ON THE q-EXTENSIONS OF THE BERNOULLI AND EULER NUMBERS, RELATED IDENTITIES AND LERCH ZETA FUNCTION 9 0 0 TAEKYUNKIM,YOUNG-HEEKIM,ANDKYUNG-WONHWANG 2 n a Abstract Recently, λ-Bernoulli and λ-Euler numbers are studied in [5, 10]. The purpose J of this paper is to present a systematic study of some families of the q-extensions of the λ- 2 Bernoulliandtheλ-Eulernumbersbyusingthebosonicp-adicq-integralandthefermionicp-adic q-integral. The investigation of these λ-q-Bernoulli and λ-q-Euler numbers leads to interesting ] T identitiesrelatedtotheseobjects. Theresultsofthepresentpapercoverearlierresultsconcerning N q-Bernoulliandq-Euler numbers. Byusing derivative operator to the generating functions of λ- q-Bernoulliandλ-q-Eulernumbers,wegivetheq-extensionsofLerchzetafunction. . h t 2000 Mathematics Subject Classification : 11S80, 11B68? a m Key words and phrases : λ-Bernoulli numbers, λ-Euler numbers, p-adic q-integral, Lerch [ zetafunction 1 1. Introduction, Definitions and Notations v 9 Throughout this paper, the symbols Z ,Q ,C and C denote the ring of p-adic 4 p p p rationalintegers,thefieldofp-adicrationalnumbers,thecomplexnumberfieldand 2 0 the completion of algebraic closure of Qp, respectively. Let N be the set of natural . numbers. 1 0 Thesymbolq canbetreatedasacomplexnumber,q ∈C,orasap-adicnumber, 9 q ∈ Cp. If q ∈ C, then we always assume that |q| < 1. If q ∈ Cp, then we 0 usually assume that |1−q| < 1. Here |·| stands for the p-adic absolute value p p v: in Cp with |p|p = 1p. The q-basic natural numbers are defined by [n]q = 11−−qqn = Xi 1+q+q2 +···+qn−1 (n ∈ N) and [n]−q = 1−1(+−qq)n. In this paper, we use the r notation a 1−qx 1−(−q)x [x] = and [x] = , see [1-19]. q 1−q −q 1+q Hence lim[x] =x for any x with |x| ≤1 in the present p-adic case. q p q→1 Forx∈Z ,wesaythatg isauniformlydifferentiablefunctionatapointa∈Z , p p andwrite g ∈UD(Z ),the setofuniformlydifferentiable function, ifthe difference p quotients g(y)−g(x) F (x,y)= g y−x have a limit l=g′(a) as (x,y)→(a,a). For f ∈UD(Z ), the q- deformed bosonic p p-adic integral is defined as pN−1 qx (1) I (f)= f(x)dµ (x)= lim f(x) , see [1-19], q Z q N→∞ [pN]q Z p x=0 X 1 2 TAEKYUNKIM,YOUNG-HEEKIM,ANDKYUNG-WONHWANG and the q-deformed fermonic p-adic integral is defined by pN−1 (−q)x I (f)= f(x)dµ (x)= lim f(x) , (see [1-19]). −q ZZp −q N→∞x=0 [pN]−q X For n∈N, let f (x)=f(x+n). Then n n−1 (2) qnI (f )=(−1)nI (f)+[2] (−1)n−1−lqlf(l). −q n −q q l=0 X The classicalBernoullipolynomials B (x) and the Euler polynomialsE (x) are n n defined as t ∞ tn 2ext ∞ tn (3) ext = B (x) and = E (x) . et−1 n n! et+1 n n! x=0 x=0 X X The Bernoulli numbers B and the Euler numbers E are defined as B = B (0) n n n n and E =E (0), (see [1-19]). n n From (1), we note that q−1 (4) qI (f )=I (f)+(q−1)f(0)+ f′(0), q 1 q logq for f (x)=f(x+1). By (4), we see that I (f )=I (f)+f′(0), (see [7]). 1 1 1 1 Let u be algebraic in C (or C). Then the Frobenius-Euler polynomials are p defined as 1−u ∞ tn (5) ext = H (u,x) , (see [5]). et−u n n! n=0 X In case x=0, H (u,0)=H (u), which are called the Frobenius Euler numbers. n n Let Cpn be the cyclic group consisting of all pn-th roots of unity in Cp for any n ≥ 0 and Tp be the direct limit of Cpn with respect to the natural morphisms, hence Tp is the union of all Cpn with discrete topology. For λ∈T with λ6=1, if we use (4), then we have p t (6) etxλxdµ (x)= . Z 1 λet−1 Z p From (6), the λ−Bernoulli numbers are defined as t ∞ tn (7) =eB(λ)t = B (λ) , (see [5]) λet−1 n n! n=0 X with the usual convention of replacing Bi(λ) by B (λ). Thus, B (λ) can be deter- i k mined inductively by 1, if k =1, (8) λ(B(λ)+1)k−B (λ)= k (0, if k >1, (see [5]). By the definition of the Frobenius-Euler numbers, we see that t ∞ 1 (m+1)H (λ−1) (9) = · m tm+1, ( see [7]). λet−1 (m+1)! λ−1 m=0 X 3 For m≥1 and λ6=1, we have m (10) B (λ)= xmλxdµ (x)= H (λ−1), (see [5]). m Z 1 λ−1 m−1 Z p We can also easily see that λxdµ (x)=0 and Z 1 p tλx R ∞ tn etx= lim = lim xmλxdµ (x)λx. m→∞ λet−1 n!m→∞ Z 1 λ∈XCpm nX=0 λ∈XCpmZ p Consequently, we have 1 xn = B (1)+ H (λ−1)λx n λ−1 n−1 λX∈Tp λ6=1 B (λ) = B (1)+ n λx, ( see [5]). n n λX∈Tp λ6=1 From (6) and (8), we note that 1 2λ B (λ)=0, B (λ)= , B (λ)=− , ··· . 0 1 λ−1 2 (λ−1)2 The Genocchi numbers are defined by the generating function 2t ∞ tn = G . et+1 nn! n=0 X These numbers satisfy the relation G = 0,G = 1,G = G = ··· = G = 0, 0 1 3 5 2k+1 and the even coefficients are G =2(1−2n)B . n n For λ∈C with |λ|<1, by (2), we have p 2 (11) λxextdµ (x)= . Z −1 λet+1 Z p By (11), we define the λ-Euler numbers as follows : ∞ 2 E (λ) (12) = n tn, (see [7,9,10]). λet+1 n! n=0 X Note that E (λ)= 2 H (−λ−1). n λ+1 n From (12), we can easily derive 2 (13) xnλxdµ (x)=E (λ)= H (−λ−1). Z −1 n λ+1 n Z p The λ-Genocchi numbers are also defined as 2t ∞ tn t xnλxdµ (x)= = G (x) . Z −1 λet+1 n n! Z p n=0 X Thus, we have G (λ)=0, G (λ)= 2 , ··· , E (λ)= Gn+1(λ). 0 1 λ+1 n n+1 Inthispaper,westudy theq-extensionofλ-Bernoullinumberandλ-Eulernum- bers related to Lerch zeta function. The purpose of this paper is to present a systematicstudy ofsomefamilies ofthe q-extensionofthe λ-Bernoulliandλ-Euler numbersbyusingthebosonicp-adicq-integralandtheferminionicp-adicq-integral. 4 TAEKYUNKIM,YOUNG-HEEKIM,ANDKYUNG-WONHWANG Theinvestigationoftheseλ-q-Bernoulliandλ-q-Eulernumbersleadstointeresting identities related to these objects. The results of the present paper cover earlier results concerning q-Bernoulli and q-Euler numbers. By using derivative operator to the generating functions of λ-q-Bernoulli and λ-q-Euler numbers, we can give the q-extension of Lerch zeta function. 2. q-extension of λ-Bernoulli numbers and polynomials For λ∈T , let us consider the q-extension of λ-Bernoulli numbers as follows. p (14) β (λ)= λx[x]kdµ (x). k,q q q Z Z p From (14), we note that pN−1 1 β (λ) = lim λx[x]kqx k,q N→∞[pN]q q x=0 X pN−1 k 1 k 1 = lim (λq)x( (−1)lqlx) N→∞[pN]q x=0 l=0(cid:18)l(cid:19) (1−q)k X X 1−q k k 1−(λql+1)pN = (−1)l (1−q)k l 1−λql+1 l=0(cid:18) (cid:19) X k 1 k l+1 = (−1)l . (1−q)k−1 l 1−λql+1 l=0(cid:18) (cid:19) X Therefore, we obtain the following theorem. Theorem 1. For k ∈N∪{0} and λ∈T , we have p k 1 k l+1 β (λ)= (−1)l . k,q (1−q)k−1 l 1−λql+1 l=0(cid:18) (cid:19) X Let F(t,λ:q) be the generating functions of β (λ) with n,q ∞ tn F(t,λ:q)= β (λ) . n,q n! n=0 X 5 Then we have ∞ tn F(t,λ:q) = β (λ) = λxe[x]qtdµ (x) n,q n! Z q n=0 Z p X ∞ tn = λx[x]ndµ (x) Z q q n! n=0Z p X ∞ 1 k k ∞ tk (15) = { (−1)l(l+1) λmq(l+1)m} (1−q)k−1 l k! k=0 l=0(cid:18) (cid:19) m=0 X X X ∞ 1 ∞ k k tk = λm (−1)l(l+1)q(l+1)m (1−q)k−1 l k! k=0 m=0 l=0(cid:18) (cid:19) X X X ∞ 1 ∞ k k tk = qmλm l (−1)lqlm (1−q)k−1 l k! k=0 m=0 l=0 (cid:18) (cid:19) X X X ∞ 1 ∞ k k tk + qmλm (−1)lqlm . (1−q)k−1 l k! k=0 m=0 l=0(cid:18) (cid:19) X X X Since l k =k k−1 , the first term of the last equation in (15) equals l l−1 (cid:0) (cid:1) (cid:0) (cid:1) ∞ ∞ 1 k k−1 tk qmλm{ (−1)lqlm} (1−q)k−1 l−1 (k−1)! m=0 k=0 l=1(cid:18) (cid:19) X X X ∞ ∞ 1 k−1 k−1 tk (16) =− q2mλm{ (−1)lqlm} (1−q)k−1 l (k−1)! m=0 k=1 l=0(cid:18) (cid:19) X X X ∞ ∞ tk ∞ =−t q2mλm [m]k =−t q2mλme[m]qt. qk! m=0 k=0 m=0 X X X The second term of the last equation in (15) equals ∞ 1 ∞ tk qmλm(1−qm)k (1−q)k−1 k! k=0 m=0 X X ∞ ∞ tk ∞ (17) =(1−q) qmλm [m]k =(1−q) qmλme[m]qt. qk! m=0 k=0 m=0 X X X From (15), (16) and (17), we obtain the following proposition. Proposition 2. Let F(t,λ:q)= ∞ β (λ)tn . Then we have n=0 n,q n! P ∞ ∞ F(t,λ:q)=−t q2mλme[m]qt+(1−q) qmλme[m]qt. m=0 m=0 X X 6 TAEKYUNKIM,YOUNG-HEEKIM,ANDKYUNG-WONHWANG Since q2m =qm{[m] (q−1)+1}, it follows that q dkF (t,λ:q) β (λ) = q | k,q (dt)k t=0 ∞ ∞ = −k q2mλm[m]k−1+(1−q) qmλm[m]k q q m=0 m=0 X X ∞ ∞ ∞ = −k(q−1) qmλm[m]k−k qmλm[m]k−1+(1−q) qmλm[m]k q q q m=0 m=0 m=0 X X X ∞ ∞ = (1−q)(k+1) qmλm[m]k−k qmλm[m]k−1. q q m=0 m=0 X X Therefore, we obtain the following theorem. Theorem 3. For k ∈N∪{0} and λ∈T , we have p ∞ ∞ β (λ)=(1−q)(k+1) qmλm[m]k−k qmλm[m]k−1. k,q q q m=0 m=0 X X Now we consider another q-extension of λ-Bernoulli numbers as follows. (18) B (λ)= q−xλx[x]ndµ (x). n,q q q Z Z p From (18), we can derive B (λ) = q−xλx[x]ndµ (x) n,q q q Z Z p n 1 n = q−x(−1)lλxqlxdµ (x) (1−q)n l Z q l=0(cid:18) (cid:19)Z p X n 1 n l = (−1)l . (1−q)n−1 l 1−λql l=0(cid:18) (cid:19) X Thus, we obtain the following theorem. Theorem 4. For n∈N∪{0} and λ∈T , we have p n 1 n l B (λ)= (−1)l . n,q (1−q)n−1 l 1−λql l=0(cid:18) (cid:19) X Let F∗(t,λ:q) be the generating functions of B (λ) with n,q ∞ tn F∗(t,λ:q)= B (λ) . n,q n! n=0 X 7 Then we have ∞ tn F∗(t,λ:q) = B (λ) = q−xλxe[x]qtdµ (x) n,q n! Z q n=0 Z p X ∞ tn = { q−xλx[x]ndµ (x)} Z q q n! n=0 Z p X ∞ 1 n n l tn = { (−1)l } (1−q)n−1 l 1−λql n! n=0 l=0(cid:18) (cid:19) X X ∞ 1 n n ∞ tn = { (−1)ll λmqlm} (1−q)n−1 l n! n=0 l=0(cid:18) (cid:19) m=0 X X X ∞ ∞ n n n−1 tn = λm{ (−1)lqlm} (1−q)n−1 l−1 n! m=0 n=1 l=1(cid:18) (cid:19) X X X ∞ ∞ n tn = − λmqm (1−qm)n−1 (1−q)n−1 n! m=0 n=1 X X ∞ ∞ (1−qm)ntn+1 = − λmqm (1−q)n n! m=0 n=0 X X ∞ = −t λmqme[m]qt. m=0 X Therefore we obtain the following lemma. Lemma 5. Let F∗(t,λ:q)= ∞ B (λ)tn . Then we have n=0 n,q n! ∞ P F∗(t,λ:q)=−t λmqme[m]qt. m=0 X We also have dkF (t,λ:q) ∞ B (λ)= q | =−k qmλm[m]k−1. k,q (dt)k t=0 q m=0 X Therefore we obtain the following theorem. Theorem 6. For k ∈N∪{0} and λ∈T , we have p ∞ B (λ)=−k qmλm[m]k−1. k,q q m=0 X 3. q-extension of λ-Euler numbers and polynomials In this section, we assume that p is an odd prime number and λ ∈ C with p |1 − λ| < 1. By using the fermionic p-adic q-integral on Z , we consider the p p q-extensions of λ-Euler numbers as follows. For n∈N∪{0}, we define the q−extension of λ-Euler numbers as (19) E (λ)= q−xλx[x]ndµ (x). n,q q −q Z Z p 8 TAEKYUNKIM,YOUNG-HEEKIM,ANDKYUNG-WONHWANG From (19), we note that E (λ) = q−xλx[x]ndµ (x) n,q q −q Z Z p pN−1 1+q = lim (−1)x[x]nλx N→∞ 1+qpN q x=0 X [2] 1 n n 1+qpNλpN = q (−1)l lim 2 (1−q)n l N→∞ 1+qlλ l=0(cid:18) (cid:19) X n [2] 1 n 2 = q (−1)l 2 (1−q)n l 1+qlλ l=0(cid:18) (cid:19) X n [2] n 1 = q (−1)l . (1−q)n l 1+qlλ l=0(cid:18) (cid:19) X Therefore we obtain the following theorem. Theorem 7. For n∈N∪{0}, we have n [2] n 1 E (λ)= q (−1)l . n,q (1−q)n l 1+qlλ l=0(cid:18) (cid:19) X Let g(t,λ:q) be the generating function of E (λ) with n,q ∞ tn g(t,λ:q)= E (λ) . n,q n! n=0 X Then we have ∞ tn g(t,λ:q) = E (λ) = q−xλxe[x]qtdµ (x) n,q n! Z −q n=0 Z p X ∞ tn = { q−xλx[x]ndµ (x)} Z q −q n! n=0 Z p X ∞ 1 n n 1 tn = [2] { (−1)l } q (1−q)n l 1+λql n! n=0 l=0(cid:18) (cid:19) X X ∞ 1 n n ∞ tn = [2] (−1)l{ (−1)mλmqlm} q (1−q)n l n! n=0 l=0(cid:18) (cid:19) m=0 X X X ∞ ∞ 1 n n tn = [2] (−1)mλm (−1)lqlm q (1−q)n l n! m=0 n=0 l=0(cid:18) (cid:19) X X X ∞ ∞ tn = [2] (−1)mλm [m]n q q n! m=0 n=0 X X ∞ = [2] (−1)mλme[m]qt. q m=0 X Thus, we have the following lemma. 9 ∞ Lemma 8. Let g(t,λ:q)= E (λ)tn. Then we have n,q n! n=0 P ∞ (20) g(t,λ:q)=[2] (−1)mλme[m]qt. q m=0 X By (20), we can also consider the λ-q-Genocchi numbers as follows. ∞ ∞ tn (21) t q−xλxe[x]qtdµ (x)=[2] t (−1)mλme[m]qt = G (λ) . Z −q q n,q n! Z p m=0 n=0 X X From (21), we note that G (λ)=0 and 0,q G (λ) q−xλx[x]ndµ (x)= n+1,q . Z q −q n+1 Z p Thus, we see that G (λ) E (λ)= q−xλx[x]ndµ (x)= n+1,q . n,q Z q −q n+1 Z p Hence n−1 n n−1 1 G (λ)=[2] (−1)l , n,q q(1−q)n−1 l 1+qlλ l=0(cid:18) (cid:19) X where n=1,2,3, ···. Indeed, [2] G (λ) = q , 1,q 1+λ 1 2[2] 1 1 [2] 2 2 G (λ) = q (−1)l = q ( − ) 2,q 1−q l 1+qlλ 1−q 1+λ 1+qλ l=0(cid:18) (cid:19) X λ = −2[2] ( ). q (1+λ)(1+qλ) Now, we consider the q-extension of λ-Euler polynomials as follows. (22) E (λ,x)= q−yλy[x+y]ndµ (y). n,q q −q Z Z p From (22), we can easily derive n [2] n 1 E (λ,x)= q (−1)lqlx . n,q (1−q)n l 1+qlλ l=0(cid:18) (cid:19) X ∞ Let g(x,λ:q)= E (λ,x)tn. Then we have n,q n! n=0 P ∞ tn g(x,λ:q) = En,q(λ,x)n! = Z q−yλye[x+y]tqdµ−q(y) n=0 Z p X ∞ [2] n n ∞ tn = q (−1)lqlx( (−1)mqmlλm) (1−q)n l n! n=0 l=0(cid:18) (cid:19) m=0 X X X ∞ = [2] (−1)mλme[m+x]qt. q m=0 X 10 TAEKYUNKIM,YOUNG-HEEKIM,ANDKYUNG-WONHWANG It follows that dn(g(x,λ:g)) ∞ E (λ,x)= | =[2] (−1)mλm[m+x]n. n,q (dt)n t=0 q q m=0 X Then we obtain the following theorem. Theorem 9. For n∈N∪{0}, we have ∞ E (λ,x)=[2] (−1)mλm[m+x]n. n,q q q m=0 X By the same method, we consider the λ-q-Genocchi polynomials as follows. ∞ (23) t q−xλxe[x+y]qtdµ (x) = [2] t (−1)mλme[m+x]qt −q q Z Z p m=0 X ∞ tn = G (λ,x) . n,q n! n=0 X By (23), we see G (λ,x) (24) E (λ,x)= q−yλy[x+y]ndµ (y)= n+1,q n,q Z q −q n+1 Z p and G (λ,x)=0. From the definition of λ-q-Euler polynomials, we derive 0,q E (λ,x) = q−yλy[x+y]ndµ (y) n,q q −q Z Z p n n (25) = [x]n−lqlx q−yλy[y]ldµ (y) l q q −q Z l=0(cid:18) (cid:19) Z p X d−1 [2] x+a = q [d]n (−1)aλa q−dyλdy[ +y]n dµ (y,) [2]qd q a=0 ZZp d qd −q X for d∈N with d≡1( mod 2). By (25), we see that n n E (λ,x)= = [x]n−lqlxE (λ,x) n,q l q l,q l=0(cid:18) (cid:19) X d−1 [2] x+a = q [d]n (−1)aλaE (λd, ). [2] q n,qd d qd a=0 X It is easy to see that qI (f )+I (f)=[2] f(0), −q 1 −q q where f (x)=f(x+1). Thus, we have 1 q q−y−1λy+1[x+1+y]ndµ (y)+ q−yλy[x+y]ndµ (y)=[2] [x]n. q −q q −q q q Z Z Z p Z p Therefore, we obtain the following theorem. Theorem 10. For n∈N∪{0}, we have λE (λ,x+1)+E (λ,x)=[2] [x]n. n,q n,q q q By Theorem 10 and (24), we have the following result.