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Kummer congruence for the Bernoulli numbers of higher order
Applied Mathematics and Computation 151 (2004) 589–593 www.elsevier.com/locate/amc
Kummer congruence for the Bernoulli numbers of higher order Leechae Jang a, Taekyun Kim
b,*
, Dal-Won Park
c
a
b
Department of Mathematics and Computer Science, KonKuk University, Chungju 380-701, South Korea Institute of Science Education, Kongju National University, Konju 314-701, South Korea c Department of Mathematics, Education Kongju National University, South Korea
Abstract The authors studied the properties of Bernoulli numbers of higher order [Appl. Math. Comput., in press; Bull. Aust. Math. 65 (2002) 59]. For q ¼ 1, we can also find their results [Proc. Jangjeon Math. Soc. 1 (2000) 97; Arch. Math. 76 (2001) 190; Proc. Jangjeon Math. Soc. 1 (2000) 161; Adv. Stud. Contemp. Math. 2 (2000) 9; Proc. Jangjeon Math. Soc. 2 (2001) 23; J. Math. Phys. A 34 (2001) L643; Proc. Jangjeon Math. Soc. 2 (2001) 19; Proc. Jangjeon Math. Soc. 2 (2001) 9; Proc. Jangjeon Math. Soc. 3 (2001) 63]. The authors suggested the question to inquire the proof of Kummer congruence for Bernoulli numbers of higher order [Appl. Math. Comput., in press]. In this paper we give a proof of Kummer type congruence for the Bernoulli numbers of higher order, which is an answer to a part of the question in [Appl. Math. Comput., in press]. Ó 2003 Elsevier Inc. All rights reserved. Keywords: Kummer congruences; Non-Archimedean integration; Volkenborn integrals; Bernoulli numbers
*
Corresponding author. E-mail addresses: [email protected] (L. Jang), [email protected], taekyun64@hotmail. com (T. Kim), [email protected] (D.-W. Park). 0096-3003/$ - see front matter Ó 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0096-3003(03)00314-X
590
L. Jang et al. / Appl. Math. Comput. 151 (2004) 589–593
1. Introduction Throughout this paper Zp , Qp and Cp will respectively denote the ring of padic rational integers, the field of p-adic rational numbers and the completion of algebraic closure of Qp . Let vp be the normalized exponential valuation of Cp with jpjp ¼ pvp ðpÞ ¼ p1 . Let l be a fixed integer and let p be a fixed prime number. We set X ¼ limðZ=lpN ZÞ; N
X ¼
[
ða þ lpZp Þ;
0
a þ lpN Zp ¼ fx 2 X j x aðmod lpN g; where a 2 Z lies in 0 6 a < lpN . For any positive integer N , we set l1 ða þ lpN ZÞ ¼
1 lpN
and this can be extended to a distribution on X (see [3–5]). This distribution yields an integral for non-negative integer m: Z xm dl1 ðxÞ ¼ Bm ; X
where Bm is called Bernoulli number. The Bernoulli numbers with order k, BðkÞ n , were defined by 1 t k X n t ; cf: ½5; 6: ¼ BðkÞ n t e 1 n! n¼0
ð1Þ
In [1,2], the authors studied the properties of Bernoulli numbers of higher order. But their results are special case for q ¼ 1 in [4–11]. In [1], the authors suggested the question which is related to Kummer congruence for Bernoulli numbers of higher order. In this paper we give a proof of Kummer type congruence for the Bernoulli numbers of higher order, which is an answer to a part of the their question for authors in [1]. 2. Kummer congruence for the Bernoulli numbers of higher order By using Volkenborn integral, it was well known that 1 Z X t tn n ¼ ; cf: ½4–11: x dl ðxÞ 0 et 1 n¼0 Zp n!
L. Jang et al. / Appl. Math. Comput. 151 (2004) 589–593
591
In [4], note that
1 t k X ¼ et 1 n¼0
Z Z
Z tn ðx þ x1 þ þ xk Þn dl1 ðx1 Þ dl1 ðx2 Þ dl1 ðxk Þ : n! X X X |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} ktimes
ð2Þ The Bernoulli polynomials with order k, BðkÞ n ðxÞ, were defined by BðkÞ n ðxÞ
¼
Z Z
Z
ðx þ x1 þ þ xk Þn dl1 ðx1 Þ dl1 ðx2 Þ dl1 ðxk Þ; X X X |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} ktimes
cf: ½3–7: ðkÞ In the case x ¼ 0, we write BðkÞ n ð0Þ ¼ Bn , cf. [4]. In [1], it was remained to prove the Kummer type congruence for the Bernoulli numbers of higher order. Now, we give the proof of Kummer type congruence for BðkÞ n by using T. KimÕs multidimensional q-Volkenborn integral at q ¼ 1 [3–11]. Let v be a Dirichlet character with conductor f . We set p ¼ p for p P 2, and p ¼ 4 for p ¼ 2. Let f ¼ ðf ; p Þ be denoted by the least common multiple of the conductor f of v and p . We define the generalized Bernoulli numbers of higher order with v as Z Z n BðmÞ ¼ vðx1 þ þ xm Þðx1 þ þ xm Þ dl1 ðx1 Þ dl1 ðxm Þ: ð3Þ n;v X
X
We easily get in (3) nm BðmÞ n;v ¼ l
l1 X x1 ;;xm ¼0
BðmÞ n
x þ þ x 1 m vðx1 þ þ xm Þ; l
ð4Þ
where Bn;v is the generalized ordinary Bernoulli number with v. By (3), we have 1 lim q m BðmÞ n;v ¼ q!1 ðf p Þ
X 1 6 x1 6 f pq
X
n
vðx1 þ þ xm Þðx1 þ þ xm Þ ;
1 6 xm 6 f pq
ð5Þ holds for n P 0. Herein as usual we set vðx1 þ þ xm Þ ¼ 0 if x1 þ þ xm is not prime to the conductor f . Thus
592
L. Jang et al. / Appl. Math. Comput. 151 (2004) 589–593
1 ð1 pnm vðpÞÞBðmÞ lim q m n;v ¼ q!1 ðf p Þ
X
X
1 6 x1 6 f pq
vðx1 þ þ xm Þ
1 6 xm 6 f pq
ðx1 þ þ xm Þn :
ð6Þ
We choose a rational number c 2 Z with ðc; f pÞ ¼ 1, c 6¼ 1. m Let x1 þ x2 þ þ xm run over the range 1 6 x1 þ þ xm 6 ðf pq Þ , ðx1 þ þ xm ; pÞ ¼ 1, xq1 þ xq2 þ þ xqm run over the range 1 6 xq1 þ m xq2 þ þ xqm 6 ðf pq Þ , ðxq1 þ xq2 þ þ xqm ; pÞ ¼ 1, and determine a number m rq ðxÞ 2 Z by xq1 þ xq2 þ þ xqm ¼ cx1 þ þ cxm þ rq ðxÞðf pq Þ . In [4], it is not difficult to prove that n n m ðxq þ þ xq Þ ðx1 þ þ xm Þ þ ðf pq Þ rq ðxÞ ððcx1 þ 1
m
þ cxm Þn þ nðcx1 þ þ cxm Þn1 Þ ðmodðf pq Þ2m Þ: Hence, we have X 1 ðf pq Þm 1 6 xq1 6 f pq
vðcÞ q m ðf p Þ þ
X
1 6 xq1
X
6 f pq
1 6 xq1 6 f pq
vðxq1 þ þ xqm Þðxq1 þ þ xqm Þn
1 6 xqm 6 f pq
X
X
ð7Þ
1 6 xqm
X
vðx1 þ þ xm Þðcx1 þ þ cxm Þ
n
6 f pq
vðcÞvðx1 þ þ xm Þrq ðxÞ
1 6 xqm 6 f pq
nðcx1 þ þ cxm Þ
n1
2m
ðmodðf pq Þ Þ:
ð8Þ
Now we define hxi ¼ x=wðxÞ, where wðxÞ nis the Teichm€ uller character. It is P easy to see that hxi 1 (mod p). Hence hxi 1 (mod pn ). We set vk ¼ vwk . By using (6)–(8) we have X
ð1 vn ðpÞÞð1 cm vm ðcÞÞBðmÞ n;vn ¼ lim
q!1
1 6 x1 6 f pq
X
rq ðxÞnvw1
1 6 xm 6 f pq
ððcx1 þ þ cxm Þhcx1 þ þ cxm i
n1 n1
Þ
:
ð9Þ By (9), we obtain the following congruence: Theorem A. For k1 k (mod pn ), we have ðmÞ
ð1 vk ðpÞÞð1 cm vk ðcÞÞ ðmod pnmp ðkk1 Þ Þ:
ðmÞ
Bk1 ;vk Bk;vk 1 ð1 vk1 ðpÞÞð1 cm vk1 ðcÞÞ k k1
L. Jang et al. / Appl. Math. Comput. 151 (2004) 589–593
593
Remark. The question to prove p-adic congruence of Kummer type for Bernoulli numbers of higher order in [1] is still open. Theorem A is an answer to a part of question in [1].
Acknowledgement This work was supported by KonKuk University in 2002.
References [1] Y. Jang, D.S. Kim, On higher order generalized Bernoulli numbers, Appl. Math. Comput. 137 (2003) 387–398. [2] M.S. Kim, J.W. Son, On a multidimensional Volkenborn integral and higher order Bernoulli numbers, Bull. Aust. Math. 65 (2002) 59–71. [3] T. Kim, S.H. Rim, Explicit formulas for the q-Bernoulli numbers of higher order, Proc. Jangjeon Math. Soc. 1 (2000) 97–107. [4] T. Kim, Sums of products of q-Bernoulli numbers, Arch. Math. 76 (2001) 190–195. [5] T. Kim, Remark on p-adic q-L-functions, Proc. Jangjeon Math. Soc. 1 (2000) 161–169. [6] T. Kim, S.H. Rim, Generalized CarlitzÕs q-Bernoulli numbers in the p-adic number field, Adv. Stud. Contemp. Math. 2 (2000) 9–19. [7] T. Kim, Some q-Bernoulli numbers of higher order associated with the p-adic q-integrals, Proc. Jangjeon Math. Soc. 2 (2001) 23–34. [8] T. Kim, A note on q-multiple zeta functions, J. Math. Phys. A 34 (2001) L643–L646. [9] T. Kim, A note on the q-analogs of Genocchu numbers, Proc. Jangjeon Math. Soc. 2 (2001) 19–22. [10] T. Kim, Remark on p-adic proofs for q-Bernoulli and Eulerian numbers of higher order, Proc. Jangjeon Math. Soc. 2 (2001) 9–15. [11] T. Kim, A note on the values of p-adic q-L-functions, Proc. Jangjeon Math. Soc. 3 (2001) 63– 68.
Kummer congruence for the Bernoulli numbers of higher order Leechae Jang a, Taekyun Kim
b,*
, Dal-Won Park
c
a
b
Department of Mathematics and Computer Science, KonKuk University, Chungju 380-701, South Korea Institute of Science Education, Kongju National University, Konju 314-701, South Korea c Department of Mathematics, Education Kongju National University, South Korea
Abstract The authors studied the properties of Bernoulli numbers of higher order [Appl. Math. Comput., in press; Bull. Aust. Math. 65 (2002) 59]. For q ¼ 1, we can also find their results [Proc. Jangjeon Math. Soc. 1 (2000) 97; Arch. Math. 76 (2001) 190; Proc. Jangjeon Math. Soc. 1 (2000) 161; Adv. Stud. Contemp. Math. 2 (2000) 9; Proc. Jangjeon Math. Soc. 2 (2001) 23; J. Math. Phys. A 34 (2001) L643; Proc. Jangjeon Math. Soc. 2 (2001) 19; Proc. Jangjeon Math. Soc. 2 (2001) 9; Proc. Jangjeon Math. Soc. 3 (2001) 63]. The authors suggested the question to inquire the proof of Kummer congruence for Bernoulli numbers of higher order [Appl. Math. Comput., in press]. In this paper we give a proof of Kummer type congruence for the Bernoulli numbers of higher order, which is an answer to a part of the question in [Appl. Math. Comput., in press]. Ó 2003 Elsevier Inc. All rights reserved. Keywords: Kummer congruences; Non-Archimedean integration; Volkenborn integrals; Bernoulli numbers
*
Corresponding author. E-mail addresses: [email protected] (L. Jang), [email protected], taekyun64@hotmail. com (T. Kim), [email protected] (D.-W. Park). 0096-3003/$ - see front matter Ó 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0096-3003(03)00314-X
590
L. Jang et al. / Appl. Math. Comput. 151 (2004) 589–593
1. Introduction Throughout this paper Zp , Qp and Cp will respectively denote the ring of padic rational integers, the field of p-adic rational numbers and the completion of algebraic closure of Qp . Let vp be the normalized exponential valuation of Cp with jpjp ¼ pvp ðpÞ ¼ p1 . Let l be a fixed integer and let p be a fixed prime number. We set X ¼ limðZ=lpN ZÞ; N
X ¼
[
ða þ lpZp Þ;
0
a þ lpN Zp ¼ fx 2 X j x aðmod lpN g; where a 2 Z lies in 0 6 a < lpN . For any positive integer N , we set l1 ða þ lpN ZÞ ¼
1 lpN
and this can be extended to a distribution on X (see [3–5]). This distribution yields an integral for non-negative integer m: Z xm dl1 ðxÞ ¼ Bm ; X
where Bm is called Bernoulli number. The Bernoulli numbers with order k, BðkÞ n , were defined by 1 t k X n t ; cf: ½5; 6: ¼ BðkÞ n t e 1 n! n¼0
ð1Þ
In [1,2], the authors studied the properties of Bernoulli numbers of higher order. But their results are special case for q ¼ 1 in [4–11]. In [1], the authors suggested the question which is related to Kummer congruence for Bernoulli numbers of higher order. In this paper we give a proof of Kummer type congruence for the Bernoulli numbers of higher order, which is an answer to a part of the their question for authors in [1]. 2. Kummer congruence for the Bernoulli numbers of higher order By using Volkenborn integral, it was well known that 1 Z X t tn n ¼ ; cf: ½4–11: x dl ðxÞ 0 et 1 n¼0 Zp n!
L. Jang et al. / Appl. Math. Comput. 151 (2004) 589–593
591
In [4], note that
1 t k X ¼ et 1 n¼0
Z Z
Z tn ðx þ x1 þ þ xk Þn dl1 ðx1 Þ dl1 ðx2 Þ dl1 ðxk Þ : n! X X X |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} ktimes
ð2Þ The Bernoulli polynomials with order k, BðkÞ n ðxÞ, were defined by BðkÞ n ðxÞ
¼
Z Z
Z
ðx þ x1 þ þ xk Þn dl1 ðx1 Þ dl1 ðx2 Þ dl1 ðxk Þ; X X X |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} ktimes
cf: ½3–7: ðkÞ In the case x ¼ 0, we write BðkÞ n ð0Þ ¼ Bn , cf. [4]. In [1], it was remained to prove the Kummer type congruence for the Bernoulli numbers of higher order. Now, we give the proof of Kummer type congruence for BðkÞ n by using T. KimÕs multidimensional q-Volkenborn integral at q ¼ 1 [3–11]. Let v be a Dirichlet character with conductor f . We set p ¼ p for p P 2, and p ¼ 4 for p ¼ 2. Let f ¼ ðf ; p Þ be denoted by the least common multiple of the conductor f of v and p . We define the generalized Bernoulli numbers of higher order with v as Z Z n BðmÞ ¼ vðx1 þ þ xm Þðx1 þ þ xm Þ dl1 ðx1 Þ dl1 ðxm Þ: ð3Þ n;v X
X
We easily get in (3) nm BðmÞ n;v ¼ l
l1 X x1 ;;xm ¼0
BðmÞ n
x þ þ x 1 m vðx1 þ þ xm Þ; l
ð4Þ
where Bn;v is the generalized ordinary Bernoulli number with v. By (3), we have 1 lim q m BðmÞ n;v ¼ q!1 ðf p Þ
X 1 6 x1 6 f pq
X
n
vðx1 þ þ xm Þðx1 þ þ xm Þ ;
1 6 xm 6 f pq
ð5Þ holds for n P 0. Herein as usual we set vðx1 þ þ xm Þ ¼ 0 if x1 þ þ xm is not prime to the conductor f . Thus
592
L. Jang et al. / Appl. Math. Comput. 151 (2004) 589–593
1 ð1 pnm vðpÞÞBðmÞ lim q m n;v ¼ q!1 ðf p Þ
X
X
1 6 x1 6 f pq
vðx1 þ þ xm Þ
1 6 xm 6 f pq
ðx1 þ þ xm Þn :
ð6Þ
We choose a rational number c 2 Z with ðc; f pÞ ¼ 1, c 6¼ 1. m Let x1 þ x2 þ þ xm run over the range 1 6 x1 þ þ xm 6 ðf pq Þ , ðx1 þ þ xm ; pÞ ¼ 1, xq1 þ xq2 þ þ xqm run over the range 1 6 xq1 þ m xq2 þ þ xqm 6 ðf pq Þ , ðxq1 þ xq2 þ þ xqm ; pÞ ¼ 1, and determine a number m rq ðxÞ 2 Z by xq1 þ xq2 þ þ xqm ¼ cx1 þ þ cxm þ rq ðxÞðf pq Þ . In [4], it is not difficult to prove that n n m ðxq þ þ xq Þ ðx1 þ þ xm Þ þ ðf pq Þ rq ðxÞ ððcx1 þ 1
m
þ cxm Þn þ nðcx1 þ þ cxm Þn1 Þ ðmodðf pq Þ2m Þ: Hence, we have X 1 ðf pq Þm 1 6 xq1 6 f pq
vðcÞ q m ðf p Þ þ
X
1 6 xq1
X
6 f pq
1 6 xq1 6 f pq
vðxq1 þ þ xqm Þðxq1 þ þ xqm Þn
1 6 xqm 6 f pq
X
X
ð7Þ
1 6 xqm
X
vðx1 þ þ xm Þðcx1 þ þ cxm Þ
n
6 f pq
vðcÞvðx1 þ þ xm Þrq ðxÞ
1 6 xqm 6 f pq
nðcx1 þ þ cxm Þ
n1
2m
ðmodðf pq Þ Þ:
ð8Þ
Now we define hxi ¼ x=wðxÞ, where wðxÞ nis the Teichm€ uller character. It is P easy to see that hxi 1 (mod p). Hence hxi 1 (mod pn ). We set vk ¼ vwk . By using (6)–(8) we have X
ð1 vn ðpÞÞð1 cm vm ðcÞÞBðmÞ n;vn ¼ lim
q!1
1 6 x1 6 f pq
X
rq ðxÞnvw1
1 6 xm 6 f pq
ððcx1 þ þ cxm Þhcx1 þ þ cxm i
n1 n1
Þ
:
ð9Þ By (9), we obtain the following congruence: Theorem A. For k1 k (mod pn ), we have ðmÞ
ð1 vk ðpÞÞð1 cm vk ðcÞÞ ðmod pnmp ðkk1 Þ Þ:
ðmÞ
Bk1 ;vk Bk;vk 1 ð1 vk1 ðpÞÞð1 cm vk1 ðcÞÞ k k1
L. Jang et al. / Appl. Math. Comput. 151 (2004) 589–593
593
Remark. The question to prove p-adic congruence of Kummer type for Bernoulli numbers of higher order in [1] is still open. Theorem A is an answer to a part of question in [1].
Acknowledgement This work was supported by KonKuk University in 2002.
References [1] Y. Jang, D.S. Kim, On higher order generalized Bernoulli numbers, Appl. Math. Comput. 137 (2003) 387–398. [2] M.S. Kim, J.W. Son, On a multidimensional Volkenborn integral and higher order Bernoulli numbers, Bull. Aust. Math. 65 (2002) 59–71. [3] T. Kim, S.H. Rim, Explicit formulas for the q-Bernoulli numbers of higher order, Proc. Jangjeon Math. Soc. 1 (2000) 97–107. [4] T. Kim, Sums of products of q-Bernoulli numbers, Arch. Math. 76 (2001) 190–195. [5] T. Kim, Remark on p-adic q-L-functions, Proc. Jangjeon Math. Soc. 1 (2000) 161–169. [6] T. Kim, S.H. Rim, Generalized CarlitzÕs q-Bernoulli numbers in the p-adic number field, Adv. Stud. Contemp. Math. 2 (2000) 9–19. [7] T. Kim, Some q-Bernoulli numbers of higher order associated with the p-adic q-integrals, Proc. Jangjeon Math. Soc. 2 (2001) 23–34. [8] T. Kim, A note on q-multiple zeta functions, J. Math. Phys. A 34 (2001) L643–L646. [9] T. Kim, A note on the q-analogs of Genocchu numbers, Proc. Jangjeon Math. Soc. 2 (2001) 19–22. [10] T. Kim, Remark on p-adic proofs for q-Bernoulli and Eulerian numbers of higher order, Proc. Jangjeon Math. Soc. 2 (2001) 9–15. [11] T. Kim, A note on the values of p-adic q-L-functions, Proc. Jangjeon Math. Soc. 3 (2001) 63– 68.