identities on the bernoulli and genocchi numbers and polynomials

Hindawi Publishing Corporation International Journal of Mathematics and Mathematical Sciences Volume 2012, Article ID 184649, pages doi:10.1155/2012/184649 Research Article Identities on the Bernoulli and Genocchi Numbers and Polynomials Seog-Hoon Rim,1 Joohee Jeong,1 and Sun-Jung Lee2 Department of Mathematics Education, Kyungpook National University, Taegu 702-701, Republic of Korea Department of Mathematics, Kyungpook National University, Daegu 702-701, Republic of Korea Correspondence should be addressed to Seog-Hoon Rim, shrim@knu.ac.kr Received June 2012; Accepted August 2012 Academic Editor: Yilmaz Simsek Copyright q 2012 Seog-Hoon Rim et al This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited We give some interesting identities on the Bernoulli numbers and polynomials, on the Genocchi numbers and polynomials by using symmetric properties of the Bernoulli and Genocchi polynomials Introduction Let p be a fixed odd prime number Throughout this paper Zp , Qp , and Cp will denote the ring of p-adic rational integers, the field of p-adic rational numbers, and the completion of the algebraic closure of Qp Let N be the set of natural numbers and Z N ∪ {0} The p-adic norm on Cp is normalized so that |p|p p−1 Let C Zp be the space of continuous functions on Zp For f ∈ C Zp , the fermionic p-adic integral on Zp is defined by Kim as follows: pN −1 I−1 f Zp f x dμ−1 x f x −1 lim N →∞ x 1.1 x see 1–16 From 1.1 , we have I−1 f1 see 1–16 , where f1 x f x −I−1 f 2f 1.2 International Journal of Mathematics and Mathematical Sciences ext Then, by 1.2 , we get Let us take f x t Zp ∞ 2t ext dμ−1 x et Gn n tn , n! 1.3 where Gn are the nth ordinary Genocchi numbers see 8, 15 From the same method of 1.3 , we can also derive the following equation: t Zp ex y t 2t dμ−1 y et ∞ ext Gn x n tn , n! 1.4 where Gn x are called the nth Genocchi polynomials see 14, 15 By 1.3 , we easily see that n n Gl xn−l l Gn x l 1.5 see 15 By 1.3 and 1.4 , we get Witt’s formula for the nth Genocchi numbers and polynomials as follows: Zp Gn , n xn dμ−1 x Zp Gn n n x y dμ−1 y x , for n ∈ Z 1.6 From 1.2 , we have Zp x n dμ−1 x Zp xn dμ−1 x 2δ0,n , 1.7 where the symbol δ0,n is the Kronecker symbol see 4, Thus, by 1.5 and 1.7 , we get G n Gn 2δ1,n 1.8 see 15 From 1.4 , we can derive the following equation: Zp 1−x n y dμ−1 y −1 1−x −1 n Zp x n y dμ−1 y 1.9 By 1.6 and 1.9 , we see that Gn n Thus, by 1.10 , we get Gn 2/n −1 n Gn n Gn n x −1 / n 1.10 International Journal of Mathematics and Mathematical Sciences From 1.5 and 1.8 , we have Gn n Gn n 2− 1 Gn − 2δ1,n n 1.11 tn n! 1.12 The Bernoulli polynomials Bn x are defined by et t ext −1 eB ∞ x t Bn x n see 6, 9, 12 with the usual convention about replacing Bn x by Bn x In the special case, x 0, Bn Bn is called the n-th Bernoulli number By 1.12 , we easily see that n n n−l x Bl l Bn x l B x n 1.13 see Thus, by 1.12 and 1.13 , we get reflection symmetric formula for the Bernoulli polynomials as follows: −1 n Bn x , Bn − x B0 1, B n − Bn 1.14 δ1,n 1.15 see 6, 9, 12 From 1.14 and 1.15 , we can also derive the following identity: −1 n Bn −1 Bn n Bn n Bn δ1,n 1.16 In this paper, we investigate some properties of the fermionic p-adic integrals on Zp By using these properties, we give some new identities on the Bernoulli and the Euler numbers which are useful in studying combinatorics Identities on the Bernoulli and Genocchi Numbers and Polynomials Let us consider the following fermionic p-adic integral on Zp as follows: n I1 Zp n l Bn x dμ−1 x l Gl n , Bn−l l l n Bn−l l for n ∈ Z Zp xl dμ−1 x N ∪ {0} 2.1 International Journal of Mathematics and Mathematical Sciences On the other hand, by 1.14 and 1.15 , we get I1 −1 n −1 n Zp Bn − x dμ−1 x n n Bn−l l l −1 n n n n n Bn−l l l −1 n − x l dμ−1 x Gl −1 n Bn−l −1 l l l l −1 Zp Bn Gl − 2δ1,l l −1 δ1,n 2.2 n n Gl n Bn−l l l l −1 n Bn Equating 2.1 and 2.2 , we obtain the following theorem Theorem 2.1 For n ∈ Z , one has −1 n n l Gl n Bn−l l l −1 n δ1,n 2.3 By using the reflection symmetric property for the Euler polynomials, we can also obtain some interesting identities on the Euler numbers Now, we consider the fermionic p-adic integral on Zp for the polynomials as follows: I2 Zp n l n l Gn x dμ−1 x n Gn−l l Zp xl dμ−1 x Gl n , Gn−l l l for n ∈ Z On the other hand, by 1.8 , 1.10 , and 1.11 , we get I2 −1 n−1 −1 n−1 Zp n l −1 n−1 n l Gn − x dμ−1 x n Gn−l l Zp − x l dμ−1 x Gl −1 n Gn−l −1 l l l 2.4 International Journal of Mathematics and Mathematical Sciences n−1 −1 n l −1 n−1 n −1 n Gn 2δ1,n − Gn n−1 −1 Gl − 2δ1,l l n Gn−l l Gl n Gn−l l l l 2.5 Equating 2.4 and 2.5 , we obtain the following theorem Theorem 2.2 For n ∈ Z , one has −1 n n Gl n Gn−l l l l −1 n Gn −1 n δ1,n 2.6 Let us consider the fermionic p-adic integral on Zp for the product of Bn x and Gn x as follows: I3 Zp m Bm x Gn x dμ−1 x n k l m n k l m k n Bm−k Gn−l l m k Gk l n Bm−k Gn−l l k l Zp xk l dμ−1 x 2.7 On the other hand, by 1.10 and 1.14 , we get I3 Zp Bm x Gn x dμ−1 x −1 n m−1 −1 n m−1 Zp m Bm − x Gn − x dμ−1 x n m k k l −1 n m−1 −1 n Bm−k Gn−l l Bm Gn n m−1 m n k l m k −1 m n Zp 1−x k l Bm Gn Gk l n Bm−k Gn−l l k l dμ−1 x 2.8 International Journal of Mathematics and Mathematical Sciences By 2.7 and 2.8 , we easily see that −1 m n m n k l −1 m n−1 δ1,m −1 m n−1 Bm δ1,n m n −1 Gk l n Bm−k Gn−l l k l m k Bm 2δ1,n − Gn −1 m n m n Bm Gn m n−1 −1 Bm Gn m n −1 δ1,m Gn −1 2.9 δ1,m δ1,n Bm Gn Therefore, by 2.9 , we obtain the following theorem Theorem 2.3 For n, m ∈ Z , one has −1 m n m n k l −1 m n Bm Gn m n −1 Gn−l Gk l n Bm−k l n−l 1k l m k −1 m n−1 m n−1 −1 Bm δ1,n δ1,m δ1,n 2.10 δ1,m Gn Corollary 2.4 For n, m ∈ N, one has 2m 2n k l 2m k Gk l 2n B2m−k G2n−l l k l 2B2m G2n 2.11 Let us consider the fermionic p-adic integral on Zp for the product of the Bernoulli polynomials and the Bernstein polynomials For n, k ∈ Z , with ≤ k ≤ n, Bk,n x n−k n k are called the Bernstein polynomials of degree n, see 11 It is easy to show k x 1−x that Bk,n x Bn−k,n − x , I4 Zp n k n k n k Bm x Bk,n x dμ−1 x m l m Bm−l l m n−k l j m n−k l j Zp xk l 1−x n−k dμ−1 x 2.12 m l n−k j −1 Bm−l m l n−k j −1 j Bm−l j Zp k x Gk l k l j dμ−1 x l j j International Journal of Mathematics and Mathematical Sciences On the other hand, by 1.14 and 2.12 , we get I4 −1 m −1 m −1 Zp n k n k m Bm − x Bn−k,n − x dμ−1 x m k l j m k l j × − 2δ1,n−k −1 k j −1 j Bm−l m l k j −1 j Bm−l n k m n−k l j dμ−1 x 2.13 l j m 1−x Zp Gn−k l j n−k l j n B δ0,k k m m −1 m l k −1 m l l j m k j n B δ k m k,n −1 j Bm−l Gn−k l j n−k l j Equating 2.12 and 2.13 , we see that m n−k l j n−k j m l −1 −1 m −1 j Bm−l −1 Bm δ0,k m m k l j Gk k l m l k j m l j j 2.14 Bm δk,n −1 j Bm−l Gn−k l j n−k l j Thus, from 2.14 , we obtain the following theorem Theorem 2.5 For n, m ∈ N, one has 2m n l j 2m l n j −1 j B2m−l Gl j l j 2m 2B2m l Gn l 2m B2m−l l n l 2.15 Finally, we consider the fermionic p-adic integral on Zp for the product of the Euler polynomials and the Bernstein polynomials as follows: I5 Zp n k Gm x Bk,n x dμ−1 x m l m Gm−l l Zp xk l 1−x n−k dμ−1 x International Journal of Mathematics and Mathematical Sciences m n−k n k l j m n−k n k l j m l n−k j −1 j Gm−l m l n−k j −1 j Gm−l Zp k xk Gk l l j dμ−1 x l j j 2.16 On the other hand, by 1.10 and 2.12 , we get I5 −1 m−1 −1 m−1 −1 Zp m n k n k m−1 l k m k Gm−l l j j m k m l l j k j −1 −1 m−1 n k m Zp 1−x n−k l j dμ−1 x 2.17 n Gm δ0,k k m−1 j −1 j Gm−l Gn−k l j − 2δ1,n−k n−k l j × 2 −1 Gm − x Bn−k,n − x dμ−1 x k l j −1 m l k j l j n Gm δk,n k m −1 j Gm−l Gn−k l j n−k l j Equating 2.16 and 2.17 , we obtain m n−k l j n−k j m l −1 −1 m−1 −1 j Gm−l Gm δ0,k m m−1 k l j m l Gk k l −1 k j m l j j 2.18 Gm δk,n −1 j Gm−l Gn−k l j n−k l j Therefore, by 2.18 , we obtain the following theorem Theorem 2.6 For n, m ∈ N, one has 2m n l j 2m l n j −1 j G2m−l Gl j l j −2G2m − 2m l Gn l 2m G2m−l l n l 2.19 International Journal of Mathematics and Mathematical Sciences Acknowledgment This paper was supported by Kynugpook National University Research Fund, 2012 References T Kim, “q-Volkenborn integration,” Russian Journal of Mathematical Physics, vol 9, no 3, pp 288–299, 2002 L Carlitz, “Note on the integral of the product of several Bernoulli polynomials,” Journal of the London Mathematical Society, vol 34, pp 361–363, 1959 T Kim, “A note on q-Volkenborn integration,” Proceedings of the Jangjeon Mathematical Society, vol 8, no 1, pp 13–17, 2005 T Kim, “On the multiple q-Genocchi and Euler numbers,” Russian Journal of Mathematical Physics, vol 15, no 4, pp 481–486, 2008 T Kim, “On the q-extension of Euler and Genocchi numbers,” Journal of Mathematical Analysis and Applications, vol 326, no 2, pp 1458–1465, 2007 T Kim, “Symmetry p-adic invariant integral on Zp for Bernoulli and Euler polynomials,” Journal of Difference Equations and Applications, vol 14, no 12, pp 1267–1277, 2008 T Kim, “Symmetry of 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