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