RESULTS ON VALUES OF BARNES POLYNOMIALS

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1 ROCKY MOUTAI JOURAL OF MATHEMATICS Volume 43, umber 6, 2013 RESULTS O VALUES OF BARES POLYOMIALS ABDELMEJID BAYAD AD TAEKYU KIM ABSTRACT. In this paper, we investigate rationality of the Barnes numbers, and we find explicit good bounds for their denominators. In addition, we give Fourier expansion of Barnes polynomials and, from this study, we connect generalized Barnes numbers to values of Dirichlet L-function at non-negative integers. 1. Introduction and statement of results. Throughout this paper we use the following notation: = {0, 1,...} set of naturals numbers. It s well-known that Bernoulli numbers and polynomials are connected to many areas in mathematics and physics. Their most important properties are that Bernoulli polynomials have an easy Fourier series with the coefficients expressed in closed form, and Bernoulli numbers are rational numbers, and their denominators are controlled by Von Staudt-Clausen theorem and Kummer congruences and are values of Riemann zeta function at positive integers, and the values of L-Dirichlet functions at negative integers give us the generalized Bernoulli numbers. In this paper, we prove the extension of these properties to Barnes numbers and polynomials. Let us recall the definitions of Bernoulli and Barnes polynomials and numbers. The Bernoulli polynomials B n (x) are defined by (1) te tx e t 1 = B n (x) tn, t < 2π. n! B n := B n (0) is the nth Bernoulli number. Let {x} be the fractional part of the real number x. Then the Bernoulli functions B n (x) are 2010 AMS Mathematics subject classification. Primary 11B68, 11B83, 11B99, 11M32. Keywords and phrases. Bernoulli numbers and polynomials, Barnes numbers and polynomials, Von Staudt theorem, Dirichlet L-function. The present research has been conducted by a Research Grant of Kwangwoon University in Received by the editors on February 13, 2011, and in revised form on June 22, DOI: /RMJ Copyright c 2013 Rocky Mountain Mathematics Consortium 1857

2 1858 A. BAYAD AD T. KIM defined by { 0 if n =1,x Z, (2) B n (x) = B n ({x}) otherwise. For more information about properties of Bernoulli numbers and their generalization, see [1, 4, 8, 9, 10, 12, 15]. Let positive integer and a 1,...,a complex with positive real part. The Barnes polynomials and numbers are given by (3) t j=1 (eajt 1) ext = B n (x a 1,...,a ) tn n!, ( ) 2π t < min a 1,..., 2π a (see [2, 3, 5, 7, 13, 14, 16]). Writing e(x) =e 2πix, x C. The essential series is e(uv) (4) 2πi e(u) 1 = e(nv) u n. n Z As Kronecker emphasized [11, 18], this identity is the foundation of the theory of classical Bernoulli functions and their relation to special values of the Riemann zeta function and Dirichlet L-functions. More precisely, let f 2 be integer and χ be a Dirichlet character modulo f (see [6, page 253]). The generalized Bernoulli numbers B m,χ Q(χ(1), χ(2),...,) associated to χ (m =0, 1,...) are defined by the generating function (5) f t χ(a) e ft 1 eat = a=1 t n 2π B n,χ, t < n! f. The main interest of these numbers is that they give the values at negative integers of Dirichlet L-series: if L(s, χ) = n=1 ( χ(n))/(n s ) (Re (s) > 1) is the L-series attached to χ, then we have the formula (6) L( n, χ) = B n+1,χ n +1 (n 0)

3 RESULTS O VALUES OF BARES POLYOMIALS 1859 (see [17, Theorem 4.2]). Indeed, in equation (4), expanding the left and the right side into a Laurent series in u yields at once the most important property Fourier expansion of the classical Bernoulli (7) B n (x) = n! k Z e(kx) k n, n 1, where the sum k Z means that k = 0 is to be omitted (and in the non-absolutely convergent case n = 1 the sum is to be interpreted as a Cauchy principal value). From this, it is not difficult to relate Bernoulli numbers and polynomials to special values of zeta and partial zeta functions defined over the field of rational numbers. This paper can now be summarized as a generalization of these facts to the Barnes polynomials and numbers. For λ C\{0}, notethat (8) t j=1 )t 1) eλxt = λ (λt) ( (e(λaj j=1 e a j(λt) 1 )ex(λt). Then, we obtain (9) B n (λx λa 1,...,λa ) tn n! = λ B n (x a 1,...,a ) λn t n. n! Therefore, by comparing the coefficients of both sides of equation (9), it s easy to see that Proposition 1 (Homogeneity). For any a 1,...,a non nul positive real numbers and λ C\{0}, we have (10) B n (λx λa 1,...,λa )=λ n B n (x a 1,...,a ), (n 1). ow we state our main results.

4 1860 A. BAYAD AD T. KIM Theorem 2 (Explicit formula and rationality). Let a 1,...,a non nul positive real numbers. Then be (11) B n (x a 1,...,a ) n! = B m1 (X) m 1! (X) Bm, (n ) m! where X = x/a and A = a a. In addition, if a 1,...,a are rational numbers, then [B n (x a 1,...,a )]/n! is a polynomial with rational coefficients. Theorem 3 (Fourier expansion). Let a 1,...,a be non nul positive real numbers, and set A = a a and X = x/a. Then, for any n 1 and X < 1, we have B n (x a 1,...,a )= ( 1) n! k 1,...,k Z\{0} e ((k k )X). k m1 1 k m Here k 1,...,k Z\{0} means that k 1,...,k Z\{0} and, in the nonabsolutely convergent case m i = 1, for any 1 i, the sum k i Z\{0} is to be interpreted as a Cauchy principal value for each i, and means that m 1,...,m with the usual convention the sum k e(k i Z\{0} ix) = 1. Theorem 4 (Universal bounds). Let a 1,...,a be non nul positive integers. Put λ(n) = p prime n p [n/(p 1)], (n 1). Then the denominator of [B n (0 a 1,...,a )]/n! divides λ(n), where [x] denotes the integer part of the real number x.

5 RESULTS O VALUES OF BARES POLYOMIALS 1861 Let q be an integer 2andχ a Dirichlet character modulo q. Weset L(s, χ) = χ(k) q k s, Gχ = χ(t)e(t/q), k=1 where s is a complex variable and L(s, χ) convergesforr(s) > 0ifχ is non principal and for the principal character it converges for R(s) > 1. Gχ is the so-called Gauss sum attached to the character χ. By homogeneity, Proposition 1, without loss of generality, we can assume for the following theorems that: a a =1. Theorem 5 (Values of L-function at non-negative integers). Let q 2 be a natural number, a 1,...,a non nul positive real numbers with a a =1,andχ a non trivial Dirichlet character modulo q 2. Then we have q χ(t)b n (t/q a 1,...,a ) 2 ( 1) (n!) (2πi) = n G χ a m 1 1 m 1 L(m 1, χ) L(m, χ), if χ( 1) = ( 1) n, 0 otherwise, with the usual convention L(0, χ) =( 1/2), χ denotes the conjugate character of χ, and where the summation such that runs on m1,...,m 0 m m = n, χ( 1) = ( 1) mi, i =1,...,. From Theorem 5 above we immediately get the following corollaries. Corollary 6 (Values of L-function for even character). Let q 2, n be natural numbers, a 1,...,a non nul positive real numbers with a a =1, χ an even non trivial Dirichlet character modulo

6 1862 A. BAYAD AD T. KIM q 2. Then we have q χ(t)b 2n (t/q a 1,...,a )= 2 ( 1) (2n)! (2πi) 2n G χ a 2j1 1 2j 1 L(2j 1, χ) L(2j, χ), j 1+ +j =n j 1,...,j 0 with the usual convention L(0, χ) = 1/2. Corollary 7 (Values of L-function for odd character). Let q 2, n be natural numbers, a 1,...,a non nul positive real numbers with a a =1and χ an odd non trivial Dirichlet character modulo q 2. Then we have q χ(t)b 2n+1 (t/q a 1,...,a )= 2 ( 1) (2n +1)! (2πi) 2n+1 G χ L(m 1, χ) L(m, χ). m 1+ +m =2n+1 m 1,...,m odd Corollary 8. Let a 1,...,a be non nul positive real numbers with a a =1and χ a non trivial Dirichlet character modulo q 2. If χ( 1) ( 1) n, we have q χ(t)b n (t/q a 1,...,a )=0. Remark 1. Our Theorem 5 gives another way to obtain formula (6). Precisely, we show how to use Theorem 5 to obtain Set L( n, χ) = B n+1,χ n +1 δ = (n 0). { 0 if χ( 1) = 1, 1 if χ( 1) = 1.

7 RESULTS O VALUES OF BARES POLYOMIALS 1863 Using the following functional equations G χ = χ( 1)q/Gχ, ( Γ(s)cos π s δ ) ( ) n = Gχ 2π 2 2i δ L(1 s, χ), q (see [17, Chapter 4, pages 29 37]), at s = n, weobtain 2(n!)G χl(n, χ) = nq 1 n L(1 n, χ). ow we take =1anda 1 = 1 and, by using Theorem 5, we derive the following equality 2(n!)G χ L(n, χ) q = χ(t)b n (t/q). Hence, L(1 n, χ) = 1 q n qn 1 χ(t)b n (t/q), and, from definition (5), it s easy to see that q n 1 q χ(t)b n (t/q) =B n,χ. Therefore, we get the well-known equality (6). 2. Proofs of main results. ProofofTheorem2. Writing X = x/a,wehave t 1 j=1 (eajt 1) ext = a i=1 ( = a i te X(ait) e ait 1 B m1 (X) m 1! ) (X) Bm t n. m!

8 1864 A. BAYAD AD T. KIM On the other hand, t j=1 (eajt 1) ext = B n n! (x a 1,...,a )t n. By comparing the right members of both equations above we obtain B n (x a 1,...,a ) n! = B m1 (X) m 1! (X) Bm, (n ). m! Let a 1,...,a be rational numbers. By using the fact that the classical Bernoulli polynomials are in Q[x], we obtain that [B n (x a 1,...,a )]/n! is a polynomial with rational coefficients. This completes the proof of our theorem. Proof of Theorem 3. Using equation (7) and Theorem 2, we can write B n (x a 1,...,a ) n! = m 1 + +m =n m 1,...,m 0 k 1,...,k Z\{0} = ( 1) m 1! m! m 1,...,m 0 k 1,...,k Z\{0} e((k k )X) k m1 1 k m ( 1) m 1! m! (2πi) m1+ +m e((k k )X). k m1 1 k m This yields the theorem. Proof of Theorem 4. Let n be non-negative integers, p a prime number, we denote by v p (n) thep-adic valuation of n. In order to prove our theorem we need the following two lemmas:

9 RESULTS O VALUES OF BARES POLYOMIALS 1865 Lemma 9. For any n, n 1 and p a prime number, we have v p (n!) { [ n p 1 ] if p 1 n, ] 1 if p 1 n. [ n p 1 For this, we set K = [log(n)/ log(p)]. It s easy to see that (1) If p 1 n, wehave i=1 v p (n!) = K [n/p i ]. i=1 K v p (n!) = [n/p i ] n n/p i = i=1 n p 1. (2) If p 1 n, wewriten = m(p 1) with m 1. Then we have v p (n!) n This proves Lemma 9. K n/p i = m(1 p K ) <m. i=1 Lemma 10. For any n, n 1 and p a prime number, let d n denote the denominator of B n /n!. We have v p (d n ) [ ] n. p 1 From the Von Staudt theorem [6, page 233], we know that { 0 if p 1 n, v p (denominator (B n )) = 1 if p 1 n.

10 1866 A. BAYAD AD T. KIM From this and Lemma 9, we obtain v p (d n )=v p (n!) + v p (denominator (B n )) Then we get our Lemma 10. [ ] n. p 1 ow we are ready to prove our theorem. Let D n be the denominator of [B n (0 a 1,...,a )]/n!. By Theorem 2, we get B n (0 a 1,...,a ) n! = B m1 m 1! Bm m! (n ). Therefore, we can write v p (D n ) max v p(d m1 d m ). From this relation and Lemma 10, we obtain ([ ] [ ]) m1 m v p (D n ) max + + p 1 p 1 [ ] [ ] m1 + + m n max =. p 1 p 1 Hence, the denominator D n of [B n (0 a 1,...,a )]/n! divides λ(n) = p prime n p[n/(p 1)]. This gives our theorem. ProofofTheorem5. Using Theorem 3, we have q χ(t)b n (t/q a 1,...,a ) = ( 1) n! 1 k m1 1 k m k 1,...,k Z\{0} q χ(t)e ((k k )t/q).

11 RESULTS O VALUES OF BARES POLYOMIALS 1867 Since q χ(t)e (kt/q) =χ(k)g χ, then we obtain q χ(t)b n (t/q a 1,...,a ) while k i Z\{0} = ( 1) n! k 1,...,k Z\{0} = ( 1) n! k 1,...,k Z\{0} G χ 1 χ(k k m1 1 k m k ) G χ χ(k 1 ) k m1 1 ) χ(k, k m 1 χ(k i ) =(1+χ( 1)( 1) mi ) L(m i, χ). k mi i Therefore, we get the following equalities q χ(t)b n (t/q a 1,...,a ) = ( 1) (n!) G χ m 1,...,m 0 (1 + χ( 1)( 1) mi ) L(m i, χ) i=1 = 2 ( 1) (n!) G χ m 1,...,m 0, χ( 1)=( 1) m i L(m 1, χ) L(m, χ).

12 1868 A. BAYAD AD T. KIM Since the summation is over m 1,...,m 0, m m = n, χ( 1) = ( 1) mi. It s easy to see that, if χ( 1) ( 1) n, then the sum is zero. Hence, we obtain our desired theorem. Acknowledgments. We are very grateful to the anonymous referee for many helpful comments and suggestions on a previous version of this work. REFERECES 1. T. Agoh and K. Dilcher, Higher-order recurrences for Bernoulli numbers, J. um. Theor. 129 (2009), A. Bayad and Y. Simsek, Dedekind sums involving Jacobi modular forms and special values of Barnes zeta functions, Annal. Inst. Fourier 61 (2011). 3., Identities on values at negative integers of twisted Barnes zeta functions, preprint. 4. K. Dilcher and L. Louise, Arithmetic properties of Bernoulli-Padé numbers and polynomials, J. um. Theor. 92 (2002), E. Friedman and S. Ruijsenaars, Shintani-Barnes zeta and gamma functions, Adv. Math. 187 (2004), K. Ireland and M. Rosen, A classical introduction to modern number theory, Springer, ew York, K. Katayama, Barne s multiple function and Apostol s generalized Dedekind sum, TokyoJ. Math , T. Kim, on-archimedean q-integrals associated with multiple Changhee q- Bernoulli polynomials, Russ. J. Math. Phys. 10 (2003), , On a p-adic interpolation function for the q-extension of the generalized Bernoulli polynomials and its derivative, Discr. Math. 309 (2009), , On Euler-Barnes multiple zeta functions, Russ. J. Math. Phys. 10 (2003), L. Kronecker, Zur Theorie der elliptischen Modulfunktionen 4 (1929), pages L.M. avas, F.J. Ruiz and J.L. Varona, The Möbius inversion formula for Fourier series applied to Bernoulli and Euler polynomials, J. Approx. Theor. 163 (2011), K. Ota, On Kummer-type congruences for derivatives of Barnes multiple Bernoulli polynomials, J. um. Theor. 92 (2002), S. Ruijsenaars, On Barnes multiple zeta and gamma functions, Adv. Math. 156 (2000), Y. Simsek, Generating functions of the twisted Bernoulli numbers and polynomials associated with their interpolation functions, Adv. Stud. Contemp. Math. 16 (2008),

13 RESULTS O VALUES OF BARES POLYOMIALS M. Spreafico, On the Barnes double zeta and Gamma functions, J. um. Theor. 129 (2009), L.C. Washington, Introduction to cyclotomic fields, Springer, ew York, A. Weil, Elliptic functions according to Eisenstein and Kronecker, Ergeb. Math. 88, Springer-Verlag, Berlin, Département de Mathématiques, Université d Evry Val d Essonne, Bd. F. Mitterrand, Evry Cedex, France address: abayad@maths.univ-evry.fr Division of General Education-Mathematics, Kwangwoon University, Seoul , Republic of Korea address: tkkim@kw.ac.kr