Recurrence formulae for Apostol-Bernoulli and Apostol-Euler polynomials

16 Pages·2013·0.35 MB·English

HeandWangAdvancesinDifferenceEquations2012,2012:209
http://www.advancesindifferenceequations.com/content/2012/1/209
RESEARCH OpenAccess
Recurrence formulae for Apostol-Bernoulli
and Apostol-Euler polynomials
YuanHe*andChunpingWang
*Correspondence:
[email protected] Abstract
FacultyofScience,Kunming
Inthispaper,usinggeneratingfunctionsandcombinatorialtechniques,weextend
UniversityofScienceand
Technology,Kunming,Yunnan AgohandDilcher’squadraticrecurrenceformulaforBernoullinumbersin(Agohand
650500,People’sRepublicofChina DilcherinJ.NumberTheory124:105-122,2007)toApostol-Bernoulliand
Apostol-Eulerpolynomialsandnumbers.
MSC: 11B68;05A19
Keywords: Apostol-Bernoullipolynomialsandnumbers;Apostol-Eulerpolynomials
andnumbers;recurrenceformula
1 Introduction
TheclassicalBernoullipolynomialsB (x)andEulerpolynomialsE (x)haveplayedimpor-
n n
tantrolesinmanybranchesofmathematicssuchasnumbertheory,combinatorics,special
functionsandanalysis,andtheyareusuallydefinedbymeansofthefollowinggenerating
functions:
text (cid:2)∞ tn (cid:3) (cid:4) ext (cid:2)∞ tn (cid:3) (cid:4)
= B (x) |t|<π , = E (x) |t|<π . (.)
et– n n! et+ n n!
n= n=
Inparticular,B =B ()andE =nE (/)arecalledtheclassicalBernoullinumbersand
n n n n
Eulernumbers,respectively.Numerousinterestingpropertiesforthesepolynomialsand
numbershavebeenexplored;see,forexample,[–].
Recently,usingsomerelationshipsinvolvingBernoullinumbers,AgohandDilcher[]
extendedEuler’swell-knownquadraticrecurrenceformulaontheclassicalBernoullinum-
bers
(cid:5) (cid:6)
(cid:2)n
n
(B +B )n= BB =–nB –(n–)B (n≥) (.)
  i n–i n– n
i
i=
toobtainanexplicitexpressionfor(B +B )n witharbitrarynon-negativeintegersk,m,
k m
nandkandmnotbothzeroasfollows:
(cid:5) (cid:6)
(cid:2)n
n
(B +B )n= B B
k m k+i m+n–i
i
i=
k!m!(n+δ(k,m)(m+k+))
=– B
m+n+k
(m+k+)!
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(cid:7) (cid:5) (cid:6)(cid:5) (cid:6)
(cid:2)m+k
B k+ k+–i im
+ (–)i m+k+–i (–)k n–
m+k+–i i k+ k+
i=
(cid:5) (cid:6)(cid:5) (cid:6)(cid:8)
m+ m+–i ik
+(–)m n– B , (.)
n+i–
i m+ m+
whereδ(k,m)=whenk=orm=,andδ(k,m)=otherwise.Asfurtherapplications,
Agoh and Dilcher [] derived some new types of recurrence formulae on the classical
BernoullinumbersandshowedthatthevaluesofB dependonlyonB ,B ,...,B
n n n+ n
forapositiveintegernand,similarly,forB andB .
n+ n+
Inthispaper,usinggeneratingfunctionsandcombinatorialtechniques,weextendthe
abovementionedAgohandDilcherquadraticrecurrenceformulaforBernoullinumbers
toApostol-BernoulliandApostol-Eulerpolynomialsandnumbers.Theseresultsalsolead
to some known ones related to the formulae on products of the classical Bernoulli and
EulerpolynomialsandnumbersstatedinNielsen’sclassicalbook[].
2 Preliminariesandknownresults
WefirstrecalltheApostol-BernoullipolynomialswhichwereintroducedbyApostol[]
(seealsoSrivastava[]forasystematicstudy)inordertoevaluatethevalueoftheHurwitz-
Lerchzetafunction.Forsimplicity,weherestartwiththeApostol-Bernoullipolynomials
B(α)(x;λ)of(realorcomplex)orderαgivenbyLuoandSrivastava[]
n
(cid:5) (cid:6)
t α (cid:2)∞ tn (cid:3) (cid:4)
ext= B(α)(x;λ) |t+logλ|<π;α:= . (.)
λet– n n!
n=
Especiallythecase α= in(.)givestheApostol-Bernoullipolynomialswhicharede-
notedbyB (x;λ)=B()(x;λ).Moreover,B (λ)=B (;λ)arecalledtheApostol-Bernoulli
n n n n
numbers.Further,Luo[]introducedtheApostol-EulerpolynomialsE(α)(x;λ)oforderα:
n
(cid:5) (cid:6)
 α (cid:2)∞ tn (cid:3) (cid:4)
ext= E(α)(x;λ) |t+logλ|<π;α:= . (.)
λet+ n n!
n=
The Apostol-Euler polynomials E (x;λ) and Apostol-Euler numbers E (λ) are given by
n n
E (x;λ)=E()(x;λ) and E (λ)=nE (/;λ),respectively.Obviously, B (x;λ) and E (x;λ)
n n n n n n
reducetoB (x)andE (x)whenλ=.SeveralinterestingpropertiesforApostol-Bernoulli
n n
andApostol-Eulerpolynomialsandnumbershavebeenpresentedin[–].Nextwegive
somebasicpropertiesforApostol-BernoulliandApostol-Eulerpolynomialsoforderαas
statedin[,].
Proposition. DifferentialrelationsoftheApostol-BernoulliandApostol-Eulerpolyno-
mialsoforderα:fornon-negativeintegerskandnwith≤k≤n,
∂k (cid:9) (cid:10) n!B(α)(x;λ) ∂k (cid:9) (cid:10) n!E(α)(x;λ)
B(α)(x;λ) = n–k , E(α)(x;λ) = n–k . (.)
∂xk n (n–k)! ∂xk n (n–k)!
Proposition. DifferenceequationsoftheApostol-BernoulliandApostol-Eulerpolyno-
mialsoforderα:forapositiveintegern,
(cid:3) (cid:4)
λB(α)(x+;λ)–B(α)(x;λ)=nB(α–)(x;λ) B()(x;λ)=xn , (.)
n n n– n
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and
(cid:3) (cid:4)
λE(α)(x+;λ)+E(α)(x;λ)=E(α–)(x;λ) E()(x;λ)=xn . (.)
n n n n
Proposition. AdditiontheoremsoftheApostol-BernoulliandApostol-Eulerpolynomi-
alsoforderα:forasuitableparameterβ andanon-negativeintegern,
(cid:5) (cid:6)
(cid:2)n (cid:3) (cid:4)
n
B(α+β)(x+y;λ)= B(α)(x;λ)B(β)(y;λ) B()(x;λ)=xn , (.)
n i i n–i n
i=
and
(cid:5) (cid:6)
(cid:2)n (cid:3) (cid:4)
n
E(α+β)(x+y;λ)= E(α)(x;λ)E(β)(y;λ) E()(x;λ)=xn . (.)
n i i n–i n
i=
Proposition. ComplementaryadditiontheoremsoftheApostol-BernoulliandApostol-
Eulerpolynomialsoforderα:foranon-negativeintegern,
(–)n (cid:3) (cid:4) (–)n (cid:3) (cid:4)
B(α)(α–x;λ)= B(α) x;λ– , E(α)(α–x;λ)= E(α) x;λ– . (.)
n λα n n λα n
Settingα=,β=andα=λ=,β=intheformulae(.)to(.),weimmediately
getthecorrespondingformulaefortheApostol-BernoulliandApostol-Eulerpolynomials,
andtheclassicalBernoulliandEulerpolynomials,respectively.Itisworthnotingthatthe
casesα=λ=inProposition.arealsocalledthesymmetricdistributionsoftheclas-
sicalBernoulliandEulerpolynomials.Theabovepropositionswillbeveryusefultoin-
vestigatethequadraticrecurrenceformulaefortheApostol-BernoulliandApostol-Euler
polynomialsinthenexttwosections.
3 RecurrenceformulaeforApostol-Bernoullipolynomials
Inwhatfollows,weshallalwaysdenotebyδ theKroneckersymbolwhichisdefinedby
,λ
δ =oraccordingtoλ(cid:5)=orλ=,andalsodenoteB (x;λ)=E (x;λ)=foranypos-
,λ –n –n
itiveintegern.BeforestatingthequadraticrecurrenceformulafortheApostol-Bernoulli
polynomials,webeginwithasummationformulaconcerningthequadraticrecurrenceof
theApostol-Bernoullipolynomials.
Theorem. Letk,m,nbenon-negativeintegers.Then
(cid:5) (cid:6)
(cid:2)n (cid:3) (cid:4)
n
B(x;μ)+B (y;λμ) k
i m+n–i
i
i=
n+k
=–δ B (x+y;λμ)
,λ m+n+k
m+
n+k
+B (y;λ)B (x+y;μ)+ B (y;λ)B (x+y;μ)
m n+k m+ n+k–
m+
(cid:5) (cid:6)
(cid:2)m m B (x;λ–)
+(n+k) (–)m+iB (x+y;λμ) m+–i . (.)
n+k+i–
i m+–i
i=
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Proof Multiplyingbothsidesoftheidentity
(cid:5) (cid:6)
  λeu  
= + (.)
λeu–μev– λeu– μev– λμeu+v–
byv(u+v)exv+y(u+v)yields
vexv (u+v)ey(u+v) eyu ve(x+y)v e(–x)u (u+v)e(x+y)(u+v)
=(u+v) –λv . (.)
μev– λμeu+v– λeu–μev– λeu– λμeu+v–
Makingk-timesderivativefortheaboveidentity(.)withrespecttov,withthehelpof
theLeibnizrule,weobtain
(cid:5) (cid:6) (cid:5) (cid:6) (cid:5) (cid:6)
(cid:2)k k ∂j vexv ∂k–j (u+v)ey(u+v)
j ∂vj μev– ∂vk–j λμeu+v–
j=
(cid:5) (cid:6) (cid:5) (cid:6) (cid:5) (cid:6)
ueyu ∂k ve(x+y)v eyu ∂k ve(x+y)v eyu ∂k– ve(x+y)v
= +v +k
λeu–∂vk μev– λeu–∂vk μev– λeu–∂vk– μev–
(cid:5) (cid:6) (cid:5) (cid:6)
e(–x)u ∂k (u+v)e(x+y)(u+v) e(–x)u ∂k– (u+v)e(x+y)(u+v)
–λv –λk . (.)
λeu–∂vk λμeu+v– λeu–∂vk– λμeu+v–
Since B (x;λ)= when λ= and B (x;λ)= when λ(cid:5)= (see, e.g., []), so by setting
 
B (x;λ)=δ ,weget
 ,λ
exu δ (cid:2)∞ B (x;λ)um
,λ m+
– = . (.)
λeu– u m+ m!
m=
Ontheotherhand,byTaylor’stheorem,wehave
(cid:5) (cid:6)
(cid:2)∞ (cid:2)∞ (cid:2)∞
(u+v)ex(u+v) ∂n uexu vn umvn
= = B (x;λ) . (.)
λeu+v– ∂un λeu– n! m+n m! n!
n= m=n=
Applying(.),(.)and(.)to(.),inviewoftheCauchyproductandthecomplemen-
taryadditiontheoremoftheApostol-Bernoullipolynomials,wederive
(cid:11) (cid:5) (cid:6) (cid:5) (cid:6) (cid:12)
(cid:2)∞ (cid:2)∞ (cid:2)n n (cid:2)k k umvn
B (x;μ)B (y;λμ)
i+j m+n+k–i–j
i j m! n!
m=n= i= j=
(cid:13) (cid:14)
(cid:2)∞ (cid:2)∞ kB (y;λ)B (x+y;μ) umvn
= B (y;λ)B (x+y;μ)+ m+ n+k–
m n+k
m+ m! n!
m=n=
(cid:2)∞ (cid:2)∞ B (y;λ)B (x+y;μ)umvn+ kδ (cid:2)∞ vn
+ m+ n+k + ,λ B (x+y;μ)
n+k–
m+ m! n! u n!
m=n= n=
(cid:2)∞ (cid:2)∞ (cid:2)∞
δ vn+ um–vn+
+ ,λ B (x+y;μ) –δ B (x+y;λμ)
n+k ,λ m+n+k
u n! m! n!
n= m=n=
(cid:11) (cid:5) (cid:6) (cid:12)
(cid:2)∞ (cid:2)∞ (cid:2)m m B (x;λ–) umvn+
+ (–)m+iB (x+y;λμ) m+–i
n+k+i
i m+–i m! n!
m=n= i=
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(cid:11) (cid:5) (cid:6) (cid:12)
(cid:2)∞ (cid:2)∞ (cid:2)m m B (x;λ–) umvn
+k (–)m+iB (x+y;λμ) m+–i
n+k+i–
i m+–i m! n!
m=n= i=
(cid:2)∞ (cid:2)∞
um–vn
–kδ B (x+y;λμ) . (.)
,λ m+n+k–
m! n!
m=n=
Comparing the coefficients of umvn+/m!(n + )! and um/m! in (.), we conclude our
proof. (cid:2)
AsaspecialcaseofTheorem.,wehavethefollowing
Corollary. Letmandnbepositiveintegers.Then
(cid:5) (cid:6)
(cid:2)m (cid:3) (cid:4)
 m 
(–)m–iB y–x;λ– B (y;λμ)– B (x;λ)B (y;μ)
m–i n+i– m n–
m i m
i=
(cid:5) (cid:6)
(cid:2)n
 n 
=– B (y–x;μ)B (x;λμ)+ B (x;λ)B (y;μ). (.)
n–i m+i– m– n
n i n
i=
Proof Settingk=andsubstitutingyforxandxforyinTheorem.,weobtainthatfor
non-negativeintegersmandn,wehave
(cid:5) (cid:6)
(cid:2)n
n
B (y;μ)B (x;λμ)
n–i m+i
i
i=
n
=B (x;λ)B (x+y;μ)+ B (x;λ)B (x+y;μ)
m n m+ n–
m+
(cid:5) (cid:6)
(cid:2)m+ (cid:3) (cid:4)
n m+
+ (–)m+iB y;λ– B (x+y;λμ). (.)
m+–i n+i–
m+ i
i=
Hence,replacingybyy–xintheabovegivesthedesiredresult. (cid:2)
Remark. Settingλ=μ=,x=tandy=–tinCorollary.,byB (–x)=(–)nB (x)
n n
foranon-negativeintegern,weimmediatelygetthegeneralizationofWoodcock’sidentity
ontheclassicalBernoullinumbers,see[,],
(cid:5) (cid:6)
(cid:2)m
 m 
(–)iB (t)B (t)– B (t)B (t)
m–i n+i– m n–
m i m
i=
(cid:5) (cid:6)
(cid:2)n
 n 
= (–)iB (t)B (t)– B (t)B (t) (m,n≥). (.)
n–i m+i– n m–
n i n
i=
Applying the difference equation and symmetric distribution of the classical Bernoulli
polynomials, one can get (–)nB (–x)=B (x)+nxn– for a positive integer n. Combin-
n n
ingwithCorollary.,wehaveanothersymmetricexpressionontheclassicalBernoulli
numbersduetoAgoh,see[,],
(cid:5) (cid:6) (cid:5) (cid:6)
(cid:2)m (cid:2)n
 m  n
B B + B B =–B (m,n≥). (.)
m–i n+i– n–i m+i– m+n–
m i n i
i= i=
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Using Theorem ., we shall give the following quadratic recurrence formula for
Apostol-Bernoullipolynomials.
Theorem. Letk,m,nbenon-negativeintegers.Then
(cid:3) (cid:4)
B (x;μ)+B (y;λμ) n
k m
k!m!(n+δ(k,m)(m+k+))
=–δ B (x+y;λμ)
,λ m+n+k
(m+k+)!
(cid:7) (cid:5) (cid:6) (cid:5) (cid:6)(cid:8)
(cid:2)m+k m m B (x;λ–)
+ (–)m+i n –k B (x+y;λμ) m+k+–i
n+i–
i i– m+k+–i
i=
(cid:7) (cid:5) (cid:6) (cid:5) (cid:6)(cid:8)
(cid:2)m+k k k B (y;λ)
+ (–)k+i n –m B (x+y;μ) m+k+–i , (.)
n+i–
i i– m+k+–i
i=
where δ(k,m)=– when k =m=, δ(k,m)= when k =, m≥ or m=, k ≥, and
δ(k,m)=otherwise.
Proof Settingk=andsubstitutingm+kforminTheorem.,wehave
(cid:3) (cid:4)
B (x;μ)+B (y;λμ) n
 m+k
nB (x+y;λμ)
=–δ m+n+k +B (y;λ)B (x+y;μ)
,λ m+k n
m+k+
(cid:5) (cid:6)
nB (y;λ)B (x+y;μ) (cid:2)m+k m+k
+ m+k+ n– +n (–)m+k+i
m+k+ i
i=
B (x;λ–)
×B (x+y;λμ) m+k+–i , (.)
n+i–
m+k+–i
whichisjustthecaseδ(k,m)=–orinTheorem..Next,considerthecaseδ(k,m)=.
WeshalluseinductiononkinTheorem.toproveTheorem..Clearly,thecasek=
inTheorem.gives
(cid:3) (cid:4) (cid:3) (cid:4)
B (x;μ)+B (y;λμ) n+ B (x;μ)+B (y;λμ) n
 m  m+
n+
=–δ B (x+y;λμ)
,λ m+n+
m+
n+
+B (y;λ)B (x+y;μ)+ B (y;λ)B (x+y;μ)
m n+ m+ n
m+
(cid:5) (cid:6)
(cid:2)m m B (x;λ–)
+(n+) (–)m+iB (x+y;λμ) m+–i . (.)
n+i
i m+–i
i=
Notingthatforanynon-negativeintegerk,
(cid:5) (cid:6)
(cid:2)m m B (x;λ–)
(–)m+iB (x+y;λμ) m+–i
n+k+i–
i m+–i
i=
(cid:5) (cid:6)
(cid:2)m+k m B (x;λ–)
= (–)m+k+iB (x+y;λμ) m+k+–i . (.)
n+i–
i–k m+k+–i
i=
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Hence,substituting(.)and(.)to(.),wegetthecasek=ofTheorem..Now
assumethatTheorem.holdsforallpositiveintegersbeinglessthank.From(.)and
(.),weobtain
(cid:3) (cid:4)
B (x;μ)+B (y;λμ) n
k m
(n+k)B (x+y;λμ)
=–δ m+n+k
,λ
m+
(n+k)B (y;λ)B (x+y;μ)
+B (y;λ)B (x+y;μ)+ m+ n+k–
m n+k
m+
(cid:5) (cid:6)
(cid:2)m+k m B (x;λ–)
+(n+k) (–)m+k+iB (x+y;λμ) m+k+–i
n+i–
i–k m+k+–i
i=
(cid:5) (cid:6)
(cid:3) (cid:4) (cid:2)k– (cid:3) (cid:4)
k
– B (x;μ)+B (y;λμ) n– B(x;μ)+B (y;λμ) n. (.)
 m+k j m+k–j
j
j=
Since(.)holdsforallpositiveintegersbeinglessthank,wehave
(cid:5) (cid:6)
(cid:2)k– (cid:3) (cid:4)
k
B(x;μ)+B (y;λμ) n
j m+k–j
j
j=
(cid:3) (cid:4)
(cid:5) (cid:6)
(m+n+k+)B (x+y;λμ)(cid:2)k– k (cid:2)m+k (cid:2)k– k
=–δ m+n+k (cid:3) j (cid:4) + (–)m+i
,λ m+k+ m+k j
j= j i= j=
(cid:7) (cid:5) (cid:6) (cid:5) (cid:6)(cid:8)
m+k–j m+k–j B (x;λ–)
×(–)k–j n –j B (x+y;λμ) m+k+–i
n+i–
i i– m+k+–i
(cid:5) (cid:6) (cid:7) (cid:5)(cid:6) (cid:5) (cid:6)(cid:8)
(cid:2)m+k (cid:2)k–
k j j
+ (–)k+i (–)k–j n –(m+k–j)
j i i–
i= j=
B (y;λ)
×B (x+y;μ) m+k+–i . (.)
n+i–
m+k+–i
Foranynon-negativeintegersi,k,m,
(cid:3) (cid:4)
(cid:2)k k
m+k+
(cid:3) j (cid:4) = ,
m+k m+
j= j
(.)
(cid:5) (cid:6)(cid:5) (cid:6) (cid:5) (cid:6)
(cid:2)k
k m+j m
(–)j =(–)k .
j i i–k
j=
Itfollowsfrom(.)that
(cid:3) (cid:4)
(cid:2)k– k
k k!m!
(cid:3) j (cid:4) = – , (.)
m+k m+ (m+k)!
j= j
(cid:5) (cid:6)(cid:5) (cid:6) (cid:7)(cid:5) (cid:6) (cid:5) (cid:6)(cid:8) (cid:5) (cid:6)
(cid:2)k–
k m+k–j m m+k m
(–)k–j =(–)k – – , (.)
j i i–k i i
j=
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(cid:5) (cid:6)(cid:5) (cid:6) (cid:5) (cid:6) (cid:5) (cid:6)
(cid:2)k–
k m+k–j m m
(–)k–jj =(–)k–k –k , (.)
j i– i–k i–
j=
(cid:5) (cid:6)(cid:5)(cid:6) (cid:5) (cid:6) (cid:5) (cid:6) (cid:5) (cid:6)
(cid:2)k–
k j   k
(–)k–j = –(–)k – , (.)
j i i–k i i
j=
(cid:5) (cid:6)(cid:5) (cid:6)
(cid:2)k–
k j
(–)k–j(m+k–j)
j i–
j=
(cid:5) (cid:6) (cid:5) (cid:6) (cid:5) (cid:6) (cid:5) (cid:6)
k   
=–m –(–)k(m+k) +m –k . (.)
i– i– i–k– i–k
Bysubstitutingtheaboveidentities(.)-(.)to(.),weget
(cid:5) (cid:6)
(cid:2)k– (cid:3) (cid:4)
k
B(x;μ)+B (y;λμ) n
j m+k–j
j
j=
(cid:5) (cid:6)
(m+n+k+)B (x+y;λμ) k k!m!
=–δ m+n+k –
,λ
m+k+ m+ (m+k)!
(cid:7) (cid:5) (cid:6) (cid:5) (cid:6) (cid:5) (cid:6) (cid:5) (cid:6)(cid:8)
(cid:2)m+k
m m m+k m
– (–)m+i n –k +(–)kn –(–)k(n+k)
i i– i i–k
i=
B (x;λ–)
×B (x+y;λμ) m+k+–i
n+i–
m+k+–i
(cid:7) (cid:5) (cid:6) (cid:5) (cid:6)(cid:8)
(cid:2)m+k k k B (y;λ)
– (–)k+i n –m B (x+y;μ) m+k+–i
n+i–
i i– m+k+–i
i=
n+k nB (y;λ)
+ B (y;λ)B (x+y;μ)– m+k+ B (x+y;μ)
m+ n+k– n–
m+ m+k+
–B (y;λ)B (x+y;μ)+B (y;λ)B (x+y;μ). (.)
m+k n m n+k
Thus,putting(.)and(.)to(.)concludestheinductionstep.Thiscompletesthe
proofofTheorem.. (cid:2)
Obviously,thecasesλ=μ=andx=y=inTheorem.leadtotheAgoh-Dilcher
quadraticrecurrenceformula(.).WenowuseTheorem.togivethefollowingformula
onproductsoftheApostol-Bernoullipolynomials.
Corollary. Letmandnbepositiveintegers.Then
(cid:5) (cid:6)
(cid:2)m m (cid:3) (cid:4)B (y;λμ)
B (x;λ)B (y;μ)=n (–)m–iB y–x;λ– n+i
m n m–i
i n+i
i=
(cid:5) (cid:6)
(cid:2)n n B (x;λμ)
+m B (y–x;μ) m+i
n–i
i m+i
i=
m!n!
–(–)mδ B (y–x;μ). (.)
,λμ m+n
(m+n)!
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Proof Takingn=andthensubstitutingmfork,nform,λforμandμforλinTheo-
rem.,weobtainthatforpositiveintegersmandn,
(cid:5) (cid:6)
(cid:2)m m B (y;μ)
B (x;λ)B (y;λμ)=n (–)iB (x+y;λ) n+i
m n m–i
i n+i
i=
(cid:5) (cid:6)
(cid:2)n n B (x;μ–)
+m (–)iB (x+y;λμ) m+i
n–i
i m+i
i=
δ m!n!
– ,μ B (x+y;λμ). (.)
m+n
(m+n)!
Substituting –y fory and μ– for λμ in(.)andusingthecomplementaryaddition
theoremforApostol-Bernoullipolynomials,wegetourresult. (cid:2)
Remark. Thecasesλ=μ=andx=yinCorollary.togetherwiththefactB =–/

andB =forapositiveintegern(see,e.g.,[])willyieldthatforpositiveintegersm
n+
andn,see[,],
[(cid:2)m+n](cid:7) (cid:5) (cid:6) (cid:5) (cid:6)(cid:8)
 m n B (x) m!n!
B (x)B (x)= n +m B m+n–i +(–)m+ B , (.)
m n i m+n
i i m+n–i (m+n)!
i=
where[x]isthemaximumintegerlessthanorequaltotherealnumberx.
4 RecurrenceformulaeformixedApostol-BernoulliandApostol-Euler
polynomials
Inthissection,weshallusetheabovemethodstogivesomequadraticrecurrenceformu-
laeformixedApostol-BernoulliandApostol-Eulerpolynomials.AsintheproofofTheo-
rem.,weneedthefollowingsummationformulaconcerningthequadraticrecurrence
ofmixedApostol-BernoulliandApostol-Eulerpolynomials.
Theorem. Letk,m,nbenon-negativeintegers.Then
(cid:5) (cid:6)
(cid:2)n (cid:3) (cid:4)
n
E(x;μ)+B (y;λμ) k
i m+n–i
i
i=
(cid:5) (cid:6)
(cid:2)m (cid:3) (cid:4)
m
= (–)m+iE x;λ– B (x+y;λμ)
m–i n+k+i
i
i=
(cid:9) (cid:10)

– mE (y;λ)E (x+y;μ)+(n+k)E (y;λ)E (x+y;μ) . (.)
m– n+k m n+k–

Proof Multiplyingbothsidesoftheidentity
(cid:5) (cid:6)
  λeu  
= – (.)
λeu+μev+ λeu+ μev+ λμeu+v–
by(u+v)exv+y(u+v),wehave
exv (u+v)ey(u+v) e(–x)u(u+v)e(x+y)(u+v) u+v eyu e(x+y)v
=λ – . (.)
μev+ λμeu+v– λeu+ λμeu+v–  λeu+μev+
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Makingk-timesderivativefortheaboveidentity(.)withrespecttov,withthehelpof
theLeibnizrule,weget
(cid:5) (cid:6) (cid:5) (cid:6) (cid:5) (cid:6)
(cid:2)k k ∂j exv ∂k–j (u+v)ey(u+v)
j ∂vj μev+ ∂vk–j λμeu+v–
j=
(cid:5) (cid:6)
e(–x)u ∂k (u+v)e(x+y)(u+v)
=λ
λeu+∂vk λμeu+v–
(cid:5) (cid:6) (cid:5) (cid:6)
u+v eyu ∂k e(x+y)v k eyu ∂k– e(x+y)v
– – . (.)
 λeu+∂vk μev+ λeu+∂vk– μev+
Applying(.)and(.)to(.),inviewoftheCauchyproductandthecomplementary
additiontheoremfortheApostol-Eulerpolynomials,weobtain
(cid:11) (cid:5) (cid:6) (cid:5) (cid:6) (cid:12)
(cid:2)∞ (cid:2)∞ (cid:2)n n (cid:2)k k um vn
E (x;μ)B (y;λμ) ·
i+j m+n+k–i–j
i j m! n!
m=n= i= j=
(cid:11) (cid:5) (cid:6) (cid:12)
(cid:2)∞ (cid:2)∞ (cid:2)m m (cid:3) (cid:4) umvn
= (–)m+iE x;λ– B (x+y;λμ)
m–i n+k+i
i m! n!
m=n= i=
(cid:2)∞ (cid:2)∞ E (y;λ)E (x+y;μ)um+vn (cid:2)∞ (cid:2)∞ E (y;λ)E (x+y;μ)umvn+
m n+k m n+k
– –
 m! n!  m! n!
m=n= m=n=
(cid:2)∞ (cid:2)∞
k umvn
– E (y;λ)E (x+y;μ) . (.)
m n+k–
 m! n!
m=n=
Thus,bycomparingthecoefficientsofum+vn+/(m+)!(n+)!andum+/(m+)!in(.),
wecompletetheproofofTheorem.. (cid:2)
WenowgiveaspecialcaseofTheorem..Wehavethefollowing
Corollary. Letmandnbepositiveintegers.Then
(cid:5) (cid:6)
(cid:2)m (cid:3) (cid:4)
m m
(–)m–iE y–x;λ– B (y;λμ)– E (x;λ)E (y;μ)
m–i n+i m– n
i 
i=
(cid:5) (cid:6)
(cid:2)n
n n
= E (y–x;μ)B (x;λμ)– E (y;μ)E (x;λ). (.)
n–i m+i n– m
i 
i=
Proof Settingk=andsubstitutingyforxandxforyinTheorem.,weobtainthatfor
positiveintegersmandn,
(cid:5) (cid:6)
(cid:2)n
n
E (y;μ)B (x;λμ)
n–i m+i
i
i=
(cid:5) (cid:6)
(cid:2)m (cid:3) (cid:4)
m
= (–)m–iE y;λ– B (x+y;λμ)
m–i n+i
i
i=
(cid:9) (cid:10)

– mE (x;λ)E (x+y;μ)+nE (x+y;μ)E (x;λ) . (.)
m– n n– m

Thus,replacingybyy–xintheabovegivesthedesiredresult. (cid:2)