A note on multiple L-function.Abstract The purpose of this paper is to construct complex analytic multiple L-function and to define the generalized gen·er·al·ized adj. 1. Involving an entire organ, as when an epileptic seizure involves all parts of the brain. 2. Not specifically adapted to a particular environment or function; not specialized. 3. multiple Bernoulli numbers In mathematics, the Bernoulli numbers are a sequence of rational numbers with deep connections in number theory. Although easy to calculate, the values of the Bernoulli numbers have no elementary description; they are closely related to the values of the Riemann zeta function at with [chi], which can be viewed in the interpolating of the multiple L-function at negative integers. Keywords and Phrases: Multiple zeta function A zeta function is a function which is composed of an infinite sum of powers, that is, which may be written as a Dirichlet series: 1. Introduction Let x, [w.sub.1], [w.sub.2],..., [w.sub.r] be complex numbers with positive real parts. Then the multiple zeta function due to E. W. Barnes are defined by [[zeta].sub.r](s, x|[w.sub.1],..., [w.sub.r]) = [[infinity infinity, in mathematics, that which is not finite. A sequence of numbers, a1, a2, a3, … , is said to "approach infinity" if the numbers eventually become arbitrarily large, i.e. ].summation summation n. the final argument of an attorney at the close of a trial in which he/she attempts to convince the judge and/or jury of the virtues of the client's case. (See: closing argument) over ([m.sub.1],..., [m.sub.r]=0)] [1/[(x + [m.sub.1][w.sub.1] + ... + [m.sub.r][w.sub.r])[.sup.s]]], for R(s) > r, cf. [1, 7]. In [1], Barnes showed that [[zeta].sub.r] has a meromorphic continuation in s, with simple poles at s = 1, 2,..., r and derived many formulas which are related to multiple zeta function. Also, he proved that the multiple zeta function interpolates multiple Bernoulli numbers at negative integers. In the usual notation notation: see arithmetic and musical notation. How a system of numbers, phrases, words or quantities is written or expressed. Positional notation is the location and value of digits in a numbering system, such as the decimal or binary system. , the n-th Bernoulli polynomials In mathematics, the Bernoulli polynomials occur in the study of many special functions and in particular the Riemann zeta function and the Hurwitz zeta function. This is in large part due to the fact that they are an Appell sequence, i.e. were defined by [t/[[e.sup.t] - 1]][e.sup.xt] = [[infinity].summation over (n=0)] [[[B.sub.n](x)]/n!][t.sup.n], |t| < 2[pi]. In particular, the numbers [B.sub.n] = [B.sub.n](0) are said to be n-th Bernoulli numbers. Let [chi] be the Dirichlet character In number theory, Dirichlet characters are certain arithmetic functions which arise from multiplicative characters on the units of  . with conductor conductorAny of various substances that allow the flow of electric current or thermal energy. A conductor is a poor insulator because it has a low resistance to such flow. f [member of] N = {1, 2,...}. Then the generalized Bernoulli numbers with [chi] were defined by [[[summation].sub.a=1.sup.f] [chi](a)[e.sup.at]t]/[[e.sup.ft] - 1] = [[infinity].summation over (n=0)] [[B.sub.n,[chi]]/n!][t.sup.n], for |t| < 2[pi], cf.[2, 3, 4, 5, 6]. It was well known that the generalized Bernoulli numbers with [chi] are closely related to the Dirichlet L-function In mathematics, a Dirichlet L-series, named in honour of Johann Peter Gustav Lejeune Dirichlet, is a function of the form L(1 - n, [chi]) = -[[B.sub.n,[chi]]/n], for n [member of] N, cf.[4, 5, 7]. In this paper we define the generalized multiple Bernoulli numbers with [chi] which are related to the multiple L-function. The main purpose of this paper is to construct complex analytic multiple L-function which interpolates generalized multiple Bernoulli numbers with [chi] at negative integers. Finally, we will discuss the Witt's type formulas for the generalized multiple Bernoulli numbers with [chi] which are newly introduced in this paper. 2. On Generalized Multiple Bernoulli Numbers with [chi] Let x, [w.sub.1], [w.sub.2],..., [w.sub.r] be complex numbers with [w.sub.i] [not equal to] = 0 for each i. Then we define the r-ple Bernoulli polynomials as follows: [[t.sup.r]/[[[PI].sub.i=1.sup.[infinity]]([e.sup.[w.sub.i]t] - 1)]][e.sup.xt] = [[infinity].summation over (n=0)] [[[B.sub.n.sup.(r)](x|[w.sub.1],..., [w.sub.r])]/n!][t.sup.n]. (1) This expression is valid for t with |t| < min(|2[pi]/[w.sub.1]|,..., |2[pi]/[w.sub.r]|), cf. [1, 7]. In the special case of x = 0, the numbers [B.sub.n.sup.(r)]([w.sub.1],..., [w.sub.r]) = [B.sub.n.sup.(r)](0|[w.sub.1],..., [w.sub.r]) will be called by the n-th r-ple Bernoulli numbers. By (1), we note that [B.sub.n.sup.(r)](x|[w.sub.1],..., [w.sub.r]) = [n.summation over (k=0)] nk[B.sub.k]([w.sub.1],..., [w.sub.r])[x.sup.n-k]. (2) Using the generating function of multiple Bernoulli numbers, we give the new formula for sums of products of Bernoulli numbers of the form [B.sub.n.sup.(r)]([w.sub.1],..., [w.sub.r]) = ([r.[product].[i=1]] [w.sub.i])[.sup.-1] [summation over ([n.sub.1]+ ... +[n.sub.r]=n)] n[n.sub.1],..., [n.sub.r] ([r.[product].[i=1]] [w.sub.i.sup.[n.sub.i]][B.sub.n.sub.i]), (3) where n[n.sub.1],..., [n.sub.r] are multinomial mul·ti·no·mi·al n. See polynomial. [multi- + (bi)nomial.] mul coefficients. Let f be the positive integer integer: see number; number theory . By Eq. (1), it is easy to see that [[t.sup.r]/[[[PI].sub.i=1.sup.r]([e.sup.-[w.sub.i]t] - 1)]][e.sup.-xt] = [f.sup.-r] [f-1.summation over ([m.sub.1],..., [m.sub.r]=0)] ([e.sup.-([1/f]([[summation].sub.i=1.sup.r]([w.sub.i][m.sub.i])+x))ft]/[[[PI].sub.i=1.sup.r](1 - [e.sup.-[w.sub.i]ft])]) (-ft)[.sup.r]. (4) Let us the Taylor expansion at t = 0 in Eq. (4), Eq. (1). Then we obtain the below distribution law for the multiple Bernoulli numbers: Proposition. For n [member of] N, we have [B.sub.n.sup.(r)](x|[w.sub.1],..., [w.sub.r]) = [f.sup.n-r] [f-1.summation over ([m.sub.1],..., [m.sub.r]=0)] [B.sub.n.sup.(r)]([[[[summation].sub.j=1.sup.r]([w.sub.j][m.sub.j]) + x]/f]|[w.sub.1],..., [w.sub.r]). (5) Definition 1. Let [chi] be the Dirichlet character with conductor f [member of] N. We now define the generalized multiple Bernoulli numbers with [chi] as follows: ([r.[product].[i=1]] [[[[summation].sub.a=1.sup.f][chi](a)[e.sup.a[w.sub.i]t]]/[[e.sup.f[w.sub.i]t] - 1]]) [t.sup.r] = [[infinity].summation over (n=0)] [[[B.sub.n,[chi].sup.(r)]([w.sub.1],..., [w.sub.r])]/n!][t.sup.n], (6) where |t| < min(|2[pi]/[w.sub.1]|,..., |2[pi]/[w.sub.r]|). By (4), (5), (6), note that [[infinity].summation over (n=0)] [[[B.sub.n,[chi].sup.(r)]([w.sub.1],..., [w.sub.r])]/n!][t.sup.n] = [1/[f.sup.r]] [f.summation over ([a.sub.1],..., [a.sub.r=1])] ([r.[product].[j=1]][chi]([a.sub.j]))[[(ft)[.sup.r][e.sup.([[[summation].sub.j=1.sup.r] [w.sub.j][a.sub.j]]/f])ft]/[[[PI].sub.j=1.sup.r]([e.sup.f[w.sub.j]t] - 1)]] = [[infinity].summation over (n=0)] ([f.sup.n-r] [f.summation over ([a.sub.1],..., [a.sub.r]=1)] ([r.[product].[j=1]][chi]([a.sub.j])) [B.sub.n.sup.(r)]([[[[summation].sub.j=1.sup.r] [a.sub.j][w.sub.j]]/f]|[w.sub.1],..., [w.sub.r]))[[t.sup.n]/n!]. Thus we have the distribution relations for the generalized r-ple Bernoulli numbers with [chi] as follows: [B.sub.n,[chi].sup.(r)]([w.sub.1],..., [w.sub.r]) = [f.sup.n-r] [f.summation over ([a.sub.1],..., [a.sub.r]=1)] ([r.[product].[j=1]][chi]([a.sub.j])) [B.sub.n.sup.(r)]([[[a.sub.1][w.sub.1] + ... + [a.sub.r][w.sub.r]]/f]|[w.sub.1],..., [w.sub.r]) = [f.sup.n-r]/[[[PI].sub.j=1.sup.r] [w.sub.j]] [f.summation over ([a.sub.1],..., [a.sub.r]=1)] ([r.[product].[j=1]][chi]([a.sub.j])) [summation over (n=[n.sub.1]+ ... +[n.sub.r])] n[n.sub.1],..., [n.sub.r] ([r.[product].[j=1]] [w.sub.j.sup.[n.sub.j]] [f.sup.[n.sub.j]][B.sub.n.sub.j] ([a.sub.j]/f)). (7) Remark. Dirichlet's L-function was defined by L(s, [chi]) = [[infinity].summation over (n=1)] [[chi](n)]/[n.sup.s], for s [member of] C, cf.[5, 7]. It were well known that the generalized Bernoulli numbers are closely related to the Dirichlet L-function by L(1 - n, [chi]) = -[[B.sub.n,[chi]]/n], for n [member of] N. In the next section we will consider the complex analytic multiple L-function which interpolates the generalized multiple Bernoulli numbers with [chi] at negative integers. 3. On Multiple L-function Barnes' r-ple zeta function [[zeta].sub.r](s, x|[w.sub.1],..., [w.sub.r]) depends on parameters [w.sub.1],..., [w.sub.r] that will be taken positive. That is, the r-ple Barnes' zeta function with parameters [w.sub.1],..., [w.sub.r] is defined by [[zeta].sub.r](s, x|[w.sub.1],..., [w.sub.r]) = [[infinity].summation over ([m.sub.1],..., [m.sub.r]=0)] 1/[(x + [w.sub.1][m.sub.1] + ... + [w.sub.r][m.sub.r])[.sup.s]], for Re(s) > r, Re(x) > 0. (8) Analytic continuation In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new region where an infinite series and special values are given by the contour contour or contour line, line on a topographic map connecting points of equal elevation above or below mean sea level. It is thus a kind of isopleth, or line of equal quantity. integral representation of [[zeta].sub.r](s, x|[w.sub.1],..., [w.sub.r]) as follows: [[zeta].sub.r](s, x|[w.sub.1],..., [w.sub.r]) = [[GAMMA The way brightness is distributed across the intensity spectrum by a monitor, printer or scanner. Depending on the device, the gamma may have a significant effect on the way colors are perceived. ](1 - s)[e.sup.-s[pi]i]]/2[pi]i [[integral].sub.I([lambda],[infinity])] [[[e.sup.-xt][t.sup.s-1]]/[[[PI].sub.i=1.sup.r](1 - [e.sup.-[w.sub.i]t])]]dt, cf.[1, 7], (9) where 0 < [lambda] < min(|2[pi]/[w.sub.1]|,..., |2[pi]/[w.sub.r]|) and I([lambda],[infinity]) is the path from +[infinity] to [lambda] along the real axis, going along the circle around 0 of radius [lambda] counterclockwise to [lambda], and then going back to +[infinity], (see [1]). It is easy to see that 1/[[[PI].sub.j=1.sup.r](1 - [e.sup.-[w.sub.j]t])] = [[infinity].summation over ([m.sub.1],..., [m.sub.r]=0)] [e.sup.-t([m.sub.1][w.sub.1]+ ... +[m.sub.r][w.sub.r])]. (10) Definition 2. Let r [member of] N. Then we define the multiple L-function of order r as follows: [L.sub.r](s, [chi]|[w.sub.1],..., [w.sub.r]) = [[infinity].summation over ([m.sub.1],..., [m.sub.r]=1)] [[chi]([m.sub.1]) ... [chi]([m.sub.r])]/[([m.sub.1][w.sub.1] + ... + [m.sub.r][w.sub.r])[.sup.s]], for s [member of] C. (11) Note that [L.sub.r](s, [chi]|[w.sub.1],..., [w.sub.r]) is a analytic continuation for R(s) > r with simple poles at s = 1, 2,..., r. By (8), (11), we easily see that [L.sub.r](s, [chi]|[w.sub.1],..., [w.sub.r]) = [f.sup.-s] [f.summation over ([a.sub.1],..., [a.sub.r]=1)] ([r.[product].[j=1]] [chi]([a.sub.j]))[[zeta].sub.r](s, [[[[summation].sub.j=1.sup.r]([w.sub.j][a.sub.j] + [w.sub.j])]/f]|[w.sub.1],..., [w.sub.r]). By (8), (10), (11), we have [1/[GAMMA](s)] [[integral].sub.0.sup.[infinity]] [[[[summation].sub.[a.sub.1],..., [a.sub.r]=1.sup.f]([[PI].sub.j=1.sup.r] [chi]([a.sub.j]))[e.sup.-[[summation].sub.j=1.sup.r]([a.sub.j][w.sub.j]+[w.sub.j])[.sup.t]][t.sup.s-1]]/[[[PI].sub.i=1.sup.r](1 - [e.sup.-f[w.sub.i]t])]]dt = [1/[GAMMA](s)] [[infinity].summation over ([m.sub.1],..., [m.sub.r]=0)] [f.summation over ([a.sub.1],..., [a.sub.r]=1)] [[integral].sub.0.sup.[infinity]] ([r.[product].[j=1]] [chi]([a.sub.j])) [e.sup.-[[summation].sub.j=1.sup.r]([a.sub.j][w.sub.j]+[m.sub.j][w.sub.j]f+[w.sub.j])[.sup.t]][t.sup.s-1]dt = [1/[GAMMA](s)] [[infinity].summation over ([n.sub.1],..., [n.sub.r]=0)] ([r.[product].[j=1]] [chi]([n.sub.j])) [[integral].sub.0.sup.[infinity]] [e.sup.-[[summation].sub.j=1.sup.r]([n.sub.j][w.sub.j]+[w.sub.j])[.sup.t]][t.sup.s-1]dt = [1/[GAMMA](s)] [[infinity].summation over ([n.sub.1],..., [n.sub.r]=0)] [[[PI].sub.j=1.sup.r] [chi]([n.sub.j])]/[([n.sub.1][w.sub.1] + ... + [n.sub.r][w.sub.r] + [w.sub.1] + ... + [w.sub.r])[.sup.s]] [[integral].sub.0.sup.[infinity]] [e.sup.-t][t.sup.s-1]dt = [L.sub.r](s, [chi]|[w.sub.1], [w.sub.2],..., [w.sub.r]). Therefore we obtain the following: Lemma lemma (lĕm`ə): see theorem. (logic) lemma - A result already proved, which is needed in the proof of some further result. . For s [member of] C, the multiple L-function of order r can be written by [L.sub.r](s, [chi]|[w.sub.1],..., [w.sub.r]) = [1/[GAMMA](s)] [[integral].sub.0.sup.[infinity]] [[[[summation].sub.[a.sub.1],..., [a.sub.r]=1.sup.f] (([[PI].sub.j=1.sup.r] [chi]([a.sub.j]))[e.sup.-[[summation].sub.j=1.sup.r]([a.sub.j][w.sub.j]+[w.sub.j])t])]/[[[PI].sub.i=1.sup.r](1 - [e.sup.-[w.sub.i]ft])]][t.sup.s-1]dt. (12) By the contour integral representation of [L.sub.r](s, [chi]|[w.sub.1], [w.sub.2],..., [w.sub.r]), we show that the r-ple L-function interpolates the generalized r-ple Bernoulli numbers with [chi] as follows: Theorem theorem, in mathematics and logic, statement in words or symbols that can be established by means of deductive logic; it differs from an axiom in that a proof is required for its acceptance. . For n [member of] N, we have [L.sub.r](-n, [chi]|[w.sub.1],..., [w.sub.r]) = [[(-1)[.sup.n]n!]/[(n + r)!]][B.sub.n+r,[chi].sup.(r)]([w.sub.1], [w.sub.2],..., [w.sub.r]). (13) In this paper we showed that the r-ple L-function [L.sub.r](s, [chi]|[w.sub.1],..., [w.sub.r]) has a meromorphic continuation in s [member of] C with simple poles at s = 1, 2,..., r. Remark. Let [Z.sub.p], [Q.sub.p] and [C.sub.p] be denoted by the ring of p-adic integers, the field of p-adic rational numbers and the completion of algebraic closure In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed. It is one of many closures in mathematics. of [Q.sub.p] and let U D([Z.sub.p], [C.sub.p]) be denoted by the space of uniformly differentiable dif·fer·en·tia·ble adj. 1. That can be differentiated: differentiable species. 2. Mathematics Possessing a derivative. functions from [Z.sub.p] to [C.sub.p], cf.[2, 3, 5]. For f [member of] U D([Z.sub.p], [C.sub.p]), the p-adic integral was defined by [[integral].sub.Z.sub.p] f(x)d[mu](x) = [lim lim abbr. Mathematics limit .[N[right arrow][infinity]]] 1/[p.sup.N] [[p.sup.N]-1.summation over (x=0)] f(x), cf.[2]. By the definition of p-adic integral, it is easy to see that [[integral].sub.Z.sub.p] [e.sup.tx]d[mu](x) = t/[[e.sup.t] - 1], cf.[3, 4, 5]. Let [w.sub.1], [w.sub.2],..., [w.sub.r] [member of] [Z.sub.p] and x [member of] [C.sub.p]. Then we easily see that [MATHEMATICAL EXPRESSION A group of characters or symbols representing a quantity or an operation. See arithmetic expression. NOT REPRODUCIBLE re·pro·duce v. re·pro·duced, re·pro·duc·ing, re·pro·duc·es v.tr. 1. To produce a counterpart, image, or copy of. 2. Biology To generate (offspring) by sexual or asexual means. IN ASCII ASCII or American Standard Code for Information Interchange, a set of codes used to represent letters, numbers, a few symbols, and control characters. Originally designed for teletype operations, it has found wide application in computers. ]. (14) By (1), (14), we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Let [chi] be the Dirichlet character with conductor f [member of] N. Then the generalized r-ple Bernoulli numbers with [chi] can be represented by p-adic integration as follows: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] = [[t.sup.r][[summation].sub.[a.sub.1],..., [a.sub.r]=1.sup.f] ([[PI].sub.j=1.sup.r] [chi]([a.sub.j])) [e.sup.([[summation].sub.j=1.sup.r][a.sub.j][w.sub.j])t]]/[[[PI].sub.i=1.sup.r] ([e.sup.[w.sub.i]ft] - 1)], (15) where [X.sub.f] [=.sub.N] Z/f[p.sup.N] Z, [X.sub.1] = [Z.sub.p], cf.[2]. By (6), (15), we obtain the Witt's formula for the generalized r-ple Bernoulli numbers with [chi] as follows: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (16) Using (5), (16), we can define the p-adic multiple distribution. However, we could not take the multiple Mellin transform In mathematics, the Mellin transform is an integral transform that may be regarded as the multiplicative version of the two-sided Laplace transform. This integral transform is closely connected to the theory of Dirichlet series, and is often used in number theory and the theory of to define multiple p-adic L-function which interpolates multiple Bernoulli numbers. Finally we would like to suggest the following question: Question. Is there multiple p-adic L-function which can be viewed as interpolating, in the same way that [L.sub.p](s, [chi]) interpolates L(s, [chi])? References [1] E. W. Barnes, On the theory of the multiple gamma functions In mathematics, the Gamma function (represented by the capitalized Greek letter Γ) is an extension of the factorial function to real and complex numbers. For a complex number z with positive real part it is defined by [2] M. Cenkci, M. Can and V. Kurt, p-adic interpolation interpolation In mathematics, estimation of a value between two known data points. A simple example is calculating the mean (see mean, median, and mode) of two population counts made 10 years apart to estimate the population in the fifth year. functions and Kummer-type congruences for q-twisted and q-generalized twisted Euler numbers
In mathematics, in the area of number theory, the Euler numbers are a sequence En , Adv. Stud stud 1. purebred. 2. a place, usually a farm, at which purebred animals are maintained and reproduced. stud animal an animal registered in a stud book. . Contemp. Math. 9 (2004), 203-216. [3] T. Kim, On explicit formulas of p-adic q--L-functions, Kyushu J. Math. 48 (1994), 73-86. [4] T. Kim, q-Volkenborn integration, Russian J. Math. Phys. 9 (2002), 288-299. [5] T. Kim, On p-adic q-L-functions and sums of powers, Discrete Math. 252 (2002), 179-187. [6] T. M. Rassias and H. M. Srivastava, Some classes of infinite series infinite series In mathematics, the sum of infinitely many numbers, whose relationship can typically be expressed as a formula or a function. An infinite series that results in a finite sum is said to converge (see convergence). One that does not, diverges. associated with the Riemann zeta and polygamma functions In mathematics, the polygamma function of order m is defined as the (m + 1)th derivative of the logarithm of the gamma function:
In mathematics, the n-th harmonic number is the sum of the reciprocals of the first n natural numbers: [7] K. Shiratani and S. Yamamoto, On a p-adic interpolation function for the Euler numbers and its derivative, Mem. Fac. Sci. Kyushu Univ. 39 (1985), 113-125. [8] E. T. Whittaker Edmund Taylor Whittaker (24 October1873 - 24 March1956) was an English mathematician, who contributed widely to applied mathematics, mathematical physics and the theory of special functions. and G. N. Watson (George) Neville Watson (31 January 1886 – 2 February 1965) was an English mathematician, a noted master in the application of complex analysis to the theory of special functions. His collaboration on the 1915 second edition of E. T. , A Course of Modern Analysis, Cambridge University Press Cambridge University Press (known colloquially as CUP) is a publisher given a Royal Charter by Henry VIII in 1534, and one of the two privileged presses (the other being Oxford University Press). , Cambridge, (1990). Taekyun Kim ([dagger]) Institute of Science Education, Kongju National University, Kongju 314-701, Republic of Korea Received April 24, 2005, Accepted June 11, 2005. * Dedicated to Professor H. M. Srivastava on the Occasion of his 65th Birthday. 2000 Mathematics Subject Classification. 11S80, 11B68, 11M99 ([dagger]) E-mail: tkimkongju.ac.kr |
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