Generating functions for special polynomials and numbers including Apostol-type and Humbert ...

16 Pages·2017·0.16 MB·English

Generating Functions for Special
Polynomials and Numbers Including
Apostol-type and Humbert-type Polynomials
GulsahOzdemir,YilmazSimsekandGradimirV.Milovanovic´
Abstract. The aim of this paper is to give generating functions and to prove
variouspropertiesforsomenewfamiliesofspecialpolynomialsandnumbers.
Severalinterestingpropertiesofsuchfamiliesandtheirconnectionswithother
polynomialsandnumbersoftheBernoulli,Euler,Apostol-Bernoulli,Apostol-
Euler,GenocchiandFibonaccitypearepresented.Furthermore,theFibonacci
typepolynomialsofhigherorderintwovariablesandanewfamilyofspecial
polynomials(x,y)(cid:55)→G (x,y;k,m,n),includingseveralparicularcases,arein-
d
troduced and studied. Finally, a class of polynomials and corresponding num-
bers,obtainedbyamodificationofthegeneratingfunctionofHumbert’spoly-
nomials,isalsoconsidered.
Mathematics Subject Classification (2010). 05A15, 11B39, 11B68, 11B73,
11B83.
Keywords.Generatingfunction,Fibonaccipolynomials,Humbertpolynomials,
Bernoullipolynomialsandnumbers,Eulerpolynomialsandnumbers,Apostol-
Bernoullipolynomialsandnumbers,Apostol-Eulerpolynomialsandnumbers,
Genocchipolynomials,Stirlingnumbers.
1. Introductionandpreliminaries
The special polynomials and numbers play an important role in many branches of
mathematicsandtheirdevelopmentisalwaysactual.Manypapersandbookswere
publishedinthisverywidearea.Wementiononlyafewbooksconnectedwithour
resultsinthiswork(cf.[4],[7],[27],[28]).
In this paper we consider some new families of numbers and polynomials,
includingtheirgeneratingfunctions,severalinterestingproperties,aswellastheir
ThesecondauthorwassupportedbytheResearchFundoftheAkdenizUniversity.Thethirdauthorwas
supportedinpartbytheSerbianAcademyofSciencesandArts(No.Φ-96)andbytheSerbianMinistry
ofEducation,ScienceandTechnologicalDevelopment(No.#OI174015).
2 G.Ozdemir,Y.SimsekandG.V.Milovanovic´
connectionswithotherpolynomialsandnumbersoftheBernoulli,Euler,Apostol-
Bernoulli,Apostol-Euler,Genocchi,FibonacciandLucastype.Inordertogiveour
results,weneedtomentionseveralspecialclassesofpolynomialsandnumberswith
theirgeneratingfunctions.
1◦TheBernoullipolynomialsofhigherorderB(h)(x)aredefinedbymeansof
d
thefollowinggeneratingfunction
(cid:18) text (cid:19)h ∞ td
F (x,t;h)= = ∑B(h)(x) . (1.1)
Bh et−1 d d!
d=0
Forh=1,(1.1)reducestothegeneratingfunctionoftheclassicalBernoullipolyno-
(1)
mials,B (x)=B (x).Furthermore,forx=0,thisgivesthewellknownBernoulli
d d
numbersB =B (0).Fordetailssee[1]–[7],[13]–[22],[29].
d d
2◦TheApostol-Bernoullipolynomialswereintroducedin1951byApostol[1]
bymeansofthefollowinggeneratingfunction
text ∞ td
F (x,t;λ)= = ∑B (x,λ) , (1.2)
AB λet−1 d d!
d=0
where|t+logλ|<2π (fordetailssee[1]–[7],[13]–[22],[29]).Severaltheirinter-
esting properties, formulas and extensions have been obtained by Srivastava [26]
(seealsotherecentbook[27]).Usingthesuitablegeneratingfunctionsseveralau-
thorshaveobtaineddifferentgeneralizationsandunificationsofthesenumbersand
polynomials(cf.[2],[5],[13],[14],[16],[17],[22],[29]).
Substitutingx=0in(1.2),forλ (cid:54)=1,wegettheApostol-Bernoullinumbers
B (λ),
d
B (λ)=B (0,λ), (1.3)
d d
and they can be expressed it terms of Stirling numbers of the second kind [1,
Eq.(3.7)].Settingλ=1in(1.2),wegettheclassicalBernoullipolynomialsB (x)=
d
B (x,1).
d
Alternatively,theApostol-Bernoullinumberscanbeexpressedintheform
λϕ (λ)
B (λ)=0, B (λ)=(−1)d−1d d−2 , d≥1, (1.4)
0 d (λ−1)d
whereϕ (λ)aremonicpolynomialsinλ andofdegreekandϕ (0)=1.Usingthe
k k
generatingfunction(1.2)forx=0and(1.4),itiseasytoprovethatthepolynomials
ϕ (λ) are self-inversive (cf. [20, pp. 16–18]), i.e., λkϕ (1/λ)≡ϕ (λ). Also, we
k k k
canprovethat
k (cid:18)k+1(cid:19)
ϕ (λ)=(1−λ)k+λ ∑ (1−λ)ν−1ϕ (λ), k≥0, (1.5)
k k−ν
ν
ν=1
GeneratingFunctionsforSpecialPolynomialsandNumbers 3
aswellasthefollowingdeterminantform
(cid:12)(cid:12) −1/λ 0 0 ··· 0 1 (cid:12)(cid:12)
(cid:12) (cid:12)
(cid:12)(cid:12) (cid:0)2(cid:1) −1/λ 0 ··· 0 ξ (cid:12)(cid:12)
(cid:12) 1 (cid:12)
(cid:12)(cid:12) (cid:0)3(cid:1)ξ (cid:0)3(cid:1) −1/λ ··· 0 ξ2 (cid:12)(cid:12)
ϕk(λ)=(−1)kλk(cid:12)(cid:12)(cid:12)(cid:12) 1... 2... ... ... ... ... (cid:12)(cid:12)(cid:12)(cid:12),
(cid:12) (cid:12)
(cid:12)(cid:12) (cid:0)k(cid:1)ξk−2 (cid:0)k(cid:1)ξk−3 (cid:0)k(cid:1)ξk−4 ··· −1/λ ξk−1 (cid:12)(cid:12)
(cid:12) 1 2 3 (cid:12)
(cid:12)(cid:12) (cid:0)k+1(cid:1)ξk−1 (cid:0)k+1(cid:1)ξk−2 (cid:0)k+1(cid:1)ξk−3 ··· (cid:0)k+1(cid:1) ξk (cid:12)(cid:12)
1 2 3 k
whereξ =1−λ.Forexample,wehave
ϕ (λ)=1, ϕ (λ)=λ+1, ϕ (λ)=λ2+4λ+1,
0 1 2
ϕ (λ)=λ3+11λ2+11λ+1, ϕ (λ)=λ4+26λ3+66λ2+26λ+1,
3 4
ϕ (λ)=λ5+57λ4+302λ3+302λ2+57λ+1,
5
ϕ (λ)=λ6+120λ5+1191λ4+2416λ3+1191λ2+120λ+1,
6
etc.Using(1.5)wecanconcludethatϕ (1)=(k+1)!.
k
3◦ The Apostol-Euler polynomials of the first kind E (x,λ) are defined by
d
meansofthegeneratingfunction
2ext ∞ td
F (x,t;λ)= = ∑E (x,λ) , (1.6)
AE λet+1 d d!
d=0
where |2t+logλ|<π (cf. [1]–[7], [22], [29]). For λ (cid:54)=1, substituting x=1/2 in
(1.6)andmakingsomearrangement,weobtaintheApostol-Eulernumbers.Setting
λ =1in(1.6),wegetthefirstkindEulerpolynomialsE (x)=E (x,1).
d d
4◦ TheApostol-Eulerpolynomialsofthesecondkindaredefinedbymeansof
thegeneratingfunction
2 ∞ td
etx= ∑E∗(x,λ) (1.7)
λet+λ−1e−t d d!
d=0
(cf. [25]). A special kind of these polynomials for λ =1 are denoted by E∗(x)=
d
E∗(x,1),andthecorrespondingnumbersbyE∗=E∗(0).Byusing(1.6)and(1.7),
d d d
forx=0,wehavethefollowingrelation
(cid:18) (cid:19)
1
E∗(0,λ)=2dλE ,λ2 .
d d 2
ThesecondkindEulernumbersE∗ aredefinedbythespecialcaseofthefirst
d
kindEulerpolynomials,E∗=2dE (1/2).
d d
5◦TheEulerpolynomialsofhigherorderE(h)(x)aredefinedbymeansofthe
d
followinggeneratingfunction
(cid:18) 2ext (cid:19)h ∞ td
F (x,t;h)= = ∑E(h)(x) , (1.8)
Eh et+1 d d!
d=0
4 G.Ozdemir,Y.SimsekandG.V.Milovanovic´
(1)
sothat,obviously,E (x)=E (x).
d d
6◦ The Genocchi numbers and polynomials and their generalizations. The
GenocchinumbersG aredefinedbythegeneratingfunction
d
2t ∞ td
F (t)= = ∑G , (1.9)
g et+1 dd!
d=0
where|t|<π (cf.[13],[16],[22],[29]).
In general, for these numbers we have G =0, G =1, and G =0 for
0 1 2d+1
d∈N.SomerelationsbetweentheGenocchi,BernoulliandEulernumbersaregiven
byG =2(cid:0)1−22d(cid:1)B andG =2dE .ThesequenceofGenocchinumbersis
2d 2d 2d 2d−1
{g } ={0,1,−1,0,1,0,−3,0,17,0,−155,0,...}.
d d≥0
The Genocchi polynomials G (x) are defined by the following generating
d
function
∞ td
F (x;t)=F (t)ext = ∑G (x) , (1.10)
g g d
d!
d=0
where|t|<π.Using(1.10),itiseasytoseethat
d (cid:18)d(cid:19)
G (x)= ∑ G xd−k
d k
k
k=0
ThefirstsevenGenocchipolynomialsare
G (x)=0, G (x)=1, G (x)=2x−1, G (x)=3x2−3x,
0 1 2 3
G (x)=4x3−6x2+1, G (x)=5x4−10x3+5x,
4 5
G (x)=6x5−15x4+15x2−3.
6
The Apostol-Genocchi polynomials g (x,λ) are defined by the generating
d
function
2t ∞ td
ext = ∑G (x,λ) , (1.11)
λet+1 d d!
d=0
where |2t+logλ| < π. Setting λ = 1 in (1.11), we get the classical Genocchi
polynomials G (x) = G (x,1), which reduce to the classical Genocchi numbers
d d
G =G (0)forx=0.
d d
Substitutingx=0in(1.11),forλ (cid:54)=1,weobtaintheApostol-Genocchinum-
bers G (λ)=G (0,λ). For some details, properties and other generalizations see
d d
[11],[13],[16],[22],[26],[27],[29].
7◦TheStirlingnumbersofthesecondkindS (n,k;λ)aredefinedbymeansof
2
thefollowinggeneratingfunction(cf.[3],[24],[26]):
(λet−1)k ∞ tn
F (t,k;λ)= = ∑S (n,k;λ) , (1.12)
S 2
k! n!
n=0
wherek∈N andλ ∈C.
0
GeneratingFunctionsforSpecialPolynomialsandNumbers 5
ThegeneralizedStirlingnumbersandpolynomialshavebeendefinedbymeans
ofthefollowinggeneratingfunction(cf.[3]):
(et−1)k ∞ tn
etα = ∑S(α)(n,k) . (1.13)
k! n!
n=0
Severalcombinatorialpropertiesofthesepolynomialshavebeenprovedin[3].
Simsek[24]hasmodifiedthegeneratingfunction(1.13),definingtheso-called
λ-arraypolynomialsSn(x;λ)bymeansofthefollowinggeneratingfunction
k
(λet−1)k ∞ tn
F (t,x,k;λ)= etx= ∑Sn(x;λ) , (1.14)
A k! k n!
n=0
wherek∈N andλ ∈C.Substitutingλ =1,theλ-arraypolynomialsreducetothe
0
arraypolynomials,S(α)(n,k)=Sn(α;1)(cf.[3],[24]).
k
8◦ The Humbert polynomials (cid:8)Πλ (cid:9)∞ were defined in 1921 by Humbert
n,m n=0
[12].Theirgeneratingfunctionis
∞
(1−mxt+tm)−λ = ∑Πλ (x)tn. (1.15)
n,m
n=0
Thisfunctionsatisfiesthefollowingrecurrencerelation(cf.[7],[18],[19]andref-
erencestherein):
(n+1)Πλ (x)−mx(n+λ)Πλ (x)−(n+mλ−m+1)Πλ (x)=0.
n+1,m n,m n−m+1,m
AspecialcaseofthesepolynomialsaretheGegenbauerpolynomialsgivenas
follows[8]:
Cλ(x)=Πλ (x)
n n,2
andalsothePincherlepolynomialsgivenasfollows(see[23],[12]):
P (x)=Π−1/2(x).
n n,3
Later,Gould[9]studiedaclassofgeneralizedHumbertpolynomials,P (m,x,y,p,C),
n
definedby
∞
(C−mxt+ytm)p= ∑P (m,x,y,p,C)tn,
n
n=0
wherem≥1isanintegerandtheotherparametersareunrestrictedingeneral(cf.
[7],[10]).
SomespecialcasesofthegeneralizedHumbertpolynomials,P (m,x,y,p,C),
n
canbegivenasfollows(cf.[12]):
P (cid:0)2,x,1,−1,1(cid:1) = P (x) Legendre(1784),
n 2 n
P (2,x,1,−ν,1) = Cν(x) Geganbauer(1874),
n n
P (cid:0)3,x,1,−1,1(cid:1) = P (x) Pincherle(1890),
n 2 n
P (m,x,1,−ν,1) = hν (x) Humbert(1921).
n n,m
6 G.Ozdemir,Y.SimsekandG.V.Milovanovic´
9◦ The Fibonacci type polynomials in two variables (x,y)(cid:55)→G (x,y;k,m,n)
j
hasbeenrecentlydefinedbyOzdemirandSimsek[21]bythefollowinggenerating
function
∞ 1
H(t;x,y;k,m,n)= ∑G (x,y;k,m,n)tj= , (1.16)
j 1−xkt−ymtm+n
j=0
where k,m,n∈N . An explicit formula for the polynomials G (x,y;k,m,n), j =
0 j
0,1,...,canbedoneinthefollowingform[21]
(cid:104) j (cid:105)
m+n (cid:18)j−c(m+n−1)(cid:19)
G (x,y;k,m,n)= ∑ ymcxjk−mck−nck,
j
c
c=0
where[a]isthelargestinteger≤a.
Inthispaperwegivesomenewidentitiesforthepreviousclassesofpolyno-
mialsandinvestigatesomenewpropertiesofthesepolynomials.Moreover,byusing
theirgeneratingfunctions,wegivesomeapplicationswhichareassociatedwiththe
Fibonaccitypepolynomialsofhigherorderintwovariables.
Thepaperisorganizedasfollows.Fibonaccitypepolynomialsofhigherorder
intwovariablesandanewfamilyofspecialpolynomials(x,y)(cid:55)→G (x,y;k,m,n)
d
areintroducedandstudiedinSections2and3,respectively.Specialcasesofpoly-
nomialsG (x,y;k,m,n)areinvestigatedinSection4.Finally,Section5isdevoted
d
toaclassofpolynomialsandcorrespondingnumbers,obtainedbyamodificationof
thegeneratingfunctionofHumbert’spolynomials.
2. Fibonaccitypepolynomialsofhigherorderintwovariables
In this section we give a new generalization of the Fibonacci type polynomials in
twovariables.
Definition2.1. TwovariableFibonaccitypepolynomialsofhigherorder(x,y)(cid:55)→
G(h)(x,y;k,m,n)aredefinedbythefollowinggeneratingfunction
j
∞ 1
∑G(h)(x,y;k,m,n)tj= (2.1)
j (1−xkt−ymtn+m)h
j=0
wherehisapositiveinteger.
Observethat
G(1)(x,y;k,m,n)=G (x,y;k,m,n).
j j
We give now a computation formula of two variable Fibonacci type polyno-
mialsofhigherorderhinthefollowingstatement.
Theorem2.2. Wehave
j
G(h1+h2)(x,y;k,m,n)= ∑G(h1)(x,y;k,m,n)G(h2)(x,y;k,m,n). (2.2)
j (cid:96) j−(cid:96)
(cid:96)=0
GeneratingFunctionsforSpecialPolynomialsandNumbers 7
Proof. Settingh=h +h into(2.1),westartwith
1 2
∞ 1 1
∑G(h1+h2)(x,y;k,m,n)tj= · ,
j (1−xkt−ymtn+m)h1 (1−xkt−ymtn+m)h2
j=0
andthen,usingagain(2.1),weget
∞ ∞ ∞
∑G(h1+h2)(x,y;k,m,n)tj= ∑G(h1)(x,y;k,m,n)tj ∑G(h2)(x,y;k,m,n)tj.
j j j
j=0 j=0 j=0
Now,byusingtheCauchyproductintheright-handsideoftheaboveequation,we
obtain
∞ ∞ j
∑G(h1+h2)(x,y;k,m,n)tj= ∑ ∑G(h1)(x,y;k,m,n)G(h2)(x,y;k,m,n)tj.
j (cid:96) j−(cid:96)
j=0 j=0(cid:96)=0
Finally,comparingthecoefficientsoftj onbothsidesinthepreviousequality,we
arriveatthedesiredresult(2.2). (cid:3)
Remark 2.3. Setting h =h =1 in (2.2), we obtain the following formula for
1 2
computingtwovariableFibonaccitypepolynomialsofthesecondorder,
j
G(2)(x,y;k,m,n)= ∑G (x,y;k,m,n)G (x,y;k,m,n).
j (cid:96) j−(cid:96)
(cid:96)=0
Ifwetakex:=ax,y=−1,k=1,m=1,n=a−1,(2.1)reducesto
∞ 1
∑G(h)(ax,−1;1,1,a−1)tj =
j (1−axt+ta)h
j=0
∞
= ∑Πh (x)tj.
j,a
j=0
Comparingthecoefficientsoftjonbothsidesoftheaboveequality,weobtain
thefollowingresult:
Corollary 2.4. A relation between two variable Fibonacci type polynomials of
higherorderG(h)(x,y;k,m,n)andHumbertpolynomialsΠh (x)isgivenby
j n,m
G(h)(ax,−1;1,1,a−1)=Πh (x).
j j,a
3. SpecialpolynomialsincludingtwovariableFibonaccitype
polynomialsandBernoulliandEulertypepolynomials
In this section, in order to introduce a new family of polynomials, we modify and
unifythegeneratingfunctionsoftheFibonaccitypepolynomialsintwovariables.
Byusingthesegeneratingfunctions,wederivesomerelationsandidentitiesinclud-
ing the Apostol-Bernoulli numbers, the Bernoulli type polynomials, the Humbert
polynomialsandtheGenocchipolynomials.Theserelationsandidentitiesalsoin-
cludetheFibonaccitypepolynomialsintwovariables.
8 G.Ozdemir,Y.SimsekandG.V.Milovanovic´
Now, we introduce the generating function for these new special polynomi-
alsintwovariables(x,y)(cid:55)→G (x,y;k,m,n),d≥0,withthethreefreeparameters
d
k,m,n.
Definition 3.1. The polynomials G (x,y;k,m,n) are defined by means of the fol-
d
lowinggeneratingfunction
1−xk−ym
F(z;x,y;k,m,n) =
1−xkez−ymez(m+n)
∞ G (x,y;k,m,n)(cid:18) z (cid:19)d
= ∑ d . (3.1)
d! 1−xk−ym
d=0
ArecurrencerelationforthepolynomialsG (x,y;k,m,n)canbeproved.
d
Theorem3.2. LetG (x,y;k,m,n)=1andd beapositiveinteger.Thenwehave
0
G (x,y;k,m,n) = xk∑d (cid:18)d(cid:19)G (x,y;k,m,n)(cid:0)1−xk−ym(cid:1)d−j
d j
j
j=0
+ym∑d (cid:18)d(cid:19)G (x,y;k,m,n)(m+n)d−j(cid:0)1−xk−ym(cid:1)d−j.
j
j
j=0
Proof. Byapplyingtheumbralcalculusmethodsto(3.1),weget
∞ zd
1−xk−ym= ∑G (x,y;k,m,n)
d (1−xk−ym)dd!
d=0
−xk ∑∞ (cid:0)G(x,y;k,m,n)+1−xk−ym(cid:1)d zd
(1−xk−ym)dd!
d=0
−ym ∑∞ (cid:0)G(x,y;k,m,n)+(m+n)(cid:0)1−xk−ym(cid:1)(cid:1)d zd ,
(1−xk−ym)dd!
d=0
withtheusualconventionofreplacingGd(x,y;k,m,n)byG (x,y;k,m,n).Compar-
d
ingthecoefficientsofzd onthebothsidesofthepreviousequality,wearriveatthe
desiredresult. (cid:3)
Afewfirstpolynomialsare
G (x,y;k,m,n)=1, G (x,y;k,m,n)=xk+(m+n)ym,
0 1
G (x,y;k,m,n)=[xk+(m+n)ym]2−(m+n−1)2xkym+xk+(m+n)2ym,
2
etc.
3.1. Relationsbetweenthepolynomialsandnumbers
Here, we consider some special cases of the polynomials G (x,y;k,m,n). By us-
d
ing the generating function from (3.1), we derive some new identities and rela-
tions,whichincludethepolynomialsG (x,y;k,m,n),theApostol-Bernoulliandthe
d
Apostol-Euler polynomials and numbers, as well as the classical Bernoulli, Euler
andGenocchipolynomialsandnumbers.
GeneratingFunctionsforSpecialPolynomialsandNumbers 9
Theorem 3.3. Let d ≥1. The polynomials G (x,y;k,m,n), d ≥1, are connected
d
withtheApostol-BernoullinumbersB (λ)inthefollowingway
d
(cid:0)1−xk−y(cid:1)d
G (x,y;k,1,0)=− B (cid:0)xk+y(cid:1), d≥1. (3.2)
d−1 d d
Proof. First,accordingto(3.1)and(1.2),wehavethefollowingrelation
1−xk−y
F(z;x,y;k,1,0)=− F (0,z;xk+y),
AB
z
i.e.,
∑∞ Gd(x,y;k,1,0)zd =−1−xk−y ∑∞ B (cid:0)xk+y(cid:1)zd,
(1−xk−y)d d! z d d!
d=0 d=0
wherewealsoused(1.3).However,since
z ∞ G (x,y;k,1,0) zd ∞ (d+1)G (x,y;k,1,0) zd+1
∑ d · = ∑ d ·
1−xk−y (1−xk−y)d d! (1−xk−y)d+1 (d+1)!
d=0 d=0
∞ dG (x,y;k,1,0) zd
= ∑ d−1 · ,
(1−xk−y)d d!
d=1
wehave
∑∞ dGd−1(x,y;k,1,0)·zd =−∑∞ B (cid:0)xk+y(cid:1)zd, (3.3)
(1−xk−y)d d! d d!
d=1 d=1
becauseB (λ)=0.
0
Comparingthecoefficientsofzd/d!onbothsidesin(3.3),weobtain(3.2). (cid:3)
ByusingtheApostol-Bernoullinumbersandtheequality(3.2)wegetanother
computationformulaforthepolynomialsG (x,y;k,m,n).Thus,
d
G (x,y;k,1,0)=−(cid:0)1−xk−y(cid:1)B (cid:0)xk+y(cid:1)=1,
0 1
(cid:0)1−xk−y(cid:1)2
G (x,y;k,1,0)=− B (cid:0)xk+y(cid:1)=xk+y,
1 2
2
(cid:0)1−xk−y(cid:1)3
G (x,y;k,1,0)=− B (cid:0)xk+y(cid:1)=x2k+2xky+xk+y2+y,
2 3
3
(cid:0)1−xk−y(cid:1)4
G (x,y;k,1,0)=− B (cid:0)xk+y(cid:1)=(cid:0)xk+y(cid:1)(cid:2)(cid:0)xk+y(cid:1)2+4(cid:0)xk+y(cid:1)+1(cid:3),
3 4
4
(cid:0)1−xk−y(cid:1)5
G (x,y;k,1,0)=− B (cid:0)xk+y(cid:1)
4 5
5
=(cid:0)xk+y(cid:1)(cid:2)(cid:0)xk+y(cid:1)3+11(cid:0)xk+y(cid:1)2+11(cid:0)xk+y(cid:1)+1(cid:3), etc.
Theorem3.4. Letd≥0.TherelationbetweenthepolynomialsG (x,y;k,m,n)and
d
theApostol-BernoullipolynomialsB (x,λ)isgivenby
d
B (cid:0)x,xk(cid:1)=−dd∑−1(cid:18)d−1(cid:19) xd−1−j G (x,0;k,m,n). (3.4)
d j (1−xk)j+1 j
j=0
10 G.Ozdemir,Y.SimsekandG.V.Milovanovic´
Proof. Startingwith(1.6)and(3.1)fory=0m(cid:54)=0,weconcludethat
zexzF(z;x,0;k,m,n)=zexz 1−xk =(cid:0)xk−1(cid:1)F (x,z;xk), (3.5)
1−xkez AE
i.e.,
z∑∞ (xz)d ∑∞ Gd(x,0;k,m,n)zd =−∑∞ B (cid:0)x,xk(cid:1)zd,
d! (1−xk)d+1 d! d d!
d=0 d=0 d=0
afterreplacingbythecorrespondingseriesrepresentations.Now,usingtheCauchy
productontheleft-handsideoftheaboveequality,weobtain
∑∞ ∑d (cid:18)d(cid:19)xd−jGj(x,0;k,m,n)zd+1 =−∑∞ B (cid:0)x,xk(cid:1)zd,
j (1−xk)j+1 d! d d!
d=0j=0 d=0
i.e.,(3.4). (cid:3)
Remark 3.5. By using (3.5), the equality (3.4) can be also given in the following
form
B (cid:0)x,xk(cid:1)=−∑d (cid:18)d(cid:19)xd−j jGj−1(x,0;k,m,n).
d j=1 j (cid:0)1−xk(cid:1)j
Theorem3.6. TheEulerpolynomialsE (x)canbeexpressedintermsofthepoly-
d
nomialsG (x,y;k,m,n)as
d
d (cid:18)d(cid:19) G (−1,0;1,m,n)
E (x)= ∑ xd−j j . (3.6)
d j 2j
j=0
Proof. AsintheproofofTheorem3.4,weassumethatm(cid:54)=0andstartwithaspecial
caseofthegeneratingfunctionin(3.1),withx=−1,y=0andk=1,i.e.,
F(z;−1,0;1,m,n)=F (0,t;1),
AE
Then,bythegeneratingfunctionoftheEulerpolynomialsE (x)givenby(1.8)(for
d
h=1),weconcludethat
exzF(z;−1,0;1,m,n)=F (x,z;1).
Eh
i.e.,
∞ (xz)d ∞ G (−1,0;1,m,n)zd ∞ zd
∑ ∑ d = ∑E (x)
d! 2d d! d d!
d=0 d=0 d=0
or
∞ d (cid:18)d(cid:19) G (−1,0;1,m,n)zd ∞ zd
∑ ∑ xd−j j = ∑E (x) ,
j 2j d! d d!
d=0j=0 d=0
fromwhichweobtain(3.6). (cid:3)
Theorem3.7. TherelationbetweenthepolynomialsG (x,y;k,m,n)andtheGenoc-
d
chipolynomialsG (x)isgivenby
d
d−1(cid:18)d−1(cid:19) G (−1,0;1,m,n)
G (x)=d ∑ xd−1−j j . (3.7)
d j 2j
j=0