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arXiv:1001.0034v1 [math.NT] 4 Jan 2010NEW IDENTITIES INVOLVING q-EULER POLYNOMIALS OF HIGHER ORDER T. Kim AND Y. H. Kim Abstract. In this paper, we present new generating functions which are relat ed to q-Euler numbers and polynomials of higher order. From these genera ting functions, we give new identities involving q-Euler numbers and polynomials of higher order. §1. Introduction/ Preliminaries LetCbe the complex number field. We assume that q∈Cwith|q|<1 and theq-number is defined by [ x]q=1−qx 1−qin this paper. The q-factorial is given by [n]q! = [n]q[n−1]q···[2]q[1]qand theq-binomial formulae are known that (x:q)n=n/productdisplay i=1(1−xqi−1) =n/summationdisplay i=0/parenleftbiggn i/parenrightbigg qq(i 2)(−x)i,(see [3, 14, 15]) , and 1 (x:q)n=n/productdisplay i=1/parenleftbigg1 1−xqi−1/parenrightbigg =∞/summationdisplay i=0/parenleftbiggn+i−1 i/parenrightbigg qxi,(see [3, 5, 14, 15]) , where/parenleftbign i/parenrightbig q=[n]q! [n−i]q![i]q!=[n]q[n−1]q···[n−i+1]q [i]q!. The Euler polynomials are defined by2 et+1ext=/summationtext∞ n=0En(x)tn n!, for|t|< π. In the special case x= 0,En(=En(0)) are called the n-th Euler numbers. In this paper, we consider the q-extensions of Euler numbers and polynomials of higher orde r. Barnes’ multiple Bernoulli polynomials are also defined by (1) tr /producttextr j=1(eajt−1)ext=∞/summationdisplay n=0Bn(x,r|a1,···,ar)tn n!,where|t|<max 1≤i≤r2π |ai|, (see [1, 14]). Key words and phrases. : multiple q-zeta function, q-Euler numbers and polynomials, higher order q-Euler numbers, Laurent series, Cauchy integral. 2000 AMS Subject Classification: 11B68, 11S80 The present Research has been conducted by the research Grant of Kw angwoon University in 2010 Typeset by AMS-TEX 1In one of an impressive series of papers (see [1, 6, 14]), Barn es developed the so-called multiple zeta and multiple gamma function. Let a1,···,aNbe positive parameters. Then Barnes’ multiple zeta function is defined by ζN(s,w|a1,···,aN) =/summationdisplay m1,···,mN=0(w+m1a1+···+mNaN)−s,(see [1]), whereℜ(s)> N,ℜ(w)>0. Form∈Z+, we have ζN(−m,w|a1,···,aN) =(−1)mm! (N+m)!BN+m(w,N|a1,···,aN). In this paper, we consider Barnes’ type multiple q-Euler numbers and polynomials. The purpose of this paper is to present new generating functi ons which are related toq-Euler numbers and polynomials of higher order. From the Mel lin transformation of these generating functions, we derive the q-extensions o f Barnes’ type multiple zeta functions, which interpolate the q-Euler polynomials of higher order at negative integer. Finally, we give new identities involving q-Euler numbers and polynomials of higher order. §2.q-Euler numbers and polynomials of higher order In this section, we assume that q∈Cwith|q|<1. Letx,a1,... ,a rbe complex numbers with positive real parts. Barnes’ type multiple Eul er polynomialsare defined by (2)2r /producttextr j=1(eajt+1)ext=∞/summationdisplay n=0E(r) n(x|a1,... ,a r)tn n!,for|t|<max 1≤i≤rπ |wi|,(see [6]), andE(r) n(a1,... ,a r)(=E(r) n(0|a1,... ,a r)) are called the n-th Barnes’ type multiple Euler numbers. First, we consider the q-extension of Euler polynomials. The q-Euler polynomials are defined by (3)Fq(t,x) =∞/summationdisplay n=0En,q(x)tn n!= [2]q∞/summationdisplay m=0(−q)me[m+x]qt,(see [8, 11, 13, 14, 15]) . From (3), we have En,q(x) =[2]q (1−q)nn/summationdisplay l=0/parenleftbiggn l/parenrightbigg(−1)lqlx (1+ql+1). In the special case x= 0,En,q(=En,q(0)) are called the n-thq-Euler numbers. From (3), we can easily derive the following relation. E0,q= 1,andq(qE+1)n+En,q= 0 ifn≥1,(see [8, 16, 17]) , 2where we use the standard convention about replacing EkbyEk,q.It is easy to show that lim q→1Fq(t,x) =2 et+1ext=∞/summationdisplay n=0En(x)tn n!,(see [2, 3, 19-23]) , whereEn(x) are the n-th Euler polynomials. For r∈N, the Euler polynomials of orderris defined by (4)/parenleftbigg2 et+1/parenrightbiggr ext=∞/summationdisplay n=0E(r) n(x)tn n!,for|t|< π. Now we consider the q-extension of (4). (5)F(r) q(t,x) = [2]r q∞/summationdisplay m1,...,m r=0(−q)m1+···+mre[m1+···+mr+x]qt=∞/summationdisplay n=0E(r) n,q(x)tn n!, whereE(r) n,q(x) are called the n-thq-Euler polynomials of order r(see [10-15]). From (5), we can derive (6) E(r) n,q(x) =[2]r q (1−q)nn/summationdisplay l=0/parenleftbiggn l/parenrightbigg(−1)lqlx (1+ql+1)r. By (5) and (6), we see that (7) F(r) q(t,x) = [2]r q∞/summationdisplay m=0/parenleftbiggm+r−1 m/parenrightbigg (−q)me[m+x]qt. Thus, we note that lim q→1F(r) q(t,x) =/parenleftBig 2 et+1/parenrightBigr ext=/summationtext∞ n=0E(r) n(x)tn n!.In the special casex= 0,E(r) n,q(=E(r) n,q(0)) are called the n-thq-Euler numbers of order r. By (5), (6) and (7), we obtain the following proposition. Proposition 1. Forr∈N, let F(r) q(t,x) = [2]r q/summationdisplay m1,...,m r=0(−q)m1+···+mre[m1+···+mr+x]qt=∞/summationdisplay n=0E(r) n,q(x)tn n!. Then we have E(r) n,q(x) =[2]r q (1−q)nn/summationdisplay l=0/parenleftbiggn l/parenrightbigg(−1)lqlx (1+ql+1)r= [2]r q∞/summationdisplay m=0/parenleftbiggm+r−1 m/parenrightbigg (−q)m[m+x]n q. 3From the Mellin transformation of F(r) q(t,x), we can derive the following equation. 1 Γ(s)/integraldisplay∞ 0F(r) q(−t,x)ts−1dt= [2]r q∞/summationdisplay m1,...,m r=0(−q)m1+···+mr [m1+···+mr+x]sq = [2]r q∞/summationdisplay m=0/parenleftbiggm+r−1 m/parenrightbigg (−q)m1 [m+x]sq, (8) wheres∈C,x/negationslash= 0,−1,−2,.... By (8), we can define the multiple q-zeta function related to q-Euler polynomials. Definition 2. Fors∈C,x∈Rwithx/negationslash= 0,−1,−2,..., we define the multiple q-zeta function related to q-Euler polynomials as ζq,r(s,x) = [2]r q∞/summationdisplay m1,...,m r=0(−q)m1+···+mr [m1+···+mr+x]sq. Note that ζq,r(s,x) is a meromorphic function in whole complex s-plane. From (8), we also note that ζq,r(s,x) = [2]r q∞/summationdisplay m=0/parenleftbiggm+r−1 m/parenrightbigg (−q)m1 [m+x]sq. By Laurent series and the Cauchy residue theorem in (5) and (8 ), we see that ζq(−n,x) =E(n) n,q(x),forn∈Z+. Therefore, we obtain the following theorem. Theorem 3. Forr∈N,n∈Z+, andx∈Rwithx/negationslash= 0,−1,−2,..., we have ζq(−n,x) =E(r) n,q(x). Letχbe the Dirichlet’s character with conductor f∈Nwithf≡1 (mod 2). Then the generalized q-Euler polynomial attached to χare considered by Fq,χ(x) =∞/summationdisplay n=0En,χ,q(x)tn n!= [2]q∞/summationdisplay m=0(−q)mχ(m)e[m+x]qt. From (3) and (9), we have En,χ,q(x) =[2]q [2]qff−1/summationdisplay a=0(−q)aχ(a)En,qf(x+a f). 4In the special case x= 0,En,χ,q=En,χ,q(0) are called the n-th generated q-Euler number attached to χ. It is known that the generalized Euler polynomials of order rare defined by (10) (2/summationtextf−1 a=0(−1)aχ(a)eat eft+1)rext=∞/summationdisplay n=0E(r) n,χ(x)tn n!, for|t|<π f. We consider the q-extension of (10). The generalized q-Euler polynomials of order rattached to χare defined by F(r) q,χ(t,x) = [2]r q∞/summationdisplay m1,...,m r=0(−q)m1+···+mr(r/productdisplay i=1χ(mi))e[m1+···+mr+x]qt =∞/summationdisplay n=0E(r) n,χ,q(x)tn n!,(see [14, 15]) . (11) Note that lim q→1F(r) q,χ(t,x) = (2/summationtextf−1 a=0(−1)aχ(a)eat eft+1)r. By (11), we easily see that E(r) n,χ,q(x) =[2]r q (1−q)nn/summationdisplay l=0/parenleftbiggn l/parenrightbigg (−qx)lf−1/summationdisplay a1,...,ar=0(r/productdisplay j=1χ(aj))(−ql+1)/summationtextr i=1ai (1+q(l+1)f)r = [2]r q∞/summationdisplay m1,...,m r=0(−q)m1+···+mr(r/productdisplay i=1χ(mi))[m1+···+mr+x]n q. Fors∈C,x∈Rwithx/negationslash= 0,−1,−2,..., we have 1 Γ(s)/integraldisplay∞ 0F(r) q,χ(−t,x)ts−1dt = [2]r q∞/summationdisplay m1,...,m r=0(−q)m1+···+mr(/producttextr i=1χ(mi)) [m1+···+mr+x]sq,(see [15]) . (12) From (12), we can consider the Dirichlet’s type multiple q-l-function as follows : Definition 4. Fors∈C,x∈Rwithx/negationslash= 0,−1,−2,..., we define the Dirichlet’s type multiple q-l-function as lq(s,x|χ) = [2]r q∞/summationdisplay m1,...,m r=0(−q)m1+···+mr(/producttextr i=1χ(mi)) [m1+···+mr+x]sq,(see [15]) . By Laurent series and the Cauchy residue theorem in (11) and ( 12), we obtain the following theorem. 5Theorem 5. Forn∈Z+, we have lq(−n,x|χ) =E(r) n,χ,q(x). Forh∈Zandr∈N, we consider the extended r-pleq-Euler polynomials. F(h,r) q(t,x) = [2]r q∞/summationdisplay m1,...,m r=0q/summationtextr j=1(h−j+1)mj(−1)/summationtextr j=1mje[m1+···+mr+x]qt =∞/summationdisplay n=0E(h,r) n,q(x)tn n!. (13) Note that lim q→1F(h,r) q(t,x) = (2 et+1)rext=∞/summationdisplay n=0E(r) n(x)tn n!. From (13), we note that E(h,r) n,q(x) =[2]r q (1−q)nn/summationdisplay l=0/parenleftbiggn l/parenrightbigg(−qx)l (−qh−r+l+1:q)r = [2]r q∞/summationdisplay m=0/parenleftbiggm+r−1 m/parenrightbigg q(−qh−r+1)m[m+x]n q. (14) By (14), we easily see that (15)F(h,r) q(t,x) = [2]r q∞/summationdisplay m=0/parenleftbiggm+r−1 m/parenrightbigg q(−qh−r+1)me[m+x]qt,(see [11, 13, 14]) . Using the Mellin transform for F(h,r) q(t,x), we have 1 Γ(s)/integraldisplay∞ 0F(r) q(−t,x)ts−1dt = [2]r q∞/summationdisplay m1,...,m r=0(−1)m1+···+mrq/summationtextr j=1(h−j+1)mj [m1+···+mr+x]sq,(see [13, 14, 15]) ,(16) fors∈C,x∈Rwithx/negationslash= 0,−1,−2,.... Now we can define the extended q-zeta function associated with E(h,r) n,q(x). 6Definition 6. Fors∈C,x∈Rwithx/negationslash= 0,−1,−2,..., we define the (h, q)-zeta function as ζ(h) q,r(s,x) = [2]r q∞/summationdisplay m1,...,m r=0(−1)m1+···+mrq/summationtextr j=1(h−j+1)mj [m1+···+mr+x]sq. Notethat ζ(h) q,r(s,x)isalsoa meromorphic function inwholecomplex s-plane. From (16) and (15), we note that (17) ζ(h) q,r(s,x) = [2]r q∞/summationdisplay m=0/parenleftbiggm+r−1 m/parenrightbigg q(−qh−j+1)m1 [m+x]sq. Using the Cauchy residue theorem and Laurent series in (16), we obtain the following theorem. Theorem 7. Forn∈Z+, we have ζ(h) q,r(−n,x) =E(h,r) n,q(x). We consider the extended r-ple generalized q-Euler polynomials as follows : F(h,r) q,χ(t,x) = [2]r q∞/summationdisplay m1,...,m r=0q/summationtextr j=1(h−j+1)mj(−1)/summationtextr j=1mj(r/productdisplay j=1χ(mj))e[m1+···+mr+x]qt(18) =∞/summationdisplay n=0E(h,r) n,χ,q(x)tn n!. By (18), we see that E(h,r) n,χ,q(x) =[2]r q (1−q)nf−1/summationdisplay a1,...,ar=0(−1)/summationtextr j=1aj(r/productdisplay j=1χ(aj))n/summationdisplay l=0/parenleftbiggn l/parenrightbigg(−1)lqlxq(h−j+l+1)aj (−q(h−r+l+1)f:qf)r =[2]r q [2]r qf[f]n qf−1/summationdisplay a1,...,ar=0(−1)/summationtextr j=1aj(r/productdisplay j=1χ(aj))q/summationtextr j=1(h−j+1)ajζ(h) qf,r(−n,x+/summationtextr j=1aj f).(19) Therefore, we obtain the following theorem. 7Theorem 8. Forn∈Z+, we have E(h,r) n,χ,q(x) =[2]r q [2]r qf[f]n qf−1/summationdisplay a1,...,ar=0(−1)/summationtextr j=1aj(r/productdisplay j=1χ(aj))q/summationtextr j=1(h−j+1)ajζ(h) qf,r(−n,x+/summationtextr j=1aj f). From (18), we note that 1 Γ(s)/integraldisplay∞ 0F(h,r) q,χ(−t,x)ts−1dt = [2]r q∞/summationdisplay m1,...,m r=0q/summationtextr j=1(h−j+1)mj(/producttextr j=1χ(mj))(−1)m1+···+mr [m1+···+mr+x]sq, (20) wheres∈C,x∈Rwithx/negationslash= 0,−1,−2,.... From (20), we define the Dirichlet’s type multiple ( h,q)-l-function associated with the generalized multiple q-Euler polynomials attached to χ. Definition 9. Fors∈C,x∈Rwithx/negationslash= 0,−1,−2,..., we define the Dirichlet’s type multiple q-l-function as follows : l(h) q(s,x|χ) = [2]r q∞/summationdisplay m1,...,m r=0q/summationtextr j=1(h−j+1)mj(/producttextr i=1χ(mi))(−1)m1+···+mr [m1+···+mr+x]sq. Note that l(h) q(s,x|χ) is a meromorphic function in whole complex plane. It is easy to show that l(h) q(s,x|χ) =[2]r q [2]r qf1 [f]sqf−1/summationdisplay a1,...,ar=0(−1)/summationtextr j=1aj(r/productdisplay j=1χ(aj))q/summationtextr j=1(h−j+1)ajζ(h) qf,r(s,x+/summationtextr j=1aj f). By (19) and (20), we obtain the following theorem. Theorem 10. Forn∈Z+, we have l(h) q(−n,x|χ) =E(h,r) n,χ,q(x). Finally, we give the q-extension of Barnes’ type multiple Euler polynomials in (2 ). Forx,a1,... ,a r∈Cwith positive real part, let us define the Barnes’ type mutipl e 8q-Euler polynomials in Cas follows : F(r) q(t,x|a1,... ,a r;b1,... ,b r) = [2]r q∞/summationdisplay m1,...,m r=0(−1)m1+···+mrq(b1+1)m1+···+(br+1)mre[a1m1+···+armr+x]t(21) =∞/summationdisplay n=0E(r) n,q(x|a1,... ,a r;b1,... ,b r)tn n!, whereb1,... ,b r∈Z. By (21), we see that E(r) n,q(x|a1,... ,a r;b1,... ,b r) =[2]r q (1−q)nn/summationdisplay l=0/parenleftbiggn l/parenrightbigg(−1)lqlx (1+qla1+b1+1)···(1+qlar+br+1) = [2]r q∞/summationdisplay m1,...,m r=0(−1)m1+···+mrq(b1+1)m1+···+(br+1)mr[a1m1+···+armr+x]n q. From (21), we note that 1 Γ(s)/integraldisplay∞ 0F(r) q(−t,x|a1,... ,a r;b1,... ,b r)ts−1dt = [2]r q∞/summationdisplay m1,...,m r=0(−q)m1+···+mrqb1m1+···+brmr [a1m1+···+armr+x]sq. (22) By (22), we define the Barnes’ type multiple q-zeta function as follows : ζq,r(s,x|a1,... ,a r;b1,... ,b r) = [2]r q∞/summationdisplay m1,...,m r=0(−q)m1+···+mrqb1m1+···+brmr [a1m1+···+armr+x]sq, wheres∈C,x∈Rwithx/negationslash= 0,−1,−2,.... By (21), (22) and (23), we obtain the following theorem. Theorem 11. Forn∈Z+, we have ζq,r(s,x|a1,... ,a r;b1,... ,b r) =E(r) n,q(x|a1,... ,a r;b1,... ,b r). Letχbe the Dirichlet’s character with conductor f∈Nwithf≡1 (mod 2). Then the generalized Barnes’ type multiple q-Euler polynomials attached to χare defined 9by F(r) q,χ(t,x|a1,... ,a r;b1,... ,b r) = [2]r q∞/summationdisplay m1,...,m r=0(−q)m1+···+mrqb1m1+···+brmr(r/productdisplay i=1χ(mi))e[a1m1+···+armr+x]qt(24) =∞/summationdisplay n=0E(r) n,χ,q(x|a1,... ,a r;b1,... ,b r)tn n!, From (24), we note that 1 Γ(s)/integraldisplay∞ 0F(r) q,χ(−t,x|a1,... ,a r;b1,... ,b r)ts−1dt = [2]r q∞/summationdisplay m1,...,m r=0(−q)m1+···+mrqb1m1+···+brmr(/producttextr i=1χ(mi)) [a1m1+···+armr+x]sq. (25) By (25), we can define Barnes’ type multiple q-l-function in C. Fors∈C,x∈Rwith x/negationslash= 0,−1,−2,..., let us define the Barnes’ type multiple q-l-function as follows : l(r) q(s,x|a1,... ,a r;b1,... ,b r) = [2]r q∞/summationdisplay m1,...,m r=0(−q)m1+···+mrqb1m1+···+brmr(/producttextr i=1χ(mi)) [a1m1+···+armr+x]sq. (26) Note that l(r) q(s,x|a1,... ,a r;b1,... ,b r) is a meromorphic function in whole complex s-plane. By (24), (25) and (26), we easily see that l(r) q(−n,x|a1,... ,a r;b1,... ,b r) =E(r) n,χ,q(x|a1,... ,a r;b1,... ,b r) forn∈Z+, (see [1-18]). References [1] E. W. Barnes, On the theory of multiple gamma function , Trans. Camb. Ohilos. Soc. A 196(1904), 374-425. [2] I. N. Cangul,V. Kurt, H. Ozden, Y. Simsek, On the higher-order w-q-Genocchi numbers , Adv. Stud. Contemp. Math. 19(2009), 39–57. [3] N. K.Govil, V. Gupta, Convergence of q-Meyer-Konig-Zeller-Durrmeyer operators , Adv. Stud. Contemp. Math. 19(2009), 97–108. [4] T. Kim, On aq-analogue of the p-adic log gamma functions and related integrals , J.Number Theory76(1999), 320–329. [5] T. Kim, q-Volkenborn integration , Russ. J. Math. Phys. 9(2002), 288–299. [6] T. Kim, On Euler-Barnes multiple zeta functions , Russ. J. Math. Phys. 10(2003), 261–267. 10[7] T. Kim, Analytic continuation of multiple q-zeta functions and their values at negative integers, Russ. J. Math. Phys. 11(2004), 71–76. [8] T. Kim, The modified q-Euler numbers and polynomials , Adv. Stud. Contemp. Math. 16 (2008), 161–170. [9] T. Kim, Note on the q-Euler numbers of higher order , Adv. Stud. Contemp. Math. 19 (2009), 25–29. [10] T. Kim, Note on Dedekind type DC sums , Adv. Stud. Contemp. Math. 18(2009), 249–260. [11] T. Kim, Note on the Euler q-zeta functions , J. Number Theory 129(2009), 1798–1804. [12] T. Kim, A note on the generalized q-Euler numbers , Proc. Jangjeon Math. Soc. 12(2009), 45–50. [13] T. Kim, Some identities on the q-Euler polynomials of higher order a nd q-stirling numbers by the fermionic p-adic integral on Zp, Russ. J. Math. Phys. 16(2009), 1061-9208. [14] T. Kim, Barnes type multiple q-zeta functions and q-Euler polynomials , arXiv:0912.5119v1. [15] T. Kim, Note on multiple q-zeta functions , to be appeared in Russ. J. Math. Phys., arXiv:0912.5477v1. [16] T. Kim, On theq-extension of Euler and Genocchi numbers , J. Math. Anal. Appl. 326, 1458–1465. [17] T. Kim, Onp-adicq-l-functions and sums of powers , J. Math. Anal. Appl. 329, 1472–1481. [18] T. Kim, Y. Simsek, Analytic continuation of the multiple Daehee q-l-functions associated with Daehee numbers , Russ. J. Math. Phys. 15(2008), 58–65. [19] Y. H. Kim, W. Kim, C. S. Ryoo, On the twisted q-Euler zeta function associated with twistedq-Euler numbers , Proc. Jangjeon Math. Soc. 12(2009), 93-100. [20] H.Ozden, I.N.Cangul, Y.Simsek, Remarks on q-Bernoulli numbers associated with Daehee numbers , Adv. Stud. Contemp. Math. 18(2009), 41-48. [21] K. Shiratani, S. Yamamoto, On ap-adic interpolation function for the Euler numbers and its derivatives , Mem. Fac. Sci., Kyushu University Ser. A 39(1985), 113-125. [22] Y. Simsek, Theorems on twisted L-function and twisted Bernoulli numbers , Advan. Stud. Contemp. Math. 11(2005), 205–218. [23] Z. Zhang, Y. Zhang, Summation formulas of q-series by modified Abel’s lemma , Adv. Stud. Contemp. Math. 17(2008), 119–129. Taekyun Kim Division of General Education-Mathematics, Kwangwoon Uni versity, Seoul 139-701, S. Korea e-mail: [email protected] Young-Hee Kim Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, S. Korea e-mail: [email protected] 11 |