(y,t)isdenoted
(cid:10) (cid:10)
byf(y,t)= ∞ f (y)tm,g(y,t)isdenotedbyg(y,t)= ∞ fm+r(y) · tm and{f (x)}∞ isa
m= m m! m= (cid:8)m+(cid:9)r m! n n=
sequenceofpolynomialsgivenby
(cid:2)∞
tn
f (x) =F(t)e(x–/)t, (.)
n
n!
n=
withF(t)beingaformalpowerseries.
Proof See[,(.),(.)]fordetails. (cid:2)
TherefollowsomespecialcasesofTheorem..Bysettingy=andf(y,t)=/(λet+)
in(.),weobtain,fornon-negativeintegersm,n,
(cid:11) (cid:12)(cid:5) (cid:5) (cid:6) (cid:6) (cid:5) (cid:6)
tm  (n) (cid:2)n n
ext = (–x)n–kE (x;λ), (.)
m! λet+ k m+k
k=
where[tn]f(t)standsforthecoefficientsof[tn]inf(t).Ontheotherhand,sinceB (x;λ)=

whenλ=andB (x;λ)=whenλ(cid:6)=(see,e.g.,[]),sobysettingB (x;λ)=δ weget
  ,λ
 δ (cid:2)∞ B (λ) tm
– ,λ = m+ · . (.)
λet– t m+ m!
m=
It is easy to see from the properties of the Beta and Gamma functions that, for non-
negativeintegersm,n,
(cid:9)
 m!·n!
tm(–t)ndt= . (.)
(m+n+)!

Thus,bysettingr=,y=andg(y,t)=/(λet–)–δ /tin(.),withthehelpof(.)
,λ
and(.),weobtain,fornon-negativeintegersm,n,
(cid:11) (cid:12)(cid:5) (cid:5) (cid:6) (cid:6) (cid:5) (cid:6)
tm  (n) (cid:2)n n B (x;λ)
ext = (–x)n–k m+k+ . (.)
m! λet– k m+k+
k=
Weshallmakeuseoftheaboveformulae(.),(.),(.)and(.)togivesomeclosed
formulaeofsumsofproductsofanynumberofApostol-BernoulliandApostol-Eulerpoly-
nomialsandnumbersinthenextsection.
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3 Therestatementofmainresults
(cid:13) (cid:14)
Beforestating ourmainresults,webegin byintroducingtheStirlingcyclenumbers n
k
givenbytherelation
(cid:11) (cid:12)
(cid:2)n
n
x(x+)···(x +n–)= xk. (.)
k
k=
(cid:13) (cid:14)
Itisobviousfrom(.)thats(n,k)=(–)n–k n .Infact,theStirlingcyclenumberscanbe
k
definedrecursivelyby(see,e.g.,[,])
(cid:11) (cid:12) (cid:11) (cid:12) (cid:11) (cid:12)
 n 
=, = = and
  n
(cid:11) (cid:12) (cid:11) (cid:12) (cid:11) (cid:12) (.)
n+ n n
=n + (n,k≥).
k k k–
Inthissectionwealwaysdenotebyp (x)apolynomialgivenby(see,e.g.,[,])
n,m
(cid:5) (cid:6)(cid:11) (cid:12)
(cid:2)n–
 k n
p (x)= (–x)k–m
n,m
(n–)! m k+
k=m
(cid:5) (cid:6)
(–)n–m– (cid:2)n– k
= s(n,k+)xk–m, (.)
(n–)! m
k=m
(cid:3) (cid:4)
andwealsodenoteby n themultinomialcoefficientdefinedby
r,...,rk
(cid:5) (cid:6)
n n!
= (n,r ,...,r ≥). (.)
r ,...,r r !···r !  k
 k  k
We next explore the closed formulae of sums of products of any number of Apostol-
Bernoulli and Apostol-Euler polynomials and numbers. The discovery depends on the
followingidentity:
(cid:5) (cid:6) (cid:5) (cid:6)
t r  m–r trm–reyt
e(x+x+···+xm)t= , (.)
λet– λet+ (λet–)r(λet+)m–r
whererisanon-negativeinteger,misapositiveintegerwith≤r≤mandy=x +x +
 
···+x .Ontakingr=min(.)by(.)weget
m
(cid:11) (cid:12)(cid:5) (cid:6) (cid:5) (cid:6)
(cid:2)
tn tmeyt n
= B (x ;λ)···B (x ;λ). (.)
n! (λet–)m k ,...,k k  km m
k+···+km=n  m
k,...,km≥
Ontheotherhand,from(.)wediscover
(cid:11) (cid:12)(cid:5) (cid:6)
tn tmeyt
n! (λet–)m
(cid:11) (cid:12)(cid:5) (cid:5) (cid:6) (cid:6)
n! (cid:2)m (–)k–s(m,k) tn–m  (k–)
= eyt . (.)
(n–m)! (m–)! (n–m)! λet–
k=
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Ifapplying(.)totheright-handsideof(.)then
(cid:11) (cid:12)(cid:5) (cid:6)
tn tmeyt
n! (λet–)m
(cid:5) (cid:6)
n! (cid:2)m (–)k–s(m,k)(cid:2)k– k– B (y;λ)
= (–y)k––j n+j+–m . (.)
(n–m)! (m–)! j n+j+–m
k= j=
Changingtheorderofthesummationontheright-handsideof(.)gives
(cid:11) (cid:12)(cid:5) (cid:6)
tn tmeyt
n! (λet–)m
(cid:5) (cid:6)
n! (cid:2)m– B (y;λ) (cid:2)m k– s(m,k)yk––j
= (–)j n+j+–m
(n–m)! n+j+–m j (m–)!
j= k=j+
(cid:5) (cid:6)
(–)m–n!(cid:2)m– B (y;λ) (cid:2)m k– s(m,k)yk–m+j
= (–)j n–j
(n–m)! n–j m–j– (m–)!
j= k=m–j
(–)m–n!(cid:2)m– B (y;λ)
= (–)j n–j
(n–m)! n–j
j=
(cid:5) (cid:6)
(cid:2)m–
 k
× s(m,k+)yk–(m–j–). (.)
(m–)! m–j–
k=m–j–
Thus,byequating(.)and(.)andthenapplying(.)thefollowingresultarises.
Theorem. Letmbeapositiveintegerandy=x +x +···+x .Then,forpositiveinteger
  m
n≥m,
(cid:5) (cid:6)
(cid:2)
n
B (x ;λ)···B (x ;λ)
k ,...,k k  km m
k+···+km=n  m
k,...,km≥
(–)m–n!(cid:2)m– B (y;λ)
n–j
= p (y) . (.)
m,m–j–
(n–m)! n–j
j=
TherefollowsomespecialcasesofTheorem..Obviouslythecaseλ=inTheorem.
isanequivalentversionoftheclassicalformulaofDilcher[,Theorem]:
(cid:5) (cid:6)
(cid:2)
n
B (x )···B (x )
k ,...,k k  km m
k+···+km=n  m
k,...,km≥
(cid:5) (cid:6) (cid:15) (cid:5) (cid:6) (cid:16)
(cid:2)m– (cid:2)j
n m–j–+k
=(–)m–m (–)j s(m,m–j+k)yk
m k
j= k=
B (y)
× n–j . (.)
n–j
Theorem.canalsobeusedtogivetheclosedformulaofsumsofproductsofanynumber
ofApostol-Bernoullinumbersdescribedin[].Forexample,sincetheApostol-Bernoulli
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polynomialsobeythesymmetricdistribution
(cid:5) (cid:6)

λB (–x;λ)=(–)nB x; (n≥), (.)
n n λ
andthedifferenceequation
λB (+x;λ)–B (x;λ)=nxn– (n≥), (.)
n n
which can be found in [], by setting x =x =···=x = and replacing λ by /λ in
  m
Theorem.,inviewof(.),wederive
(cid:5) (cid:6)
(cid:2)
n
(–)nλm B (λ)···B (λ)
k ,...,k k km
k+···+km=n  m
k,...,km≥
(–)m–n!(cid:2)m– B (m;)
= p (m) n–j λ . (.)
m,m–j–
(n–m)! n–j
j=
On multiplying λx– in both sides of (.) and substituting x=,...,m –, in view of
addingtheprecedingresults,wegetforpositiveintegermandnon-negativeintegern,
(cid:2)m–
λm–B (m;λ)=B (;λ)+n kn–λk–. (.)
n n
k=
Wenoticethat,from(.),wehaveλB (;λ)=+B (λ)andλB (;λ)=B (λ)forpositive
  n n
integern≥,andthefollowingrelation(see,e.g.,[]):
(cid:2)n–
p (n)mk= (m,n≥and≤m≤n–). (.)
n,k
k=
Thus, by applying (.) and (.) to (.), one can obtain the closed formula for the
Apostol-BernoullinumbersduetoKimandHu[,Theorem.],asfollows:
(cid:5) (cid:6)
(cid:2)
n
B (λ)···B (λ)
k ,...,k k km
k+···+km=n  m
k,...,k⎧m≥
(cid:10)
⎨(–)m+n+n! m–p (m)Bn–j(λ) ifn>m,
= (n–m)! j= (cid:10)m,m–j– n–j (.)
⎩n!p B (λ)–n! n–p (n)Bn–j(λ) ifn=m.
n,  j= n,n–j– n–j
Iftakingr=in(.)thenby(.)weobtain
(cid:11) (cid:12)(cid:5) (cid:6)
tn meyt
n! (λet+)m
(cid:5) (cid:6)
(cid:2)
n
= E (x ;λ)···E (x ;λ). (.)
k ,...,k k  km m
k+···+km=n  m
k,...,km≥
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Withthehelpof(.)and(.),werewritetheleft-handsideof(.)as
(cid:11) (cid:12)(cid:5) (cid:6)
tn meyt
n! (λet+)m
(cid:11) (cid:12)(cid:5) (cid:5) (cid:6) (cid:6)
(cid:2)m (–)m–ks(m,k) tn  (k–)
=m– eyt
(m–)! n! λet+
k=
(cid:5) (cid:6)
(cid:2)m (–)m–ks(m,k)(cid:2)k– k–
=m– (–y)k––jE (y;λ). (.)
n+j
(m–)! j
k= j=
Changingtheorderofthesummationontheright-handsideof(.)arises
(cid:11) (cid:12)(cid:5) (cid:6)
tn meyt
n! (λet+)m
(cid:5) (cid:6)
m– (cid:2)m– (cid:2)m k–
= (–)m–j–E (y;λ) s(m,k)yk––j. (.)
n+j
(m–)! j
j= k=j+
Thefollowingresultfollowsfrom(.),(.)and(.).
Theorem. Letmbeapositiveintegerandy=x +x +···+x .Then,forpositiveinteger
  m
n≥m,
(cid:5) (cid:6)
(cid:2) (cid:2)m–
n
E (x ;λ)···E (x ;λ)=m– p (y)E (y;λ). (.)
k ,...,k k  km m m,j n+j
k+···+km=n  m j=
k,...,km≥
Itisworthynoticingthatsince(.)canberewrittenas
(cid:11) (cid:12)(cid:5) (cid:6)
tn meyt
n! (λet+)m
(cid:5) (cid:6)
m– (cid:2)m– (cid:2)m k–
= (–)jE (y;λ) s(m,k)yk–(m–j)
n+m––j
(m–)! m––j
j= k=m–j
m– (cid:2)m–
= (–)jE (y;λ)
n+m––j
(m–)!
j=
(cid:5) (cid:6)
(cid:2)j
m+k–j–
× s(m,m+k–j)yk, (.)
k
k=
bycombining(.)and(.),weobtain
(cid:5) (cid:6)
(cid:2)
n
E (x ;λ)···E (x ;λ)
k ,...,k k  km m
k+···+km=n  m
k,...,km≥
(cid:15) (cid:5) (cid:6) (cid:16)
m– (cid:2)m– (cid:2)j m+k–j–
= (–)j s(m,m+k–j)yk E (y;λ). (.)
n+m––j
(m–)! k
j= k=
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Obviously the case λ= in (.) gives the formula of Dilcher [, Theorem ] on the
classicalEulerpolynomials.
Wenextconsiderthecase≤r≤m–in(.).Forconvenience,inthefollowingwe
alwaysdenotez=λet.Weapplythefamiliarpartialfractiondecompositionandlet
 (cid:2)r– M m(cid:2)–r– N
i j
= + . (.)
(z–)r(z+)m–r (z–)r–i (z+)m–r–j
i= j=
WearenowinthepositiontodeterminethecoefficientsM andN in(.).Bymultiply-
i j
ingbothsidesin(.)by(z–)r andtakingz→,weobtain
 
M =lim = . (.)
 z→(z+)m–r m–r
For the case ≤i≤r– in (.), by multiplying (z–)r–i in both sides of (.) and
(cid:10) (cid:3) (cid:4)
taking z→, with the help of the binomial theorem (z+)n = n n (z–)kn–k for
k= k
non-negativeintegern,weget
(cid:15) (cid:16)
(cid:2)i–
 M
M =lim(z–)r–i – k
i z→ (z–)r(z+)m–r (z–)r–k
k=
(cid:10)
 –(z+)m–r i– M (z–)k
= lim k= k
m–r z→ (z–)i
(cid:10) (cid:10) (cid:3) (cid:4)
 – mj=–r ik–= mj–r m–r–jMk(z–)k+j
= lim . (.)
m–r z→ (z–)i
Inlightof(.),wecanmaketheL’Hôspitalrulefor(.).Infact,repeatedlyapplying
theL’Hôspitalruleitimestherearises
(cid:5) (cid:6)
(cid:2)m–r(cid:2)i–
 m–r M (k+j)
M =– lim k i(z–)k+j–i, (.)
i i!z→ j j
j= k=
where(x) isthefallingfactorial(x) ofordernin(.).Hence,thecoefficientsM satisfy
n n i
(cid:5) (cid:6)
(cid:2)i–
 m–r M
M = and M =– k (≤i≤r–). (.)
 m–r i i–k i–k
k=
Itfollowsfrom(.)thatwecanclaim
(cid:5) (cid:6)
(–)i m–r+i–
M = (≤i≤r–). (.)
i m–r+i i
Weshalluseinductiononi.Obviously(.)holdstriviallywheni=.Assumethat(.)
iscompleteforanysmallervalueofi.From(.)and(.),wehave
(cid:5) (cid:6)(cid:5) (cid:6) (cid:5) (cid:6)
 (cid:2)i m–r m–r+k– (–)i m–r+i–
M =– (–)k+ . (.)
i m–r+i i–k k m–r+i i
k=