-Analogue of twisted -series and -twisted Euler numbers

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<ul><li><p>Journal of Number Theory 110 (2005) 267278www.elsevier.com/locate/jnt</p><p>q-Analogue of twisted l-series and q-twisted Eulernumbers</p><p>Yilmaz SimsekDepartment of Mathematics, Mersin University, 33342 Mersin, Turkey</p><p>Received 17 October 2002; revised 30 April 2004</p><p>Communicated by D. Goss</p><p>Abstract</p><p>We dene generating functions of q-generalized Euler numbers and polynomials and twistedq-generalized Euler numbers and polynomials. By using these functions, we give some propertiesof these numbers. We construct a complex twisted lq -functions which interpolate twisted q-generalized Euler numbers. As an application, we obtain the values of twisted lq -functions atnon-positive integers. Therefore, we nd a relation between twisted lq -functions and twistedq-generalized numbers explicitly. 2004 Elsevier Inc. All rights reserved.</p><p>MSC: primary 11B68; secondary 11M41</p><p>Keywords: Generalized Euler numbers; q-Euler numbers; Twisted q-generalized Euler numbers; lq -series</p><p>1. Introduction</p><p>In [5], Koplitz constructed a q-analogue of the p-adic L-function Lp,q(s, x) andsuggested two questions. Question (1) was solved by Satoh [6]. Satoh constructeda complex analytic qL-series which is a q-analogue of Dirichlets L-function andinterpolates q-Bernoulli numbers, which is an answer to Koblitzs question. Satoh [6]proved the distribution relation for Carlitzs q-Euler polynomials and constructed p-adicq-Euler measures to obtain a p-adic interpolation function for q-Euler numbers, on the</p><p>E-mail address: ysimsek@mersin.edu.tr</p><p>0022-314X/$ - see front matter 2004 Elsevier Inc. All rights reserved.doi:10.1016/j.jnt.2004.07.003</p></li><li><p>268 Y. Simsek / Journal of Number Theory 110 (2005) 267278</p><p>same way that Koblitz constructed Lp,q(s, ). By using p-adic invariant measure, Kim[2] answered to part of question (1) of Koblitz.In [12], Tsumura constructed q-analogues of the Dirichlet series which related to</p><p>algebraic number eld. He also constructed q-analogues of the p-adic log--functions,similar to [5]. But he did not give explicit relations between these functions and alge-braic number eld. By applying q-analogues of the Dirichlet L-series, he obtained theq-representations for the class number formulae.In [3], by using the p-adic q-integral due to Kim [2], Kim and Son [3] dened p-</p><p>adic q-analogue of the ordinary Bernoulli numbers and generalized q-Euler numbers.They gave a different construction of Tsumuras [11] p-adic function lp(u, s, ).The objective of this paper is to construct a complex analytic twisted lq( u , s, ) series</p><p>which is q-analogue of lq -series and interpolates twisted q-Euler numbers, which isan answer to Koblitzs question (1) related to Euler numbers. We induce the twistedlq -series from the generating function of twisted generalized q-Euler numbers.The generating function of q-Euler numbers Fu,q is given by [7]: if | u |&gt; 1,</p><p>Fu,q(t) = (1 1u)</p><p>n=0</p><p>e[n]t</p><p>un,</p><p>where q is a complex number with |q| &lt; 1. In the complex case, the generatingfunction of q-generalized Euler numbers Fu,q,(t) is given as follows:</p><p>Fu,q,(t) =f</p><p>a=1(a)ufae[a]tFuf ,qf (t[f ]qa)</p><p>= (1 1uf</p><p>)</p><p>fa=1</p><p>(a)ufan=0</p><p>e[f n+a]t</p><p>uf n(1.1)</p><p>=n=0</p><p>Hn (u, q)tn</p><p>n! ,</p><p>where q is a complex number with | q |&lt; 1, Hn (u, q) is the nth q-generalized Eulernumber, and</p><p>[x] = [x; q] = 1 qx</p><p>1 q</p><p>and [a+ nf ] = [a] + qa[nf ], [nf ] = [f ][n; qf ]. Note that the series on the right-handside of (1.1) is uniformly convergent in the wider sense. Consequently, we have</p><p>Hk (u, q) =dk</p><p>dtkFu,q,(t) |t=0= (1 1</p><p>uf)</p><p>fa=1</p><p>(a)ufan=0</p><p>[f n+ a]kuf n</p><p>.</p></li><li><p>Y. Simsek / Journal of Number Theory 110 (2005) 267278 269</p><p>This is used to construct a lq(u, s, ) series. Fu,q,(t) is uniquely determined as asolution of the following q-difference equation:</p><p>Fu,q,(t) = (1 1uf</p><p>)</p><p>fa=1</p><p>(a)ufae[a]t + e[f ]tFu,q,(qf t)</p><p>uf. (1.2)</p><p>Note that if = 1, then (1.2) reduces to</p><p>Fu,q(t) = 1 1u+ 1</p><p>uetFu,q(qt). (1.3)</p><p>The above equation was given by Satoh [6]. If q 1, then Fu,q(t) = Fu(t), which isgiven by [8]</p><p>Fu(t) = 1 uet u.</p><p>The generating function of twisted q-generalized Euler numbers Fu,q,(t) is givenby</p><p>Fu ,q,</p><p>(t) =</p><p>1 a f0 b</p></li><li><p>270 Y. Simsek / Journal of Number Theory 110 (2005) 267278</p><p>Eq. (1.5) is used to construct a twisted lq( u , s, ) series. Fu ,q,(t) is uniquely deter-mined as a solution of the following q-difference equation:</p><p>Fu ,q,</p><p>(t) = (1 1urf</p><p>)</p><p>1 a f0 b 1, and for k0,</p><p>Hk(u, x, q) = (qxH + [x])k,</p><p>with the usual convention of replacing Hj by Hj . As q 1, we have Hk(u, q) Hk(u) and Hk(u, x, q) Hk(u, x), where Hk(u) and Hk(u, x) are the usual Eulernumbers and Euler polynomial, respectively.</p><p>2. Generating function of generalized q-Euler numbers</p><p>Satoh [7] constructed generating function of q-Euler numbers as follows:</p><p>Fu,q(t) = (1 1u)</p><p>n=0</p><p>e[n]t</p><p>un=</p><p>n=0</p><p>Hn(u, q)tn</p><p>n! . (2.1)</p></li><li><p>Y. Simsek / Journal of Number Theory 110 (2005) 267278 271</p><p>The generating function of q-Euler polynomials is given by</p><p>Fu,q(t, x) = Fu,q(qxt)e[x]t</p><p>= (1 1u)</p><p>n=0</p><p>e[n+x]t</p><p>un=</p><p>n=0</p><p>Hn(u, x, q)tn</p><p>n! . (2.2)</p><p>(Note that [u+ v] = [u] + qu[v] is used in the above.)We consider the generalized q-Euler numbers, Hn (u, q), which are dened by Satoh</p><p>[6]. We shall explicitly determine the generating function Fu,q,(t) of Hn (u, q) asfollows:</p><p>Fu,q,(t) =n=0</p><p>Hn (u, q)tn</p><p>n! . (2.3)</p><p>This is the unique solution of the following q-difference equation:</p><p>Fu,q,(t) = (1 1uf</p><p>)</p><p>fa=1</p><p>(a)ufae[a]t + e[f ]tFu,q,(qf t)</p><p>uf. (2.4)</p><p>Lemma 2.1. Let be a primitive Dirichlet character of conductor f N.</p><p>Fu,q,(t) =f</p><p>a=1(a)ufae[a]tFuf ,qf (t[f ]qa)</p><p>= (1 1uf</p><p>)</p><p>fa=1</p><p>(a)ufan=0</p><p>e[f n+a]t</p><p>uf n. (2.5)</p><p>Proof. The right-hand side of (2.5) is uniformly convergent in the wider sense, andsatises (2.4). </p><p>In the case when = 1, (2.5) reduces to (1.3).The generating function of generalized q-Euler polynomials is dened as follows</p><p>Fu,q,(t, x) = Fu,q,(qxt)e[x]t . (2.6)</p></li><li><p>272 Y. Simsek / Journal of Number Theory 110 (2005) 267278</p><p>We give the generalized q-Euler polynomials as follows:</p><p>Theorem 2.1. Let be a primitive Dirichlet character of conductor f . For any positiveinteger n, we have</p><p>Hn (u, x, q) = [f ]nf</p><p>a=1(a)ufaHn(uf , x + a</p><p>f, qf ).</p><p>Proof. By (2.6), we have</p><p>Fu,q,(t, x) = Fu,q,(qxt)e[x]t =n=0</p><p>Hn (u, x, q)tn</p><p>n! . (2.7)</p><p>Substituting (2.5) into (2.7), we obtain</p><p>Fu,q,(t, x) =f</p><p>a=1(a)ufae[a+x]tFuf ,qf (t[f ]qa+x).</p><p>We now use denition of Fu,q(t) in the above, then we arrive at the following:</p><p>Fu,q,(t, x) = (1 1uf</p><p>)</p><p>fa=1</p><p>(a)ufan=0</p><p>uf ne[f n+a+x]t</p><p>=f</p><p>a=1(a)ufa((1 1</p><p>uf)</p><p>n=0</p><p>e[n+ a+x</p><p>f;qf ][f ]t</p><p>(uf )n),</p><p>where we use [a + nf ] = [a] + qa[nf ], [nf ] = [f ][n; qf ]. Substituting (2.2) into theabove, we have</p><p>Fu,q,(t, x) = [f ]nf</p><p>a=1(a)ufa</p><p>n=0</p><p>Hn (u,x + af</p><p>, qf )tn</p><p>n! . (2.8)</p><p>By comparing coefcient of tnn! in the right-hand side of (2.7) and (2.8) series, we</p><p>obtain the desired result. </p><p>Note that we can give the generalized q-Euler numbers. For x = 0, the followingCorollary is the special case of Theorem 2.1.</p></li><li><p>Y. Simsek / Journal of Number Theory 110 (2005) 267278 273</p><p>Corollary 2.1. Let be a primitive Dirichlet character of conductor f . For any positiveinteger k, we have</p><p>Hn (u, q) = [f ]nf</p><p>a=1(a)ufaHn(uf , a</p><p>f, qf ).</p><p>We shall explicitly determine the twisted generating function Fu ,q,</p><p>(t) of Hn,(u, q).Let be a primitive Dirichlet character of conductor f and be rth root of 1.</p><p>Fu ,q,</p><p>(t) =n=0</p><p>Hn,(u, q)tn</p><p>n! .</p><p>This is the unique solution of the following q-difference equation:</p><p>Fu ,q,</p><p>(t) = (1 1urf</p><p>)</p><p>1 a f0 b</p></li><li><p>274 Y. Simsek / Journal of Number Theory 110 (2005) 267278</p><p>The generating function of twisted q-generalized Euler polynomials is given by</p><p>Fu ,q,</p><p>(t, x) = Fu ,q,</p><p>(qxt)e[x]t</p><p>= (1 1urf</p><p>)</p><p>1 a f0 b</p></li><li><p>Y. Simsek / Journal of Number Theory 110 (2005) 267278 275</p><p>By using denition of q-Euler numbers, we have H0(u, x, q) = 1. Thus we get</p><p>H 0,(u, q) =</p><p>1 a f0 b</p></li><li><p>276 Y. Simsek / Journal of Number Theory 110 (2005) 267278</p><p>Proof. By using (3.1), we have</p><p>lq(u,k) = (1 1u)</p><p>n=0</p><p>[n]kun</p><p>= uu 1Hk(u, q).</p><p>We obtain the desired result. </p><p>For k0, by applying Lemma 2.1, we have</p><p>Hk (u, q) =dk</p><p>dtkFu,q,(t) |t=0</p><p>= (1 1uf</p><p>)</p><p>fa=1</p><p>(a)ufan=0</p><p>[f n+ a]kuf n</p><p>. (3.2)</p><p>An interpolation function for Hk (u, q) numbers are given by</p><p>Denition 3.1 (Satoh [6]). For s C,</p><p>lq(u, s, ) =n=1</p><p>(n)un[n]s . (3.3)</p><p>The values of lq(u, s, ) at non-positive integers are given by</p><p>Proposition 3.2 (Satoh [6]). For any positive integer k0, we have</p><p>lq(u,k, ) = 1uf 1H</p><p>k (u, q).</p><p>Proof. Substituting n = a + mf with m = 1, ..., a = 1, ..., f 1 and s = k into(3.3), we obtain</p><p>lq(u,k, ) =f</p><p>a=1</p><p>(a)ua</p><p>n=0</p><p>[fm+ a]kufm</p><p>.</p><p>By substituting (3.2) into the above, we have</p><p>Hk (u, q) = (uf 1)lq(u,k, ).</p><p>Then we obtain the desired result. </p></li><li><p>Y. Simsek / Journal of Number Theory 110 (2005) 267278 277</p><p>Koblitz [4] and Simsek [9] constructed twisted L-functions which are found usefulin number theory and p-adic analysis. The author gave some fundamental propertiesof twisted L-functions and twisted Bernoulli numbers.Here, we construct a complex twisted ql-series which interpolate twisted q-</p><p>generalized Euler numbers. As applications, we obtain the values of twisted lq -seriesat non-positive integers.Let r be a positive integer, and let be rth root of 1. Then twisted L-functions are</p><p>dened as follows [4,9]:</p><p>L(s, , ) =n=1</p><p>(n)n</p><p>ns.</p><p>Since the function n (n)n has period f r , this is a special case of the DirichletL-functions. Such L-series (for r = f ) are used classically to prove the formula forL(1, ) by Fourier inversion. Finally, we dene an interpolation function for twistedq-generalized Euler numbers.</p><p>Denition 3.2. For s C,</p><p>lq(u</p><p>, s, ) =</p><p>n=1</p><p>(n)n</p><p>un[n]s , (3.4)</p><p>where is rth root of 1.</p><p>The values of lq( u , s, ) at non-positive integers are given by</p><p>Theorem 3.3. For any integer K0, we have</p><p>lq(u</p><p>,k, ) =</p><p>Hk,(u, q)</p><p>urf 1 .</p><p>Proof. Substituting n = a+bf+mrf with m = 1, ..., a = 1, ..., f, b = 0, 1, 2, ..., r1and s = k, k0 into (3.4), we obtain</p><p>lq(u</p><p>,k, ) =</p><p>1 a f0 b</p></li><li><p>278 Y. Simsek / Journal of Number Theory 110 (2005) 267278</p><p>Remark 3.1. We may construct a p-adic twisted interpolation function for twistedq-generalized Euler numbers and may prove Kummer congruence for these numbers.</p><p>References</p><p>[1] L. Carlitz, q Bernoulli numbers and polynomials, Duke Math. J. 15 (1948) 9871000.[2] T. Kim, On a q-analogue of the p-adic log Gamma functions and related Integrals, J. Number</p><p>Theory 76 (1999) 320329.[3] M.S. Kim, J.W. Son, Some remarks on a q-analogue of Bernoulli numbers, J. Korean Math. Soc.</p><p>No. 2 39 (2002) 221236.[4] N. Koblitz, A New Proof of Certain formulas for p-adic L-Functions, Duke Math. J. 46 (2) (1979)</p><p>455468.[5] N. Koblitz, On Carlitzs q-Bernoulli numbers, J. Number Theory 14 (1982) 332339.[6] J. Satoh, q-Analogue of Riemanns -function and q-Euler Numbers, J. Number Theory 31 (1989)</p><p>346362.[7] J. Satoh, A construction of q-analogue of Dedekind sums, Nagoya Math. J. 127 (1992) 129143.[8] K. Shiratani, On Euler Numbers, Memoirs Faculty of the Sciences, Kyushu University, Fukuoda,</p><p>1975, pp. 15.[9] Y. Simsek, Analytic properties of twisted L-series and twisted Bernoulli numbers, to appear Proc.</p><p>Jangjeon Math. Soc.[10] Y. Simsek, On twisted generalized Euler numbers, Bull. Korean Math. Soc. 41 (2) (2004) 299306.[11] H. Tsumura, On a p-adic interpolation of the generalized Euler numbers and its applications, Tokyo</p><p>J. Math. 10 (1987) 281293.[12] H. Tsumura, A note on q-analogues of the Dirichlet series and q-Bernoulli numbers, J. Number</p><p>Theory 39 (1991) 251256.</p><p>q-Analogue of twisted l-series and q-twisted Euler numbersIntroductionGenerating function of generalized qqqq-Euler numbersTwisted llllqqqq-seriesReferences</p></li></ul>

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