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On Explicit Formulae for Bernoulli Numbers and Their Counterparts in Positive Characteristic

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On Explicit Formulae for Bernoulli Numbers and Their Counterparts in Positive Characteristic View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Number Theory 113 (2005) 53–68 www.elsevier.com/locate/jnt On explicit formulae for Bernoulli numbers and their counterparts in positive characteristic Sangtae Jeonga,∗,1, Min-Soo Kimb,2, Jin-Woo Sonb,2 aDepartment of Mathematics, Inha University, Incheon, Republic of Korea 402-751 bDepartment of Mathematics, Kyungnam University, Masan, Republic of Korea 631-701 Received 17 December 2003; revised 18 June 2004 Available online 30 March 2005 Communicated by D. Goss Abstract We employ the basic properties for the Hasse–Teichmüller derivatives to give simple proofs of known explicit formulae for Bernoulli numbers (of higher order) and then obtain some parallel results for their counterparts in positive characteristic. © 2004 Elsevier Inc. All rights reserved. MSC: 11B68; 11R58 Keywords: Bernoulli numbers; Bernoulli–Carlitz numbers; Hasse–Teichmüller derivatives; Carlitz exponential 1. Introduction A typical way of taking the coefficient of a term of degree n in a formal power series over a field is to take the nth derivative of it evaluated at zero divided by n!. However, when the underlying field has positive characteristic, this method is no longer true. An alternative way for avoiding this obstacle is to replace the ordinary higher derivatives ∗ Corresponding author. Fax: +82 32 874 5615. E-mail addresses: [email protected] (S. Jeong), [email protected] (M.-S. Kim), [email protected] (J.-W. Son). 1 Supported by Inha University Research Grant(INHA-30359). 2 Supported by Kyungnam University Research Fund, 2004. 0022-314X/$ - see front matter © 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.jnt.2004.08.013 54 S. Jeong et al. / Journal of Number Theory 113 (2005) 53–68 with the Hasse–Teichmüller derivatives. Over a field of characteristic 0, two sets of the derivatives are closely related by (1) below. On the other hand, the Hasse–Teichmüller derivatives behave well since they are not a trivial operator over a field of positive characteristic. It is well known that the Bernoulli numbers have various arithmetical properties such as the von Staudt–Clausen theorem and the Kummer congruences and so on (see [IR]). They also occur as special values of the Riemann zeta function. Counterparts in positive characteristic of the Bernoulli numbers were first introduced by L. Carlitz. In what follows, we shall call them the Bernoulli–Carlitz numbers. Like the classical case, the Bernoulli–Carlitz numbers are very important in the arithmetic of the rational function field over a finite field. For example, they occur as special values of Goss’s zeta function and Carlitz [C1,Go1] established the von Staudt–Clausen theorem for the Bernoulli–Carlitz numbers. Various explicit formulae for Bernoulli numbers are known in the literature [Sa,Gu] but little is known about an explicit formula for Bernoulli– Carlitz numbers even if certain classes of such numbers [Ge] are explicitly known. In this paper, we first aim to give simple proofs of known formulae for Bernoulli numbers (of higher order) and then to derive some parallel results for their counterparts in positive characteristic. Throughout, as a main tool for proofs, we use the basic properties for the Hasse–Teichmüller derivatives, which will be briefly summarized in the next section for ease of reference. 2. Hasse–Teichmüller derivatives Let k be a field of any characteristic, k[z] be a polynomial ring in one variable z over k. For each integer n0 the Hasse–Teichmüller derivative H (n) of order n is defined by m H (n)(zm) = zm−n n for any integer m0 and extends to the polynomial ring k[z] by k-linearity. In this section, we will describe some basic properties of the Hasse–Teichmüller derivatives in some detail but refer to [H,HS,T,Co] for more details on H (n). When k has characteristic 0, H (n) is closely related by the ordinary higher derivative d n ( dz) : d n H (n) = 1 . n! dz (1) (n) We also see that for np, H is a non-trivial operator even when k has positive (p) p+1 p+1 characteristic p. For example, H (z ) = z since p ≡ 1 (mod p). Like the ordinary higher derivatives, the Hasse–Teichmüller derivatives satisfy the product rule [T], the quotient rule [GN] and the chain rule [H]. For later use, we first state the product rule or the Leibniz rule of H (n) in k[z] whose derivation follows immediately from the Vandermonde formula. S. Jeong et al. / Journal of Number Theory 113 (2005) 53–68 55 f g ∈ k[z] n H (n)(fg) = n H (i)(f )H (n−i)(g) Product rule I. For , and 1, i=0 . From this product rule we easily see by induction on j that it extends to j 2 factors: Product rule II. For fi ∈ k[z], n1 and j 2, (n) (i1) (ij ) H (f1 ···fj ) = H (f1) ···H (fj ). i1,...,ij 0, i1+···+ij =n In particular, Teichmüller [T] used the product rule I (with induction on n) to derive the following formula. Product rule III. For f ∈ k[z], n1 and j 2, j j(j − ) ···(j − k + ) H (n)(f j ) = f j−k 1 1 i1!···in! k=1 i1,···,in 0, i1+···+in=k i1+2i2+···+nin=n ( ) i (n) i ×(H 1 (f )) 1 ···(H (f )) n . As a special case, when j is equal to p, being prime characteristic of the base field k, we have a simpler formula: For a non-negative integer i, H (ip)(f p) = (H (i)(f ))p. By applying this formula iteratively to a pth power we derive the following formula of Brownawell [B]: H (ipe)(f pe ) = (H (i)(f ))pe . It is well known that the sequence of the Hasse–Teichmüller derivatives on k[z] extends uniquely to a higher k-derivation on any localization S−1k[z] of k[z]. In particular, H (n) on k[z] extends uniquely to a higher k-derivation on the rational function field k(z). Furthermore, the sequence of H (n) on k[z] extends uniquely to the completion of k[z] at any place v. For example, taking v = (z), H (n) extends continuously to k[[z]], the ring of formal power series, providing k[[z]] with a higher k-derivation, which then extends to a higher k-derivation on k((z)). Thus, H (n) on k(z) is extended (by 1/z-adic continuity) to the completion of k(z) at the infinite place v = 1/z, which we call the field k((z−1)) of formal Laurent series in the variable z−1 over k. We mention here that the product rule also holds over k((z−1)) hence over k(z) and it is Teichmüller [T] who gave a first proof for this rule. And the quotient rule was first 56 S. Jeong et al. / Journal of Number Theory 113 (2005) 53–68 observed by Göttfert [GN]. He derived it to investigate the characteristic polynomial associated with linear recurrence sequences over a finite field. The proof of the quotient rule follows from the product rule over k((z−1)) and induction on n. Quotient rule I. For f ∈ k[[z]] and n1, n j (n) 1 (−1) (i ) (i ) H = H 1 (f ) ···H j (f ). f f j+1 j=1 i1,···,ij 1, i1+···+ij =n There is another quotient rule for H (n) which follows from the formula in [Gu] and (1), together with the product rule II. Quotient rule II. For f ∈ k[[z]] and n1, n j (n) 1 n + 1 (−1) (i ) (i ) H = H 1 (f ) ···H j (f ). f j + 1 f j+1 j=1 i1,...,ij 0, i1+···+ij =n (n) As for the chain rule, Hx (f ) denotes the nth Hasse–Teichmüller derivative of a (n) function f in k[[z]] with respect to a variable x. The chain rule for Hz (f ) was derived by Hasse [H] but one for the ordinary higher derivatives was observed by L.F.A. Arbogast [K] in 1800. Chain Rule. For f ∈ k[[z]] and n1, n j (n) (j) (1) i (n) in Hz (f ) = Hx (f )(Hz (x)) 1 ···(Hz (x)) . i1,i2,...,in j=0 i1,...,in 0, i1+···+in=j i1+2i2+···+nin=n 3. Bernoulli numbers In this section we employ the Hasse–Teichmüller derivatives to provide simple proofs of known explicit formulas for Bernoulli numbers and their generalization of higher order. In the literature there is the old book by Saalschütz, which contains 88 explicit formulas as well as 49 recurrence relations for the Bernoulli numbers. So we recom- mend the reader to consult this book [Sa] for checking whether the formula he or she will derive is new or not. The Bernoulli numbers Bn are defined by the generating function in Q[[z]]: ∞ z Bn n = z . (2) ez − 1 n! n=0 S. Jeong et al. / Journal of Number Theory 113 (2005) 53–68 57 Alternatively they are also defined by the recursive formula n−1 n! B = 1,Bn =− Bj for n1. (3) 0 j!(n + 1 − j)! j=0 z = (ez − ) ∞ Bn zn By comparing the coefficients in the expansion of 1 n=0 n! , one can derive the recursive formula above. We now give a simple proof of the following formula for the Bernoulli numbers, which is equivalent to formula LXXI in [Sa]. Theorem 3.1. For n1, n j B = n! (− )j 1 .
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