SUMS OF PRODUCTS OF BERNOULLI NUMBERS OF THE SECOND KIND

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1 SUMS OF PRODUCTS OF BEROULLI UMBERS OF THE SECOD KID SUMS OF PRODUCTS OF BEROULLI UMBERS OF THE SECOD KID Ming Wu Deparmen of Mahemaics, JinLing Insiue of Technology, anjing , People s Republic of China mingwu1996@yahoo.com.cn Hao Pan Deparmen of Mahemaics, Shanghai Jiaoong Universiy, Shanghai , People s Republic of China haopan79@yahoo.com.cn Submied June 2006-Final Revision April 2007 ABSTRACT The Bernoulli numbers of he second ind b n are defined by b n n log1 +. In his paper, we give an explici formula for he sum b j1 b j2 b j. We also esablish a q-analogue for j 1 +j 2 + +j n j 1,j 2,...,j 0 n b b n n 1b n n 2b. The Bernoulli numbers B n are defined by I is well-nown cf. [5] ha for n > 1 B n n! n e 1. 2n B 2j B 2n 2j 2n + 1B 2n. 1 2j As a generalizaion of 1, in [2] Dilcher proved ha for n > /2 j 1 +j 2 + +j n j 1,j 2,...,j 0 2n 2j 1, 2j 2,..., 2j 2n! 1/2 2n! B 2j1 B 2j2 B 2j c B 2n 2 2n 2, 2 146

2 SUMS OF PRODUCTS OF BEROULLI UMBERS OF THE SECOD KID where he array {c } is given by c and c +1 1 c c 1 1 wih c 0 for < 0 and > 1/2. On he oher hand, he Bernoulli numbers of he second ind b n are given by b n n log1 +. And we se b 0 whenever < 0. I is easy o chec ha n 1 b n + 1 δ n,0, 3 where δ n,0 1 or 0 according o wheher n 0 or no. In [3], Howard used he Bernoulli numbers of he second ind o give an explici formula for degenerae Bernoulli numbers. And some 2-adic congruences of b n have been invesigaed by Adelberg in [1]. In his shor noe, we shall give an analogue of 2 for he Bernoulli numbers of he second ind. Define an array of polynomials { x} by and if > 0. a 1 0 x 1, a x 0 for < 0 and, x 1 x + 1a 1 x + x a 1 1 x 1 1 Theorem 1: Le 1 be an ineger. Then for any non-negaive ineger n j 1 +j 2 + +j n j 1,j 2,...,j 0 b j1 b j2 b j 1 nb n. 4 Proof: Le: s n j 1 +j 2 + +j n j 1,j 2,...,j 0 b j1 b j2 b j. Clearly s 1 n b n, whence 4 holds for 1. ow we mae an inducion on. For arbirary power series f, le [ n ]f deno he coefficien of n in f. I is easy o see ha log 1 + b j j j0 s n n. 147

3 SUMS OF PRODUCTS OF BEROULLI UMBERS OF THE SECOD KID Therefore ow s +1 n [ n ] log [n 1 ] log [ n 1 d 1 ] d log 1 + [ n 1 ] 1 d 1 d log [ n 2 ] d 1 d log d s n n 1 + d Thus by he inducion hypohesis on, d d 1 log 1 + n s n n 1.. s +1 n 1 n s n + n 1s n 1 n 1 nb n n 1 1 n 1b n 1 1 nb n n 1 n a +1 nb n. 1 1 n 1b n n + n 1 1 n 1 b n We are done. For example, subsiuing 2, 3 in 4, we obain ha and s 2 n n 1b n n 2b, 5 s 3 n 1 2 n 1n 2b n n 22n 5b n 32 b n 2. 6 For arbirary ineger n, le [n] q 1 qn 1 q. 148

4 SUMS OF PRODUCTS OF BEROULLI UMBERS OF THE SECOD KID We say ha [n] q is a q-analogue of he ineger n since lim q 1 [n] q n. Then [1] q 1 and [n a] q [n] q q n a [a] q. Define he q-logarihm funcion by log q n n1 which is convergen for < 1. Also define a q-analogue of he Bernoulli numbers of he second ind by b n q n Clearly we ge he q-analogue of 3 n ow we can give a q-analogue of 5. Theorem 2: For any ineger n 0, we have [n] q log q b n q [ + 1] q δ n,0. 7 n q 1 b qb n q [n 1] q b n q [n 2] q b q, 8 where we se b q 0 if < 0. Proof: We mae an inducion on n. When n 0, noing ha [ 1] q q 1 and b 0 q 1 by 7, so boh sides of 8 coincide wih q 1. ow assume ha n > 0 and 8 holds for smaller values of n. In view of 7, we have when < n. Then n b n q 1 j b n j q n q 1 b qb n q q b n q q b n q q b n q + n q 1 b q 1 j b n j q n 1 j n j q 1 b qb n j q n 1 j [n j 1] q b n j q + [n j 2] q b n j 1 q 149

5 SUMS OF PRODUCTS OF BEROULLI UMBERS OF THE SECOD KID where we apply he inducion hypohesis in he las sep. ow we now ha n 1 j n j q 1 b qb n j q n n [n] q n 1 j [n j 1] q b n j q + [n j 2] q b n j 1 q 1 j [n] q q n j 1 b n j q + [n 1] q q n j 2 b n j 1 q 1 j q n j 2 b n j 1 q Thus 1 j b n j q 1 j b n j 1 q + [n 1] q [j + 1] q [n] q b n q [n 1] q b q n 1 j q n j 1 b n j q n 1 j q n j 1 b n j q + n 1 j q n j 1 b n j q. n q 1 b qb n q q b n q [n] q b n q [n 1] q b q + q n 2 b q [n 1] q b n q [n 2] q b q. Remar: A q-analogue of 1 has been given by Saoh in [2]. ACKOWLEDGMET We are graeful o he anonymous referee for his/her valuable commens on his paper. We also han our advisor, Professor Zhi-Wei Sun, for his helpful suggesions. REFERECES [1] A. Adelberg. 2-adic Congruences of ölund umbers and of Bernoulli umbers of he Second Kind. J. umber Theory : [2] K. Dilcher. Sums of Producs of Bernoulli umbers. J. umber Theory : [3] F. T. Howard. Explici Formulas for Degenerae Bernoulli umbers. Discree Mah : [4] J. Saoh. Sums of Producs of Two q-bernoulli umbers. J. umber Theory : [5] R. Siaramachandrarao and B. Davis. Some Ideniies Involving he Riemann Zeafuncion, II. Indian J. Pure Appl. Mah : AMS Classificaion umbers: 11B68, 05A30, 11B j2