THE BERNOULLI PERIODIC FUNCTIONS MIGUEL A. LERMA Abstrct. We study slightly modified versions of the Bernoulli periodic functions with nicer structurl properties, nd use them to give very simple proof of the Euler-McLurin summtion formul. 1. Definitions The Bernoulli polynomils B n(x) cn be defined in vrious wys. 1 The following re two of them ([4, ch. 1], [1]): (1.1) (1.2) (1) By generting function: t e xt e t 1 = n= B n(x) tn n! (2) By the following recursive formuls (n 1): B (x) = 1, B n (x) = n B n 1(x), (1.3) 1 (1.4) B n(x) dx =. The first Bernoulli polynomils re: B (x) = 1 B 1(x) = x 1 2 B 2(x) = x 2 x + 1 6 B 3(x) = x 3 3 2 x2 + 1 2 x Dte: June 19, 22. 1 Here we use the nottion B n for the Bernoulli polynomils, nd reserve the nottion B n for the Bernoulli periodic functions. 1
THE BERNOULLI PERIODIC FUNCTIONS 2 The Bernoulli numbers re B n = B n(), nd the Bernoulli periodic functions re usully defined B n (x) = B n( x ). However here we normlize B 1 defining B 1 (k) = insted of 1/2 for k integer, so tht B 1 coincides with the normlized swtooth function: { x 1 if x Z (1.5) B 1 (x) = 2 if x Z, where x = x x = frctionl prt of x, x = integer prt of x. Also we will leve B (k) undefined for k integer in fct B should be defined s the distribution B (x) = 1 δ per (x), where δ per (x) = k= δ(x k) is the periodic Dirc s delt. 1..1. Properties of the Bernoulli Periodic Functions. (n 1): 1. B 1 (x) = swtooth function (eq. 1.5). 2. B n(x) = n B n 1 (x) for n > 2 or x Z. 3. 1 B n (x) dx =. 1..2. Fourier expnsions. The Fourier expnsion for the Bernoulli periodic functions is (n 1): (1.6) B n (x) = n! (2πi) n k= k e 2πikx k n, so: if k =, (1.7) Bn (k) = n! otherwise. (2πik) n This result lso holds in the distributionl sense for n =. 1.1. Polylogrithms. The Bernoulli periodic functions pper nturlly in expressions involving polylogrithms, together with the so
THE BERNOULLI PERIODIC FUNCTIONS 3 clled Clusen functions (see [3]): (1.8) Cl 2n 1 (θ) = (1.9) Cl 2n (θ) = for n 1. cos(kθ) k 2n 1, sin(kθ) k 2n, To be more precise, the polylogrithms cn be defined by the series: z k (1.1) Li n (z) = k n for n, z < 1, or by the following recursive reltions: (1.11) Li (z) = z 1 z, (1.12) Li n (z) = z Li n 1 (ξ) ξ dξ (n 1), in C \ [1, ). Note tht Li 1 (z) = log(1 z) is the usul logrithm. Li 2 (z) is the dilogrithm. 2 A generting function is (1.13) z e (t+1)u (e u z) du = 2 n= Li n (z) t n. The Bernoulli periodic functions nd the Clusen functions re relted to the polylogrithms in the following wy: (1.14) 2in! (2πi) n Li n(e 2πix ) = A n (x) + i B n (x), for x Z, where (1.15) A n (x) = ( 1) n+1 2 2n! (2π) n Cl n(2πx). We will cll the A n (x) conjugte Bernoulli periodic functions. The first ones re A (x) = cot πx, A 1 (x) = 2 log (2 sin πx ),... π The series (1.1) converges for z = 1 if n 2. 2 For some uthors the dilogrithm is Li 2 (1 z).
THE BERNOULLI PERIODIC FUNCTIONS 4 For n = 1 both Li 1 (e 2πix ) nd Cl 1 (2πx) diverge t x =, but (1.16) πi B 1 (x) = Li 1 (e 2πix sin 2πkx ) Cl 1 (2πx) = i, k nd the series becomes zero for x =, so our definition B 1 () = llows (1.16) to hold lso for x =. For n =, x Z, we esily compute (1.17) Cl (x) = i Li (e 2πix ) i 2 = 1 2 cot(πx). Also by definition Cl (k) = for k Z. Hence, (1.18) I{Li (e 2πix )} = Cl (x) for every x R. Finlly we observe tht for y > { } x (1.19) R Li (e 2πi(u+yi) ) du 1 2 = 1 2π rg { e 2πy e 2πix}, which tends to 1 2 B 1(x) s y + for every x R, hence (1.2) lim R { Li (e 2πi(x+yi) ) } = 1 y + 2 B (x) = 1 2 + 1 2 δ per(x) (where δ per is the periodic Dirc s delt) in the distributionl sense. We lso note tht the Bernoulli periodic functions nd their conjugtes hve hrmonic extensions to the upper hlf plne, given by the formul: (1.21) 2in! (2πi) n Li n(e 2πiz ) = A n (z) + i B n (z), for I(z) >.
THE BERNOULLI PERIODIC FUNCTIONS 5 2. The Euler-Mclurin Summtion Formul Theorem 2.1. Let f : [, b] C be q times differentible, b f (q) (x) dx <. Then for 1 m q: (2.1) n b b f(n) = f(x) dx + m ( 1) k k! + ( 1)m+1 m! ( Bk (b) f (k 1) (b) B k () f (k 1) () ) b B m (x) f (m) (x) dx, where f(k) for < b represents summtion modified by tking k b only hlf of f(k) when k = or k = b. Proof. (See [2]) We hve (2.2) b f(n) = k b = b f(x) d(x B 1 (x)) f(x) dx b f(x) d B 1 (x) Next, integrte by prts successively the lst integrl on the right hnd side of (2.2). 2..1. Sum of Powers. As n exmple of ppliction of the Euler- Mclurin summtion formul, we give the sum of the first m rth powers: S(m, r) = m n r = 1 r + 2 r + 3 r + + m r. n=1
THE BERNOULLI PERIODIC FUNCTIONS 6 Here f(x) = x r, so f (k) (x) = r!x (r k) /(r k)! for k =, 1,..., r, f (k) (x) = for k > r, nd m n r = x r dx n m Hence + r+1 ( 1) k k! = mr+1 r+1 + ( Bk (m) f (k 1) (m) B k () f (k 1) () ) ( 1) k B k () k! r! (r k + 1)! mr k+1 B r+1() = mr+1 + 1 r+1 ( 1) ( ) k r+1 Bk () m r k+1 B r+1() k S(m, r) = n m = 1 n r + mr 2 { ( r+1 k= ( 1) ( } ) ) k r+1 Bk m r k+1 B k r+1 where B k re the Bernoulli numbers B = 1, B 1 = 1/2, B k = B k () for k > 1.
THE BERNOULLI PERIODIC FUNCTIONS 7 References [1] Tom M. Apostol. Introduction to Anlytic Number Theory. Springer-Verlg, New York, 1976. [2] Rlph P. Bos, Jr. Prtil sums of infinite series, nd how they grow. Amer. Mth. Monthly, 84:237 258, 1977. [3] Leonrd Lewin. Polylogrtihms nd Associted Functions. North Hollnd, 1981. [4] Hns Rdemcher. Topics in Anlytic Number Theory. Springer-Verlg, 1973.