1. Basic properties of Bernoulli and Euler polynomials. n 1. B k (n = 1, 2, 3, ). (1.1) k. k=0. E k (n = 1, 2, 3, ). (1.2) k=0

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1 A ecture given in Taiwan on June 6, 00. INTRODUCTION TO BERNOULLI AND EULER POLYNOMIALS Zhi-Wei Sun Departent of Matheatics Nanjing University Nanjing The Peope s Repubic of China E-ai: zwsun@nju.edu.cn Hoepage: Abstract. In this ecture note we deveop the theory of Bernoui and Euer poynoias in an eeentary way so that idde schoo students can understand ost part of the theory. 1. Basic properties of Bernoui and Euer poynoias Definition 1.1. The Bernoui nubers B 0, B 1, B, are given by B 0 1 and the recursion B 0, i.e. B n 1 B (n 1,, 3,. (1.1 n The Euer nubers E 0, E 1, E, are defined by E 0 1 and the recursion 0 0 n E 0, i.e. E n 0 n E (n 1,, 3,. (1. By induction, a the Bernoui nubers are rationas and a the Euer nubers are integers. Beow we ist vaues of B n and E n with n 10. n B n 1 1/ 1/6 0 1/30 0 1/4 0 1/30 0 5/66 E n

2 ZHI-WEI SUN Definition 1.. For n N {0, 1,, }, the nth Bernoui poynoia B n (x and the nth Euer poynoia E n (x are defined as foows: B n (x 0 B x n and E n (x 0 ( E x 1 n. (1.3 Ceary both B n (x and E n (x are onic poynoias with rationa coefficients. Note that B n (0 B n and E n (1/ E n / n. Here we ist B n (x and E n (x for n 5. n B n (x 1 x 1 x x x 3 3 x + x x 4 x 3 + x 1 30 x 5 5 x x3 x 6 E n (x 1 x 1 x x x 3 3 x x 4 x x x5 5 x x 1 Lea 1.1. Let, N and. Then ( ( x. Proof. Ceary ( ( x x 0 i< (x i! 0 r< (x r! 0 j< (x j (!!!(! (. This ends the proof. Lea.. Let n N, and δ n, be 1 or 0 according as n or not. Then B n (1 B n (0 δ n,1 and E n (1 + E n (0 δ n,0. (1.3 Proof. By Definition 1., B n (1 B n (0 0 B B n 0 <n B δ n,1

3 INTRODUCTION TO BERNOULLI AND EULER POLYNOMIALS 3 and E n (1 + E n (0 0 1 E 0 n ( ( 1 1 n ( n E δ n,0. We are done. Theore 1.1. Let n N. Then we have B n (x + y B (xy n, and E n (x + y Aso, 0 0 E (xy n. (1.4 B n (x + 1 B n (x nx and E n (x E n (x x n. (1.5 Proof. By the binoia theore and Lea 1.1, B n (x + y B (x + y n B Siiary, 0 0 n 0 E n (x + y E E B x y n ( B x y n E 0 n 0 n x y n ( B (xy n. ( x + y 1 n ( x 1 y n ( 0 ( x 1 y n 0 B x y n E (xy n. In view of the above and Lea 1., B n (x + 1 B n (x (B (1 B (0 x n nx

4 4 ZHI-WEI SUN and E n (x E n (x This concudes the proof. 0 0 (E (1 + E (0 x n x n. Theore 1.. Let n N. Then we have the recursion B (x (n + 1x and E (x + E n (x x n. (1.6 Aso, 0 B n (1 x ( 1 n B n (x and E n (1 x ( 1 n E n (x. (1.7 Proof. By Theore 1.1, ( n+1 n + 1 B (x B (x1 n+1 B n+1 (x and 0 0 E (x + E n (x 0 0 B n+1 (x + 1 B n (x (n + 1x n In view of the above and Theore 1.1, (B (1 x ( 1 B (x 0 B (1 x + ( 1 n 0 0 E (x1 n + E n (x E n (x E n (x x n. n 0 B (x( 1 n+1 (n + 1(1 x n + ( 1 n (B n+1 (x 1 B n+1 (x ( 1 n ((n + 1(x 1 n (B n+1 ((x B n+1 (x 1 0. Siiary, (E (1 x ( 1 E (x + E n (1 x ( 1 n E n (x 0 ( n ( n E (1 x + E n (1 x ( 1 n E (x( 1 n + E n (x 0 (1 x n ( 1 n (E n (x 1 + E n (x On the basis of these two recursions, (1.7 foows by induction.

5 INTRODUCTION TO BERNOULLI AND EULER POLYNOMIALS 5 Coroary 1.1. Let n > 1 be an integer. (i When n is odd, we have B n (1/ E n 0, and B n 0 if n > 1. (ii If n is even, then E n (0 0. Proof. When n is odd, taing x 1/ in (1.7 we find that B n (1/ E n (1/ 0. Reca that E n n E n (1/. By (1.7, B n (1 ( 1 n B n (0 and E n (1 ( 1 n E n (0. This, together with (1.3, shows that B n 0 if n > 1 and n, and that E n (0 0 if n.. On the sus r0 r and r0 ( 1r r For N {0, 1, } and n Z + {1,, 3, }, we set It is we nown that S (n r and T (n ( 1 r r. (.1 r0 r0 S 0 (n n, S 1 (n n(n 1 and S (n n(n 1(n 1. 6 In 1713 J. Bernoui introduced the Bernoui nubers, and used the to express S (n as a poynoia in n with degree + 1. Later Euer introduced the Euer nubers to study the su T (n. Theore.1. Let and n be positive integers. Then S (n B +1(n B n n + 1< ( B 1 n +1 (. and T (n E (0 ( 1 n E (n ([ ] n S S (n. (.3

6 6 ZHI-WEI SUN Proof. By Theore.1, B +1 (x + 1 B +1 (x ( + 1x. Therefore ( + 1S (n (B +1 (r + 1 B +1 (r r0 ( + 1 B +1 (n B +1 B n +1 0 n +1 ( + 1 n ( B + ( n +1. 1< By Coroary 1.1 B 0 for 3, 5,, so (. foows. In view of Theore.1, E (x E (x x. Thus T (n ( 1 r (E (r + E (r + 1 We aso have T (n r0 r r0 ( ( 1 r E (r ( 1 r+1 E (r + 1 E (0 ( 1 n E (n. r0 r This ends our proof. r0 [(/] r +1 j0 Exape.1 As B 4 (x x 4 x 3 + x 1/30, we have Siiary, S 3 (n B 4(n B 4 4 S 4 (n B 5(n B 5 5 n4 n 3 + n 4 B 5(n 5 ([ ] n + 1 j S (n +1 S S (n. n (n 1 4 S 1 (n. n5 5 n4 + n3 3 n 30. Since E 3 (x x 3 (3/x + 1/6 and E 4 (x x 4 x 3 + (/3x, we have T 3 (n E 3(0 ( 1 n E 3 (n 1 ( 1 ( 1n n 3 3 n and 1 ( 1n 1 ( 1 n n (n 3 4 T 4 (n E 4(0 ( 1 n E 4 (n ( 1 ( n 4 n n.

7 INTRODUCTION TO BERNOULLI AND EULER POLYNOMIALS 7 Coroary.1. For any N we have ( ( E (x +1 x + 1 B +1 B +1. ( Proof. Whenever n {, 4, 6, } we have E (0 E (n T (n +1 S (n/ S (n B +1 (n/ B +1 (n + (1 +1 B Since both sides are poynoias in n, it foows that E (0 E (x +1 B +1 (x/ B +1 (x + (1 +1 B +1. ( + 1 If n {1, 3, 5, }, then E (0 + E (n T (n +1 S ((n + 1/ S (n B +1 ((n + 1/ B +1 (n + (1 +1 B So E (0 + E (x +1 B +1 ((x + 1/ B +1 (x + (1 +1 B +1. ( + 1 ( inus ( yieds (.4 iediatey. 3. Raabe s theore and its appications The foowing theore of Raabe pays iportant roes in the theory of Bernoui poynoias. Theore 3.1. Let > 0 and n 0 be integers. Then r0 ( x + r B n 1 n B n (x. (3.1

8 8 ZHI-WEI SUN Proof. For any 0, 1,, we have ( + 1 r0 (x + r r0 (B +1 (x + r + 1 B +1 (x + r B +1 (x + B +1 (x ( + B (x +1. This, together with Lea 1.1 and the recursion for Bernoui nubers, yieds that r0 ( x + r B n r n n n n + 1 Bn Bn n 0 0 (x + r 0 ( + 1 B (x 1 0 ( + 1 B n B (x ( ( n + 1 n B (xb n B (x B n n 1 B (xδ,n 1 n B n (x. This copetes the proof. Coroary 3.1. For n N we have E n (x ( B n+1 (x n+1 B n+1. (3. n + 1 Proof. Appying Theore 3.1 with, we obtain that B n+1 On the other hand, by Coroary.1, ( x B n+1 B n+1(x n. ( x + 1 B n+1 B n+1 n + 1 n+1 E n(x.

9 INTRODUCTION TO BERNOULLI AND EULER POLYNOMIALS 9 The first equation inus the second one yieds that B n+1 B n+1(x (n + 1E n (x n+1. which is equivaent to (3.. Fro Theore 3.1 we can deduce the foowing resut. Theore 3.. Let n N. Then ( 1 B n ( 1 n 1 B n. (3.3 When n, we have ( ( 1 B n B n (3 1 n 1 B n 3 3, (3.4 ( ( 1 3 B n B n n ( 1 n 1B n, ( ( ( 1 5 B n B n ( 1 n 1(3 1 n 1 B n 6 6. (3.6 Proof. Taing x 0 and in (3.1, we find that ( ( ( B n + B n 1 n B n (0, i.e. B n ( 1 n 1B n. Now we et n be even. Note that B n (1 x ( 1 n B n (x B n (x. (3.1 in the case x 0 and 3, yieds that ( ( 1 B n (0 + B n + B n 3 1 n B n, 3 3 which is equivaent to (3.4. Taing x 1/ and in (3.1, we get that ( ( ( 1/ + 0 1/ B n + B n 1 n B n. So ( ( ( B n B n n B n n ( 1 n 1B n. 4 4

10 10 ZHI-WEI SUN (3.1 in the case x 1/3 and, gives that therefore ( ( ( 1/ / B n + B n 1 n B n ; 3 ( ( ( ( B n B n 1 n B n B n ( 1 n 1(3 1 n 1 B n. This copetes the proof. Coroary 3.. Let n N. Then If n is odd, then ( ( 1 E n E n 3 3 E n (0 (1 n+1 B n+1 n + 1. (3.7 ( n+1 1(3 n 1 B n+1 n + 1. (3.8 Proof. Taing x 0 in (3. we obtain (3.7. Now et n be odd. Then E n (/3 ( 1 n E n (1/3 E n (1/3. By Coroary 3.1 and Theore 3., we have ( 1 E n 3 n + 1 ( 3 n 1 n + 1 ( ( ( 1 1 B n+1 n+1 B n ( n+1 1(3 n 1 B n+1 n + 1. Theore 3.3. Let Z + and n N. If then ( x + r ( 1 r B n+1 r0 B n+1 n+1 ( n 1(3 n 1 B n+1 n + 1 n E n(x; (3.9 if then ( x + r ( 1 r E n r0 E n(x n. (3.10

11 INTRODUCTION TO BERNOULLI AND EULER POLYNOMIALS 11 Proof. We use Coroary.1 and Theore 3.1. When is even, we have ( x + r ( 1 r B n+1 r0 / 1 s0 / 1 ( If is odd, then ( / 1 x + s B n+1 s0 ( x s B n+1 ( / 1 x/ + s ( (x + 1/ + s B n+1 B n+1 / / s0 s0 n ( ( n x + 1 Bn+1 Bn+1 ( ( x + 1 B n+1 B n+1 n + 1 n+1 r0 ( 1 r E n + r ( ( (x + r/ + 1 ( 1 r B n+1 r0 ( ( x + r ( 1 r B n+1 r0 ( x + r ( 1 r B n+1 r0 So (3.10 aso hods. n + 1 n E n(x. ( (x + r/ B n+1 + ( 1 r+ B n+1 + r + n + 1 ( n E n(x. 4. Nuber-theoretic properties of Bernoui nubers and Bernoui poynoias Let p be a prie. A rationa a/b with a, b Z and (b, p 1, wi be caed a p-integer. We et Z p denote the set of a p-integers. For x, y Z p and n N, by x y (od p n we ean that x y p n Z p. Lea 4.1. Let be a positive integer and p be a prie. Then pb Z p and S (p pb p pb 1 (od p. (4.1

12 1 ZHI-WEI SUN Furtherore, if p > 3 then S (p pb p pb 1 (od p. (4. Proof. By Theore.1, S (p B +1(p B ( + 1 j j1 ( + 1 p +1 B pb p j B +1 j 1 ( p + 1 pb. Ceary p ( and hence p /( + 1 Z p. So pb Z p by induction on. Observe that S (p pb 1 1 ( p p pb 1 + p + 1 pb 1< 1 ( 1 p 1 ( + 1 pb ( 1 p 1 1 ( + 1 pb. Obviousy p 1 /( 3 p/6 Z p, and p/6 pz p if p > 3. When {3, 4, }, we have p 1 ( ( , and p ( ( + 1 providing p 5. Thus, if {3, 4, }, then p 1 /((+1 p 1 /(+1 p 1 / Z p, oreover p 1 /(( + 1 pz p providing p > 3. In view of the above, (4.1 hods, and (4. is aso vaid if p > 3. Theore 4.1 (von Staudt-Causen. We have B + p 1 1 p Z for, 4, 6,. (4.3

13 INTRODUCTION TO BERNOULLI AND EULER POLYNOMIALS 13 Proof. Let > 0 be an even integer. Reca that B 1 0 if >. So, by Lea 4.1, we have S (p pb δ, p B 1 0 (od p. If p 1, then by Ferat s itte theore p 1 p 1 S (p r 1 1 (od p r1 r1 and hence B +1/p Z p. If p 1, then there is a g Z such that g 1 (od p, as (g 1S (p p 1 r1 (gr p 1 r1 r 0 (od p we have p S (p and hence B Z p. By the above, B + p 1 p 1 Z q for any prie q. So B + p 1 p 1 Z. We are done. Theore 4. (Beeger, Let p > 3 be a prie. Then (p 1! pb p 1 p (od p. (4.4 Proof. Wison s theore asserts that w p ((p 1! + 1/p Z. For any integer a 0 (od p et q p (a denote the Ferat quotient (a p 1 1/p. Then and hence (pw p 1 p 1 p 1 r1 p 1 p 1 r p 1 (1 + pq p (r 1 + p q p (r (od p r1 p 1 1 (p 1pw p (pw p 1 p (r p 1 1 S p 1 (p p +. By Theore 4., S p 1 (p pb p 1 (od p. So (p 1! pw p 1 pb p 1 r1 r1 p (od p. Theore 4.3. Let p be a prie and n > 0 be an even integer. (i (E. Kuer If p 1 n, then B n /n Z p, oreover B / B n /n (od p whenever n (od p 1. (ii (L. Caritz If p and p 1 n then (B n + p 1 1/n Z p.

14 14 ZHI-WEI SUN Theore 4.4 (Voronoi, Let n > 0 be even, p Z +, Z and (p, 1. Then p 1 [ j ( n 1 B n n j p j1 ] (od p. (4.5 Theore 4.5 (L. Euer. We have tan x ( 1 ( 1B (! and n1 1 x for x 1 (π ( 1 n (! B for 1,, 3,. ( π, π, Theore 4.6 (Kuer, Let p > 3 be a prie such that p does not divide the nuerator of B, B 4,, B p 3. Then x p + y p z p has no integer soutions with p xyz. Theore 4.7 (A. Granvie and Z. W. Sun, Let p be an odd prie reativey prie to a fixed q {5, 8, 10, 1}. Then we can deterine B p 1 (a/q B p 1 od p (with 1 a q and (a, q 1 as foows: ( a B p 1 B p 1 5 ( (ap p F p ( 5 p + 5p 1 1 (od p; p ( a ( B p 1 B p 1 8 ap p P p ( p + 4 p 1 1 (od p; p ( a B p 1 B p 1 15 ( ap p F p ( 5 p p (p 1 1 (od p; p p ( a ( 3 3 B p 1 B p 1 1 a p S p ( 3 p + 3(p p 3p 1 1 (od p; p where ( is the Jacobi sybo, and we define the foowing second-order inear recurrence sequences: F 0 0, F 1 1, and F n+ F n+1 + F n for a n 0 P 0 0, P 1 1, and P n+ P n+1 + P n for a n 0 S 0 0, S 1 1, and S n+ 4S n+1 S n for a n 0.