Some Results for the Apostol-Genocchi Polynomials of Higher Order

16 Pages·2013·0.17 MB·English

BULLETINofthe Bull.Malays.Math.Sci.Soc.(2)36(2)(2013),465–479
MALAYSIANMATHEMATICAL
SCIENCESSOCIETY
http://math.usm.my/bulletin
SomeResultsfortheApostol-GenocchiPolynomialsofHigherOrder
1HASSANJOLANY,2HESAMSHARIFIAND3R.EIZADIALIKELAYE
1SchoolofMathematics,StatisticsandComputerScience,UniversityofTehran,Iran
2DepartmentofMathematics,FacultyofScience,UniversityofShahed,Tehran,Iran
3FacultyofManagementandAccounting,QazvinIslamicAzadUniversity,Qazvin,Iran
[email protected],2hsharifi@shahed.ac.ir,[email protected]
Abstract. ThepresentpaperdealswithmultiplicationformulasfortheApostol-Genocchi
polynomialsofhigherorderanddeducessomeexplicitrecursiveformulas. Someearlier
resultsofCarlitzandHowardintermsofGenocchinumberscanbededuced.Weintroduce
the2-variableApostol-Genocchipolynomialsandthenweconsiderthemultiplicationtheo-
remfor2-variableGenocchipolynomials.AlsoweintroducegeneralizedApostol-Genocchi
polynomialswitha,b,cparametersandweobtainseveralidentitiesongeneralizedApostol-
Genocchipolynomialswitha,b,cparameters.
2010MathematicsSubjectClassification:11B68,05A10,05A15
Keywordsandphrases:Apostol-Genocchinumbersandpolynomials(ofhigherorder),gen-
eralization of Genocchi numbers and polynomials, Raabe’s multiplication formula, mul-
tiplicationformula,Bernoullinumbersandpolynomials,Eulernumbersandpolynomials,
Stirlingnumbers.
1. Preliminariesandmotivation
TheclassicalGenocchinumberscanbedefinedinanumberofways. Thewayinwhichit
is defined is often determined by which sorts of applications they are intended to be used
for. TheGenocchinumbershavewide-rangingapplicationsfromnumbertheoryandCom-
binatorics to numerical analysis and other fields of applied mathematics. There exist two
important definitions of the Genocchi numbers: the generating function definition, which
is the most commonly used definition, and a Pascal-type triangle definition, first given by
PhilipLudwigvonSeidel,anddiscussedin[38]. Assuch,itmakesitveryappealingforuse
incombinatorialapplications. Theideabehindthisdefinition, asinPascal’striangle, isto
utilizearecursiverelationshipgivingsomeinitialconditionstogeneratetheGenocchinum-
bers. The combinatorics of the Genocchi numbers were developed by Dumont in [8] and
variousco-authorsinthe70sand80s.DumontandFoataintroducedin1976athree-variable
symmetricrefinementofGenocchinumbers,whichsatisfiesasimplerecurrencerelation. A
six-variable generalization with many similar properties was later considered by Dumont.
CommunicatedbyLeeSeeKeong.
Received:September20,2010;Revised:June16,2011.
466 H.Jolany,H.SharifiandR.E.Alikelaye
In[13],Jangetal. definedanewgeneralizationofGenocchinumbers,polyGenocchinum-
bers. Kim in [14] gave a new concept for the q-extension of Genocchi numbers and gave
somerelationsbetweenq-Genocchipolynomialsandq-Eulernumbers. In[36], Simseket
al. investigatedtheq-GenocchizetafunctionandL-functionbyusinggeneratingfunctions
andMellintransformation. Genocchinumbersareknowntocountalargevarietyofcom-
binatorial objects, among which numerous sets of permutations. One of the applications
of Genocchi numbers that was investigated by Jeff Remmel in [29] is counting the num-
ber of up-down ascent sequences. Another application of Genocchi numbers is in Graph
Theory. Forinstance,BooleannumbersoftheassociatedFerrersGraphsaretheGenocchi
numbers of the second kind [5]. A third application of Genocchi numbers is in Automata
Theory. One of the generalizations of Genocchi numbers that was first proposed by Han
in [7] proves useful in enumerating the class of deterministic finite automata (DFA) that
accept a finite language and in enumerating a generalization of permutations counted by
Dumont. Recently S. Herrmann in [10], presented a relation between the f-vector of the
boundaryandtheinteriorofasimplicialballdirectlyintermsofthe f-vectors. Themost
interesting point about this equation is the occurrence of the Genocchi numbers G . In
2n
the last decade, a surprising number of papers appeared proposing new generalizations of
the classical Genocchi polynomials to real and complex variables or treating other topics
relatedtoGenocchipolynomials. Qiu-MingLuoin[25]introducednewgeneralizationsof
Genocchipolynomials,hedefinedtheApostol-Genocchipolynomialsofhigherorderandq-
Apostol-Genocchi polynomials and he obtained a relationship between Apostol-Genocchi
polynomials of higher order and Goyal-Laddha-Hurwitz-Lerch Zeta function. Next Qiu-
MingLuoandH.M.Srivastavain[27]byApostol-Genocchipolynomialsofhigherorder
derivedvariousexplicitseriesrepresentationsintermsoftheGaussianhypergeometricfunc-
tion and the Hurwitz (or generalized) zeta function which yields a deeper insight into the
effectiveness of this type of generalization. Also it is clear that Apostol-Genocchi poly-
nomials of higher order are in a class of orthogonal polynomials and we know that most
suchspecialfunctionsthatareorthogonalaresatisfiedinmultiplicationtheorem,sointhis
present paper we show this property is true for Apostol-Genocchi polynomials of higher
order.
The study of Genocchi numbers and their combinatorial relations has received much
attention [2,8,10,14,17,19,25,30,31,34,35,38]. In this paper we consider some com-
binatorial relationships of the Apostol-Genocchi numbers of higher order. The unsigned
Genocchinumbers{G2n}n(cid:62)1canbedefinedthroughtheirgeneratingfunction:
∞ x2n (cid:16)x(cid:17)
∑G =x.tan
2n
(2n)! 2
n=1
andalso
t2n (cid:16)t(cid:17)
∑(−1)nG =−ttanh
2n
(2n)! 2
n(cid:62)1
So,bysimplecomputation
(cid:16)t(cid:17) (cid:0)t(cid:1)2s+1 (cid:0)t(cid:1)2m (−1)m E t2m+2s+1
tanh = ∑ 2 . ∑(−1)mE 2 = ∑ 2m
2 (2s+1)! 2m(2m)! 22m+2s+1(2m)!(2s+1)!
s(cid:62)0 m(cid:62)0 s,m(cid:62)0
SomeResultsfortheApostol-GenocchiPolynomials 467
n−1(cid:18)2n−1(cid:19)(−1)mE t2n−1
= ∑ ∑ 2m ,
2m 22n−1(2n−1)!
n(cid:62)1m=0
weobtainforn(cid:62)1,
n−1 (cid:18)2n(cid:19) E
G = ∑(−1)n−k−1(n−k) 2k
2n 2k 22n−2
k=0
whereE areEulernumbers. AlsotheGenocchinumbersG aredefinedbythegenerating
k n
function
2t ∞ tn
G(t)= = ∑G , (|t|<π).
et+1 nn!
n=0
Ingeneral,itsatisfiesG =0,G =1,G =G =G =...G =0,andevencoefficients
0 1 3 5 7 2n+1
aregivenG =2(1−22n)B =2nE ,whereB areBernoullinumbersandE areEuler
2n 2n 2n−1 n n
numbers. The first few Genocchi numbers for even integers are -1, 1, -3, 17, -155, 2073,
....ThefirstfewprimeGenocchinumbersare-3and17,whichoccuratn=6and8.There
arenootherswithn<105. Forx∈R,weconsidertheGenocchipolynomialsasfollows
2t ∞ tn
G(x,t)=G(t)ext = ext = ∑G (x) .
et+1 n n!
n=0
Inspecialcasex=0,wedefineG (0)=G . Becausewehave
n n
n (cid:18)n(cid:19)
G (x)= ∑ G xn−k,
n k
k
k=0
ItiseasytodeducethatG (x)arepolynomialsofdegreek. Here, wepresentsomeofthe
k
firstGenocchi’spolynomials:
G (x)=1, G (x)=2x−1, G (x)=3x2−3x, G (x)=4x3−6x2+1,
1 2 3 4
G (x)=5x4−10x3+5x, G (x)=6x5−15x4+15x2−3, ...
5 6
The classical Bernoulli polynomials (of higher order) B(α)(x) and Euler polynomials (of
n
higher order) E(α)(x),(α ∈C), are usually defined by means of the following generating
n
functions[15,16,19,21,28,32,33]
(cid:16) z (cid:17)α ∞ zn
exz= ∑B(α)(x) , (|z|<2π)
ez−1 n n!
n=0
and
(cid:16) 2 (cid:17)α ∞ zn
exz= ∑E(α)(x) , (|z|<π)
ez+1 n n!
n=0
Sothat,obviously,
B (x):=B1(x) and E (x):=E(1)(x).
n n n n
In2002,Q.M.Luoetal.(see[9,23,24])definedthegeneralizationofBernoullipolynomials
andEulernumbers,asfollows
tcxt ∞ B (x;a,b,c) b
= ∑ n tn, (|tln |<2π)
bt−at n! a
n=0
2 ∞ tn b
= ∑E (a,b) , (|tln |<π).
bt+at n n! a
n=0
468 H.Jolany,H.SharifiandR.E.Alikelaye
Here,wegiveananalogousdefinitionforgeneralizedApostol-Genocchipolynomials.
Leta,b>0,TheGeneralizedApostol-GenocchiNumbersandApostol-Genocchipoly-
nomialswitha,b,cparametersaredefinedby
2t ∞ tn
= ∑G (a,b;λ)
λbt+at n n!
n=0
2t ∞ tn
ext = ∑G (x,a,b;λ)
λbt+at n n!
n=0
2t ∞ tn
cxt = ∑G (x,a,b,c;λ)
λbt+at n n!
n=0
respectively.
Forarealorcomplexparameterα, TheApostol-Genocchipolynomialswitha,b,cpa-
rametersoforderα,G(α)(x;a,b;λ),eachofdegreenisxaswellasinα,aredefinedbythe
n
followinggeneratingfunctions
(cid:16) 2t (cid:17)α ∞ tn
exz= ∑G(α)(x,a,b;λ) ,
λbt+at n n!
n=0
(1)
Clearly,wehaveG (x,a,b;λ)=G (x;a,b;λ).
n n
Now, weintroducethe2-variableApostol-Genocchipolynomialsandthenweconsider
themultiplicationtheoremfor2-variableApostol-GenocchiPolynomials. Westartwiththe
definitionofApostol-GenocchipolynomialsG (x;λ). TheApostol-GenocchiPolynomials
n
G (x;λ)invariablexaredefinedbymeansofthegeneratingfunction
n
2zexz ∞ zn
= ∑G (x;λ) (|z|<2π whenλ =1,|z|<|logλ|whenλ (cid:54)=1),
λez+1 n n!
n=0
with,ofcourse,
G (λ):=G (0;λ),
n n
WhereG (λ)denotestheso-calledApostol-Genocchinumbers.
n
Also(see[1,16,20,22,25,26,32])Apostol-GenocchiPolynomialsG(α)(x;λ)oforderα
n
invariablexaredefinedbymeansofthegeneratingfunction:
(cid:18) 2z (cid:19)α ∞ zn
exz= ∑G(α)(x;λ)
λez+1 n n!
n=0
with,ofcourse,G(α)(λ):=Gα(0;λ).WhereGα(λ)denotestheso-calledApostol-Genocchi
n n n
numbersofhigherorder. Ifweset,
(cid:18) 2t (cid:19)α
φ(x,t;α)= ext,
et+1
then,
∂φ
=tφ,
∂x
and,
∂φ (cid:26)α+tx αet (cid:27)∂φ
t − − =0.
∂t t et+1 ∂x
SomeResultsfortheApostol-GenocchiPolynomials 469
Next, we introduce the class of Apostol-Genocchi numbers as follows (for more infor-
mationsee[38]).
[n]
2 n!G (λ)G (λ)
G (λ)= ∑ n−2s s
H n
s!(n−2s)!
s=0
Thegeneratingfunctionof G (λ)isprovidedby
H n
4t3 ∞ tn
= ∑ G (λ)
(λet+1)(λet2+1) H n n!
n=0
andthegeneralizationof G (λ)for(a,b)(cid:54)=0,is
H n
4t3 ∞ tn
= ∑ G (a,b;λ)
(λeat+1)(λebt2+1) H n n!
n=0
where
1 [n2] n!an−2sbsG (λ)G (λ)
G (a,b;λ)= ∑ n−2s s
H n
ab s!(n−2s)!
n=0
The main object of the present paper is to investigate the multiplication formulas for the
Apostol-typepolynomials.
Luoin[22]definedthemultiplealternatingsumsas
(cid:18) (cid:19)
l
Z(l)(m;λ)=(−1)l ∑ (−λ)v1+2v2+...+mvm
k v ,v ,...,v
0≤v1,v2,...,vm≤l 1 2 m
v1+v2+...+vm=(cid:96)
m
Z (m;λ)= ∑(−1)j+1λjjk=λ−λ22k+...+(−1)m+1λmmk
k
j=1
m
Z (m)= ∑(−1)j+1jk=1−2k+...+(−1)m+1mk, (m,k,l∈N ;λ ∈C)
k 0
j=1
whereN :=N∪{0}, (N:={1,2,3,...}).
0
2. ThemultiplicationformulasfortheApostol-Genocchipolynomialsofhigherorder
In this Section, we obtain some interesting new relations and properties associated with
Apostol-Genocchipolynomialsofhigherorderandthenderiveseveralelementaryproper-
tiesincludingrecurrencerelationsforGenocchinumbers. Firstofallweprovethemultipli-
cationtheoremofthesepolynomials.
Theorem 2.1. For m∈N, n∈N , α,λ ∈C, the following multiplication formula of the
0
Apostol-Genocchipolynomialsofhigherorderholdstrue:
(cid:18) α (cid:19) (cid:16) r (cid:17)
(2.1) G(α)(mx;λ)=mn−α ∑ (−λ)rG(α) x+ ;λm
n n
v ,v ,...,v m
v1,v2,...,vm−1≥0 1 2 m−1
wherer=v +2v +...+(m−1)v ,(misodd)
1 2 m−1
Proof. Itiseasytoobservethat
1 1−λet+λ2e2t+...+(−λ)m−1e(m−1)t
(2.2) =−
λet+1 (−λ)memt−1
470 H.Jolany,H.SharifiandR.E.Alikelaye
Butwehave,ifx ∈C
i
(cid:18) (cid:19)
n
(2.3) (x +x +...+x )n= ∑ xa1xa2...xam
1 2 m a ,a ,...,a 1 2 m
a1,a2,...,am(cid:62)0 1 2 m
a1+a2+...am=n
Thelastsummationtakesplaceoverallpositiveorzerointegersa (cid:62)0suchthata +a +
i 1 2
...+a =n,where
m
(cid:18) (cid:19)
n n!
:=
a ,a ,...,a a a !...a !
1 2 m 1! 2 m
Sobyapplying(2.2)onthefollowingfirstequalitysignandsetting(x =1,x =(−λ)kekt
1 k
fork≥2)andn=α in(2.3)onthefollowingsecondequalitysign,weobtain
∞ tn (cid:18) 2t (cid:19)α (cid:18) 2t (cid:19)α(cid:32)m−1 (cid:33)α
∑G(α)(mx;λ) = emxt = ∑(−λ)kekt emxt
n n! λet+1 λmemt+1
n=0 k=0
= ∑ (cid:18) α (cid:19)(−λ)r(cid:18) 2t (cid:19)αe(x+mr)mt
v ,v ,...,v λmemt+1
v1,v2,...,vm−1(cid:62)0 1 2 m−1
∞ (cid:32) (cid:18) α (cid:19) (cid:16) r (cid:17)(cid:33)tn
= ∑ mn−α ∑ (−λ)rG(α) x+ ;λm
n
v ,v ,...,v m n!
n=0 v1,v2,...,vm(cid:62)0 1 2 m
Bycomparingthecoefficientoftn/(n!)onbothsidesoflastequation,proofiscomplete.
In terms of the generalized Apostol-Genocchi polynomials, by setting λ =1 in Theo-
rem2.1, weobtainthefollowingexplicitformulathatiscalledmultiplicationtheoremfor
Genocchipolynomialsofhigherorder.
Corollary2.1. Form∈N,n∈N ,α,∈C,wehave
0
(cid:18) α (cid:19) (cid:16) r(cid:17)
G(α)(mx)=mn−α ∑ (−1)rG(α) x+ (misodd).
n n
v ,v ,...,v m
v1,v2,...,vm−1(cid:62)0 1 2 m−1
AndusingCorollary2.1,(bysettingα =1),wegetCorollary2.2thatisthemainresult
of[37]andiscalledmultiplicationTheoremforGenocchipolynomials.
Corollary2.2. Form∈N,n∈N ,wehave
0
m−1 (cid:18) k(cid:19)
G (mx)=mn−1 ∑(−1)kG x+ (misodd).
n n
m
k=0
Now,weconsiderthemultiplicationformulafortheApostol-Genocchinumberswhenm
iseven.
Theorem2.2. Form∈N(meven),n∈N,α,λ ∈C,thefollowingmultiplicationformula
oftheApostol-Genocchipolynomialsofhigherorderholdstrue:
(cid:18) α (cid:19) (cid:16) r (cid:17)
G(α)(mx;λ)=(−2)αmn−α ∑ (−λ)rB(α) x+ ,λm ,
n n
v ,v ,...,v m
v1,v2,...,vm−1(cid:62)0 1 2 m−1
wherer=v +2v +...+(m−1)v .
1 2 m−1
SomeResultsfortheApostol-GenocchiPolynomials 471
Proof. Itiseasytoobservethat
1 1−λet+λ2e2t+...+(−λ)m−1e(m−1)t
=−
λet+1 (−λ)memt−1
So,weobtain
∞ tn
∑G(α)(mx;λ)
n
n!
n=0
(cid:18) 2t (cid:19)α (cid:18) t (cid:19)α (cid:18) t (cid:19)α(cid:32)m−1 (cid:33)α
= emxt =2α emxt =(−2)α ∑(−λet)k emxt
λet+1 λet+1 λmemt−1
k=0
=(−2)α ∑ (cid:18) α (cid:19)(−λ)r(cid:18) t (cid:19)αe(x+mr)mt
v ,v ,...,v λmem−1
v1,v2,...,vm−1(cid:62)0 1 2 m−1
∞ (cid:32) (cid:18) α (cid:19) (cid:16) r (cid:17)(cid:33)tn
= ∑ (−2)αmn−α ∑ (−λ)r×B(α) x+ ;λm
n
v ,v ,...,v m n!
n=0 v1,v2,...,vm−1(cid:62)0 1 2 m−1
Bycomparingthecoefficientsoftn/(n!)onbothsidesproofwillbecomplete.
Next, usingTheorem2.2, (withλ =1), weobtaintheGenocchipolynomialsofhigher
ordercanbeexpressedbytheBernoullipolynomialsofhigherorderwhenmiseven
Corollary2.3. Form∈N(meven),n∈N ,α ∈C,weget
0
(cid:18) α (cid:19) (cid:16) r(cid:17)
G(α)(mx)=(−2)αmn−α ∑ (−1)rBα x+ .
n v ,v ,...,v n m
v1,v2,...,vm−1(cid:62)0 1 2 m−1
Alsobyapplyingα=1,inCorollary2.3weobtainthefollowingassertionthatisoneof
themostremarkableidentitiesinareaofGenocchipolynomials.
Corollary2.4. Form∈N,n∈N ,weobtain
0
m−1 (cid:18) k(cid:19)
G (mx)=−2mn−1 ∑(−1)kB x+ miseven.
n n
m
k=0
Obviously, the result of Corollary 2.4 is analogous with the well-known Raabe’s mul-
(l) (l)
tiplication formula. Now, we present explicit evaluations of Z (m;λ), Z (λ), Z (m) by
n n n
Apostol-Genocchipolynomials.
Theorem2.3. Form,n,l∈N ,λ ∈C,wehave
0
l (cid:18)l(cid:19)(−1)j(m+1)λmj+l n+l(cid:18)n+l(cid:19)
Z(l)(m;λ)=2−l ∑ ∑ G(j)(mj+l;λ)G(l−j) (λ)
n j (n+1) k k n+l−k
j=0 l k=0
where(n) =1,(n) =n(n+1)...(n+k−1).
0 k
(l)
Proof. BydefinitionofZ (m;λ),wecalculatethefollowingsum
n
∞ tn
∑Z(l)(m;λ)
n
n!
n=0
472 H.Jolany,H.SharifiandR.E.Alikelaye
 
∞ (cid:18) l (cid:19) tn
= ∑(−1)l ∑ v ,v ,...,v (−λ)λ1+2λ2+...+mλm(v1+2v2+...+mvm)nn!
n=0 0(cid:54)v1,v2,...,vm(cid:54)l 1 2 m
v1+v2+...+vm=l
(cid:18) (cid:19)
l
=(−1)l ∑ (−λet)λ1+2λ2+...+mλm
v ,v ,...,v
0(cid:54)v1,v2,...,vm(cid:54)l 1 2 m
v1+v2+...+vm=l
(cid:32) (cid:33)l
=(cid:0)λet−λ2e2t+...+(−1)m+1λmemt(cid:1)l = (−1)m+1λm+1e(m+1)t + λet
λet+1 λet+1
l (cid:18)l(cid:19)(cid:34)2t(−1)m+1λm+1e(m+1)t(cid:35)j(cid:20) 2tλet (cid:21)l−j
=(2t)−l ∑
j λet+1 λet+1
j=0
l (cid:18)l(cid:19) ∞ tn ∞ tn
=(2t)−l ∑ (−1)j(m+1)λmj+l ∑G(j)(mj+l;λ) ∑G(l−j)(λ)
n n
j n! n!
j=0 n=0 n=0
∞ (cid:34) l (cid:18)j(cid:19)(−1)j(m+1)λmj+l n+l(cid:18)n+l(cid:19) (cid:35)tn
=2−l ∑ ∑ ∑ G(j)(mj+l;λ)G(l−j) (λ)
l (n+1) k k n+l−k n!
n=0 j=0 l k=0
bycomparingthecoefficientsoftn/(n!)onbothsides,proofwillbecomplete.
As a direct result, using λ =1 in Theorem 2.3, we derive an explicit representation of
(l)
multiple alternating sums Z (m), in terms of the Genocchi polynomials of higher order.
n
Wealsodeducetheirspecialcasesandapplicationswhichleadtothecorrespondingresults
fortheGenocchipolynomials.
Corollary2.5. Form,n,l∈N ,thefollowingformulaholdstrueintermsoftheGenocchi
0
polynimials
l (cid:18)l(cid:19)(−1)j(m+1) n+l(cid:18)n+l(cid:19)
Z(l)(m)=2−l ∑ ∑ G(j)(mj+l)Gl−j
n j (n+1) k k n+l−k
j=0 l k=0
where(n) =1,(n) =n(n+1)...(n+k−1).
0 k
Next we investigate some of the recursive formulas for the Apostol-Genocchi numbers
of higher order that are analogous to the results of Howard [3,11,12] and we deduce that
theyconstituteausefulspecialcase.
Theorem2.4. Letmbeodd,n,l∈N ,λ ∈C,thenwehave
0
n (cid:18)n(cid:19)
mnG(l)(λm)−mlG(l)(λ)=(−1)l−1∑ mkG(l)(λm)Z(l) (m−1;λ).
n n k k n−k
k=0
Proof. Bytakingx=0,α =lin(2.1),wherer=v +2v +...+(m−1)v weobtain
1 2 m−1
(cid:18) l (cid:19) (cid:16)r (cid:17)
mlG(l)(λ)=mn ∑ (−λ)rG(l) ,λm
n n
v ,v ,...,v m
v1,v2,...,vm−1(cid:62)0 1 2 m−1
Butweknow
n (cid:18)n(cid:19)
G(l)(x;λ)= ∑ G(l)(λ)xn−k
n k k
k=0
SomeResultsfortheApostol-GenocchiPolynomials 473
So,weobtain
(cid:18) l (cid:19) n (cid:18)n(cid:19) (cid:16)r(cid:17)n−k
mlG(l)(λ)=mn ∑ (−λ)r ∑ G(l)(λm)
n v ,v ,...,v k k m
v1,v2,...,vm−1(cid:62)0 1 2 m−1 k=0
n (cid:18)n(cid:19) (cid:18) l (cid:19)
= ∑ mkG(l)(λm) ∑ (−λ)rrn−k
k k v ,v ,...,v
k=0 0(cid:54)v1,v2,...,vm−1(cid:54)l 1 2 m−1
n (cid:18)n(cid:19) (cid:18) l (cid:19)
= ∑ mkG(l)(λm) ∑ (−λ)rrn−k+mnG(l)(λm)
k k v ,v ,...,v n
k=0 0(cid:54)v1,v2,...,vm−1(cid:54)l 1 2 m−1
v1+v2+...vm−1=l
n (cid:18)n(cid:19)
=(−1)l ∑ mkG(l)(λm)Z(l) (m−1;λ)+mnG(l)(λm)
k k n−k n
k=0
Soproofiscomplete.
Furthermore,wederivesomewell-knownresults(see[14])involvingGenocchipolyno-
mials of higher order and Genocchi polynomials which we state here. By setting λ =1,
l=1inTheorem2.4,wegetCorollaries2.6,2.7,respectively.
Corollary2.6. Letmbeodd,n,l∈N ,thenwehave
0
n (cid:18)n(cid:19)
(mn−ml)G(l)=(−1)l−1∑ G(l)Z(l) (m−1).
n k k n−k
k=0
Corollary2.7. Letmbeodd,n∈N ,λ ∈C,thenwehave
0
n (cid:18)n(cid:19)
mnG (λm)−mG (λ)= ∑ mkG (λm)Z (m−1;λ).
n n k n−k
k
k=0
Alsobysettingλ =1inCorollary2.7,wegetthefollowingassertionthatisanalogous
totheformulaofHowardintermsofGenocchinumbers.See[11,12]formoreinformation.
Corollary2.8. Formbeodd,n,l∈N ,λ ∈C,weobtain
0
n (cid:18)n(cid:19)
(mn−m)G = ∑ mkG Z (m−1).
n k n−k
k
k=0
Next,weinvestigatethegeneralizationofHoward’sformulaintermsofApostol-Genocchi
numbers,whenmiseven.
Theorem2.5. Letmbeeven,n,l∈N ,λ ∈C,thefollowingformula
0
n (cid:18)n(cid:19)
mlG(l)(λ)−(−2)lmnB(l)(λm)=2l ∑ mkB(l)(λm)Z(l) (m−1;λ)
n n k k n−k
k=0
holdstrue,wherer=v +2v +...+(m−1)v .
1 2 m−1
Proof. Wehave
(cid:18) l (cid:19) (cid:16)r (cid:17)
G(l)(λ)=(−2)lmn−l ∑ (−λ)rB(l) ,λm
n n
v ,v ,...,v m
v1,v2,...,vm−1(cid:62)0 1 2 m−1
Butweknow
n (cid:18)n(cid:19)
B(l)(x;λ)= ∑ B(l)(λ)xn−k
n k k
k=0
474 H.Jolany,H.SharifiandR.E.Alikelaye
Soweget
(cid:18) l (cid:19) n (cid:18)n(cid:19) (cid:16)r(cid:17)n−k
mlG(l)(λ)=(−2)lmn ∑ (−λ)r ∑ B(l)(λm)
n v ,v ,...,v k k m
v1,v2,...,vm−1(cid:62)0 1 2 m−1 k=0
n (cid:18)n(cid:19) (cid:18) l (cid:19)
=(−2)l ∑ mkB(l)(λm) ∑ (−λ)rrn−k
k k v ,v ,...,v
k=0 v1,v2,...,vm−1(cid:62)0 1 2 m−1
n (cid:18)n(cid:19)
=2l ∑ mkB(l)(λm)Z(l) (m−1;λ)+(−2)lmnB(l)(λm)
k k n−k n
k=0
Soweobtain
n (cid:18)n(cid:19)
mlG(l)(λ)−(−2)lmnB(l)(λm)=2l ∑ mkB(l)(λm)Z(l) (m−1;λ)
n n k k n−k
k=0
Sotheproofiscomplete.
Alsobylettingλ =1inTheorem2.5,weobtainthefollowingassertion.
Corollary2.9. Letmbeeven,n,l∈N ,thenweget
0
n (cid:18)n(cid:19)
mlG(l)−(−2)lmnB(l)=2l ∑ mkB(l)Z(l) (m−1)
n n k n n−k
k=0
HerewepresentarecurrencerelationforApostol-Genocchinumbersofhigherorder.
Theorem2.6. Letn,k(cid:62)1,thenwehave
(cid:16) 2k(cid:17)
(n+1) (n) (n)
G (λ)=2kG (λ)− 2− G (λ)
k k−1 n k
Proof. Let us put G (t;λ)=(2t/(λet+1))n. Then G (t;λ) is the generating function of
n n
(cid:48)
higherorderApostol-Genocchinumbers. ThederivativeG(t;λ)=(d/dt)G (t;λ)isequal
n
to
(cid:18)1 λet (cid:19) n n
n − G (t;λ)= G (t;λ)−nG (t;λ)+ G (t;λ)
t λet+1 n t n n λet+1 n
and
n
(cid:48)
tG (t;λ)=nG (t;λ)−ntG (t;λ)+ G (t)
n n n 2 n+1
soweobtain
G(n)(λ) G(n)(λ) G(n) (λ) nG(n+1)(λ)
k =n k −n k−1 + k
(k−1)! k! (k−1)! 2 k!
fork(cid:62)1. Thisformulacanwrittenas
(cid:18) (cid:19)
2k
(n+1) (n) (n)
G (λ)=2kG (λ)− 2− G (λ)
k k−1 n k
soproofiscomplete.