IDENTITIES INVOLVING BERNOULLI NUMBERS RELATED TO SUMS OF POWERS OF INTEGERS

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1 IDENTITIES INVOLVING ERNOULLI NUMERS RELATED TO SUMS OF POWERS OF INTEGERS PIERLUIGI MAGLI Abstract. Pointing out the relations between integer power s sums and ernoulli and Genocchi polynomials, several combinatorial identities are established. 1. Introduction and preliminaries In [3], the authors establish several identities involving ernoulli numbers by means of series expansion of trigonometric functions. We are able to recover some of these results and obtain new ones, with a surprisingly easier technique. The classical ernoulli and Genocchi numbers are respectively defined by exponential generating functions as follows: e 1 = n n n!, and 2 e + 1 = n G n n!. According to the identity e + 1 = e 1 2 e 2 1, G n = 2(1 2 n ) n. We can generate their associated polynomials by multiplying the generating functions by the factor e x, therefore they read as ( ) n n (x) = k x n k, (1.1) k ( ) n G n (x) = G k x n k. (1.2) k The Euler numbers arise from the hyperbolic secant s expansion 1 cosh = 2e e = n E n n!, (1.3) while the Euler polynomials are generated by 2e x e + 1 = It is easy to verify that, for all n N = 1, 2, 3,... G n (x) = n E n 1 (x) and E n (1/2) = E n 2 n. E n (x) n n!. (1.4) 140 VOLUME 46/47, NUMER 2

2 IDENTITIES INVOLVING ERNOULLI NUMERS AND SUMS OF INTEGERS Definition 1. Let P n n=0 be the sequence defined by the following exponential generating function sinh = 2 e e 2 1 = n P n n!. (1.5) This sequence is easily related to ernoulli numbers as follows: P n = 2 n n + G n = 2(1 2 n 1 ) n, (1.6) which can be confirmed by extracting the coefficient of n /n! on both sides of the functional relation e /2 e 1 e 1 = e / Denoting by P n (x) the associated polynomial ( ) n P n (x) = P k x n k, (1.7) k it follows that ( x P n (x) = 2 n n ), (1.8) 2 by comparing the generating functions of these polynomials. 2. Polynomials and sums of integer powers In the following proposition we point out how our cited polynomials are related to the sum of integer powers. Proposition 2. For λ N = 1, 2, 3,..., k n 1 = 1 n n(λ) ( 1) n n, (2.1) ( 1) k k n 1 For λ N 0 = 0, 1, 2,..., = ( 1)λ+1 2n ( 1) n 1 = 2n 1 n ( 1) k ( 1) n 1 = ( 1) λ 2n 2 n Gn (λ) ( 1) λ+n G n. (2.2) n ( 1 + λ) 2 n G n, (2.3) 2 n G n ( 1 + λ) 2 ( 1)λ n E n 1. (2.4) 2 n 1 Proof. The proof is equivalent for each item, so we just prove the first one. Let us consider the generating function of the ernoulli polynomials (x ) = e 1 ex and the sequence produced for x N. Therefore, e (1 ) = e 1 = e 1 + [n /n!] n + δ n,1 = ( 1) n n, MAY 2008/

3 THE FIONACCI QUARTERLY (2 ) = e2 e 1 =. e e 1 + e (λ ) = eλ e 1 = e(λ 1) e 1 + e(λ 1) where λ N, so (2.1) is verified by induction. [n /n!] ( 1) n n + n, [n /n!] ( 1) n n + n The last proposition leads us to establish many identities involving ernoulli numbers. In the following the more remarkable instances are displayed. Particularly, (2.5) corresponds to the results in [3, Theorem 1]. Proposition 3. For λ N, λ = 2n ( 1) k 2n λ 2n+1 1 ( 1) λ δ 0,n +, (2.5) 2λ λ = 2n 1 ( 1) k 2n 1 λ 2n ( 1) ( λ 1 2 2n) 2n. (2.6) λ2n Proof. For λ N, we obtain the following polynomial summation n (λ) + 1 ( ) n (2 2 G n(λ) = λ ) n 2 k k k λ k in view of the connection between ernoulli and Genocchi numbers. On the other hand, (2.1)-(2.2) lead us to ( ) n (2 λ ) n 2 k k = n 1 + ( 1) +1 k n 1 + ( 1) n k λ k n + ( 1)n+λ G n. 2 Splitting up this identity according to the parity of n, it is easy to obtain (2.5)-(2.6) by taking into account that 2n+1 = δ n,0 /2 and G 2n+1 = δ n,0, n N 0. k n 1 For λ = 1, 2, 3 in the previous proposition, we readily obtain the following identities. Example 4. ) (2 2 ) = δ 0,n, (2.7a) 2 = 2n n, (2.7b) 3 = 22n+1 3 2n+1 (2n + 1) + δ 0,n 3, (2.7c) 142 VOLUME 46/47, NUMER 2

4 IDENTITIES INVOLVING ERNOULLI NUMERS AND SUMS OF INTEGERS (2 2 ) = 2 2n 2n, (2.7d) 2 = 4n 2 + ( 2 2 2n) 2n, (2.7e) 2n 22n 3 = 22n 3 (2n + 2n). (2.7f) 2n Evaluating n (λ + 1/2) G n(λ + 1/2) according to (2.3)-(2.4), we obtain the following proposition. Proposition 5. For λ N 0, (2 2 ) (2λ + 1) = 4n + 2 (2λ + 1) 2n ( 1) ( 1) 2n + ( 1)λ (2n + 1) (2λ + 1) E 2n, (2.8) 2n+1 2 (2 2 ) (2λ + 1) 4n = 1 + ( 1) ( 1) 2n 1 (2λ + 1) 2n + ( 2 2 2n) 2n. (2.9) (2λ + 1) 2n For λ = 0, 1, 2 in the previous proposition we readily obtain the following identities. Example (2 2 ) = (2n + 1) E 2n, (2.10a) (2 2 ) 3 = 8n n n+1 3 E 2n, (2.10b) 2n (2 2 ) (8n + 4) 32n 5 = + 2n n+1 5 E 2n, (2.10c) 2n+1 2 (2 2 ) = (2 2 2n ) 2n, (2.10d) 2 (2 2 ) 3 = 8n 2 22n 32n 3 2n 2n, (2.10e) ( 2n ) 2 (2 2 ) 5 = 8n 5 2n 32n n 5 2n 2n. (2.10f) Evaluating n (2λ)+ 1 2 P n(2λ) according to (2.1)-(2.3), we obtain the following proposition. MAY 2008/

5 THE FIONACCI QUARTERLY Proposition 7. For λ N, 1 (2λ) = 2n + 1 (2λ) 2n+1 2λ 1 k 2n + ( 1) 2n + 2n δ 0,n, (2.11) 4λ ( ) 2n λ 1 2n (2λ) = k 2n 1 + ( 1) 2n 1 (2λ) 2n n 1 (2λ) 2n 2n + n 2λ. (2.12) For λ = 1, 2, 3 in the previous proposition we readily obtain the following identities. Example 8. ) (2 2 1 ) = 2n n (1 + 2n + δ n,0), (2.13a) 1 4 = 4n n+1 ( n n) (1 + 2n + δ n,0), (2.13b) 1 6 = 4n + 2 ( n n + 2 4n n) 6 2n (1 + 2n + δ n,0), (2.13c) (2 2 1 ) = n 2 + n 2 + ( 2 2 2n 1) 2n, (2.13d) 2n 2 22n 1 4 = n 4 + 2n 2 (1 + 4n 1 22n n 1 ) + ( 2 2 2n 1) 2n, (2.13e) 42n 1 6 = n 6 + n 3 (1 + 2n 22n n n n 1 ) + ( 2 2 2n 1) 2n. (2.13f) 62n Evaluating n (2λ+1)+ 1 2 P n(2λ+1) according to (2.1), we obtain the following proposition. Proposition 9. For λ N 0, 1 (2λ + 1) = 2 2n + 1 k 2n + 2 2n (2λ + 1) 2n+1 k 2n 144 VOLUME 46/47, NUMER 2

6 IDENTITIES INVOLVING ERNOULLI NUMERS AND SUMS OF INTEGERS + 2n δ 0,n, (2.14) 2(2λ + 1) ( ) 2n (2λ + 1) 2n = k 2n n 1 k 2n 1 (2λ + 1) 2n + (1 + 22n 1 ) 2n (2λ + 1) 2n + n 2λ + 1. (2.15) For λ = 0, 1, 2 in the previous proposition we readily obtain the following identities. Example (2 2 1 ) = 1 2 (1 + 2n + 2δ n,0), (2.16a) 1 3 = n n+1 6 (1 + 2n + 2δ n,0), (2.16b) 1 5 = 2n + 1 ( n n + 2 4n+1) 5 2n (1 + 2n + 2δ n,0), (2.16c) (2 2 1 ) = n + ( n 1 ) 2n, (2.16d) 1 3 = n 3 + 2n(1 + 22n ) n 1 3 2n 3 2n 2n, (2.16e) ( 2n ( 2n 1 5 = n 5 + 2n 5 2n ( n + 3 2n n 1) n ( n 1 ) 2n. (2.16f) References [1] L. Comtet, Advanced Combinatorics, Dordrecht-Holland, The Netherlands, [2] R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics, Addison-Wesley Publ. Company, Reading, Massachusetts, [3] G. Liu and H. Luo, Some Identities Involving ernoulli Numbers, The Fibonacci Quarterly, 43.3 (2005), MSC2000: 05A19, 1168 Department of Mathematics, Università del Salento, Lecce, Italy address: pierluigi.magli@unile.it MAY 2008/