Kim and Jang Advances in Difference Equations (2015) 2015:4 DOI 10.1186/s13662-014-0340-3

RESEARCH

Open Access

A note on the Von Staudt-Clausen?s theorem for the weighted q-Genocchi numbers Byung Moon Kim1 and Lee-Chae Jang2* *

Correspondence: [email protected] 2 General Education Institute, Konkuk University, Chungju, 138-701, Korea Full list of author information is available at the end of the article

Abstract Recently, the Von Staudt-Clausen theorem for q-Euler numbers was introduced by Kim (Russ. J. Math. Phys. 20(1):33-38, 2013) and Araci et al. have also studied this theorem for q-Genocchi numbers (see Araci et al. in Appl. Math. Comput. 247:780-785, 2014) based on the work of Kim et al. In this paper, we give the corresponding Von Staudt-Clausen theorem for the weighted q-Genocchi numbers and also prove the Kummer-type congruences for the generated weighted q-Genocchi numbers. MSC: 11B68; 11S40 Keywords: Genocchi number; weighted q-Genocchi number; weighted q-Euler number; Von Staudt-Clausen theorem

1 Introduction and preliminaries As is well known, a theorem including the fractional part of Bernoulli numbers, which is called the Von Staudt-Clausen theorem, was introduced by Karl Von Staudt and Thomas Clausen (see []). In [], Kim has studied the Von Staudt-Clausen theorem for the q-Euler numbers and Araci et al. have introduced the Von Staudt-Clausen theorem associated with q-Genocchi numbers. Let p be a fixed odd prime number. Throughout this paper, Zp , Qp and Cp will denote the ring of p-adic integers, the field of p-adic rational numbers and the completion of the –  algebraic closure Qp . Let us assume that q is an indeterminate in Cp with | – q|p < p –p x where | · |p is a p-adic norm. The q-extension of x is defined by [x]q = –q . Note that –q limq→ [x]q = x. For f ∈ C(Zp ) = the space of all continuous functions on Zp , the fermionic p-adic q-integral on Zp is defined by Kim to be 

N

Zp

p –   f (x)(–q)x N→∞ [pN ]–q x=

f (x) dμ–q (x) = lim

(see [–]).

()

From (), we note that 

 q

Zp

f (x + ) dμ–q (x) +

Zp

f (x) dμ–q (x) = []q f ().

()

© 2015 Kim and Jang; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.

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From n ∈ N, we have  qn

Zp

 f (x + n) dμ–q (x) + (–)n–

= []q

n– 

f (l)(–)n–l– ql

Zp

f (x) dμ–q (x)

(see []).

()

l=

Let d ∈ N with d ≡  (mod ) and (p, d) = . Then we set Z/dpN Z, x = xd = lim ← – N



X∗ =

a + dpZp


and a + dpN Zp = {x ∈ X|x ≡ a (mod dpN )} where a ∈ Z lies in  ≤ a < dpN . It is well known that   f (x) dμ–q (x) = f (x) dμ–q (x), where f ∈ C(Zp ) (see [–]). () Zp

X

Recently, the weighted q-Euler numbers were introduced by the generating function to be ∞ 

(α) En,q

n=

tn = n!

 Zp

e[x]qα t dμ–q (x) =

∞   n=

 Zp

[x]nqα dμ–q (x)

tn n!

(see [, ]).

()

Thus, by (), we get  (α) (x) = En,q

Zp

[x]nqα dμ–q (x) (see [, ]),

where α ∈ Cp . Many researchers have studied the weighted q-Euler numbers and qGenocchi numbers in the recent decade (see [–]). From (), Araci defined the weighted q-Genocchi numbers as follows: ∞ 

(α) Gn,q

n=

tn =t n!

 Zp

e[x]qα t dμ–q (x) =

∞   n=

Zp

 [x]nqα dμ–q (x)

t n+ . n!

()

By (), we get (α) Gn+,q

n+

 =

Zp

[x]nqα dμ–q (x),

(α) G,q = .

()

The weighted q-Genocchi polynomials are also defined by ∞ 

t (α) Gn,q (x)

n=



n

=t

n!

Zp

e[x+y]qα t dμ–q (x).

()

Thus, by (), we have (α) (x) Gn+,q

n+

 =

Zp

[x + y]nqα dμ–q (y)

(n ≥ ).

()

Kim and Jang Advances in Difference Equations (2015) 2015:4

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Let us assume that χ is a Dirichlet character with conductor d ∈ N with d ≡  (mod ). Then we defined the generalized weighted q-Genocchi numbers attached to χ as follows: 

(α) Gn+,q,χ

=

n+

X

χ(x)[x]nqα dμ–q (x).

()

From (), we have 

(α) Gn+,q,χ

=

n+

X

χ(x)[x]nqα dμ–q (x) N

dp –   χ(x)(–)x [x]nqα = lim N→∞ [dpN ]–q x= d– [d]nqα  = (–)k χ(k)qk [d]–q k=



pN –  k  x dx lim (–) q x+ N→∞ [pN ]–qd d qdα x=

d– G(α) d ( k ) [d]nqα  k k n+,q d . (–) χ(k)q = [d]–q n+

()

k=

Theorem . Let χ be the Dirichlet character with conductor d ∈ N with d ≡  (mod ). For n ∈ N∗ = N ∪ {}, we have d– [d]nqα 

(α) Gn,q,χ =

[d]–q

(α) (–)k χ(k)qk Gn,q d

k=

  k . d

Next we give a familiar theorem, which is known as the Von Staudt-Clausen theorem. Lemma . (Von Staudt-Clausen theorem) Let n be an even and positive integer. Then Bn +



 ∈ Z. p p–|n,p:prime

Notice that pBn is a p-adic integer where p is an arbitrary prime number, n is an arbitrary integer and also Bn is a Bernoulli number as in []. The purpose of this paper is to show that the weighted q-Genocchi numbers can be described by a Von Staudt-Clausentype theorem. Finally, we prove a Kummer-type congruence for the generated weighted q-Genocchi numbers.

2 Von Staudt-Clausen theorems From (), we have (α) Gn+,q

n+

 =

Zp

[x]nqα dμ–q (x) =

[]q 

 Zp

qx [x]nqα dμ– (x).

Thus, by (), we have (α) Gn+,q

Gn+ = = lim q→ n +  n+

 Zp

xn dμ– (x) (see [–, ]).

()

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In [], Kim introduced the following inequality:   p–    j j  (–) [j]qα q  ≤ .  

()

j=

Let us define the following equality: for k ≥ , k

n– pk – n– n– . L(α) n– (k) = []qα – q[]qα + · · · + p –  qα q

()

From (), we note that

qd

(α) Gn+,q d (d)

n+

+

(α) Gn+,q d

= []q

n+

d– 

[l]nqd (–)l ql ,

l=

where d ∈ N with d ≡  (mod ). By () and (), we get lim nL(α) n– (k) =

k→∞

 (α) G . []q n,q

By (), we get L(α) n– (k + ) pk+ –

=



(–)a qa [a]n– qα

a= pk – p–

=



n– k k (–)a+jp qa+jp a + jpk qα

a= j= pk – p–

=



n– k k (–)a+jp qa+jp [a]qα + qαa jpk qα

a= j=

=

pk – p– n–   a= j= l=

=

 l n– (–)a+j qaαl jpk qα qa+jpk [a]n––l qα l

pk – p– n–   a= j= l=



l n– k (–)a+j qa(αl+)+jp pk qα [j]lqα pk [a]n––l qα l

pk –

=



(–)a qa [a]n– qα

a=

+

[]qpk

pk – p– n–  



l n– k [a]n––l (–)a+j qa(αl+)+jp pk qα [j]l αpk qα q l

a= j= l=

=

[]qpk

pk – p– n–   a= j= l=



l n– k (–)a+j qa(α+l)+jp pk qα [j]lqα pk [a]n––l qα l

()

Kim and Jang Advances in Difference Equations (2015) 2015:4

pk –

=



(–)a qa [a]n– qα

a=

+

Page 5 of 8

[]qpk []qpk

pk – p– n–  



l n– k (–)a+j qa(αl+)+jp pk qα [j]l αpk . [a]n––l qα q l

a= j= l=

()

Thus, by (), we get pk –

L(α) n– (k

+ ) ≡



  a a mod pk qα . [a]n– qα (–) q

()

a=

From (), we have pk+ –



a (–)a [a]n– qα q

a= k

=

p– p –  

a+pj (–)a+pj [a + pj]n– qα q

a= j=

=

p– 

pk – a a

(–) q



a=

 n– (–)j qpj [a]qα + qαa [p]qα [j]qαp

j=

p– p – n–     n –  = qαal [p]lqα [j]lqpα (–)a+j qa+pj [a]n––l qα l a= j= k

l=

 p–

=

(–)a qa [a]n– qα

[]qpk+

a=

[]qp

p– p – n–     n –  + [p]lqα [j]lqpα (–)a+j qa+pj+αal [a]n––l qα l a= j= k

l=

 p–

=

  mod [p]qα . (–)a qa [a]n– qα

()

a=

Therefore, by () and (), we obtain the following theorem. Theorem . Let L(α) n– (k) =

pk – a=

(–)a [a]n– qα . Then we have

pk –

L(α) n– (k

+ ) =



a a [a]n– qα (–) q .

a=

Furthermore pk –

 a=

a a [a]n– qa (–) q α

     mod [p]qα . mod pk qα ≡ (–)a qa [a]n– qα p–

a=

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By Theorem ., we get p– 

 a (–)a n[a]n– qα q =

a=

X

  (α) [x]n– qα dμ–q (x) ≡ Gn,q mod [p]q .

()

Therefore, by (), we have the following theorem. Theorem . For n ≥ , we have p– 

  (α) (–)a n[a]n– qα = Gn,q mod [p]q .

a=

From () and (), we note that (α) +n Gn+,q

p– 

a (–)a+ [a]n– q α q ∈ Zp

(n ≥ ).

a=

Corollary . For n ≥ , we have (α) Gn+,q

+n

p– 

a (–)a+ [a]n– q α q ∈ Zp .

a=

Let n ≥ . Then we observe that   (α)   (α) p– p–    Gn+,q   Gn+,q   a n a a a n  =  (–) [a] q + (–) q [a] – α α q q   n+   n+  p a= a=   (α) p– G   n+,q  a n  – ≤ max  (–) [a]qα   n+  a=

p

p

 p–      a a n  ,  (–) q [a]qα  ≤ .   a=

()

p

Therefore, we obtain the following theorem. Theorem . For n ≥ , we have (α) Gn+,q

n+

∈ Zp .

Let χ be the Dirichlet character d ∈ N with d ≡  (mod ). The generalized weighted q-Genocchi numbers attached to χ are introduced as follows: ∞ 

(α) Gn,q,χ

n=

∞  tn = []q t (–)m χ(m)e[m]qα t n! m=  = t χ(x)e[x]qα t dμ–q (x).

()

X

Let f = [f , p] be the least common multiple of the conductor f of χ and p. By (), we get  (α) Gn,q,χ =n

fpN – X

χ(x)[x]n– qα dμ–q (x) = n lim

n→∞

 x=

χ(x)(–)x [x]n– qα .

()

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Thus, we have 

(α) = n lim Gn,q,χ

χ(a)(–)a qa [a]n– qα

ρ→∞ ≤a≤f pρ ,(a,p)=

f pρ –



+ n[p]n– lim qα χ(p) ρ→∞

ρ→∞

≤a≤f pρ ,(a,p)=



= n lim

χ(a)(–)a qap [a]n– qα p

(α) n– χ(a)(–)a qa [a]n– qα + a[p]qα χ(p)Gn,qp ,χ .

()

≤a≤f pp ,(a,p)=

Therefore, by (), we obtain the following theorem. Theorem . For n ≥ , we have 

n lim

(α) (α) n– χ(a)(–)a qa [a]n– qα = Gn,q,χ – [p]qα χ(p)Gn,qp ,χ .

ρ→∞

()

≤a≤f pρ ,(a,p)=

Assume that w is the Teichmüller character by mod p. For a ∈ X ∗ , set a α = a : q α = [a]qα w(a)



. Note that |a α – |p < p p– , where a s = exp(s loga ) for s ∈ Zp . For s ∈ Zp , we define

(α) the weighted p-adic l-function associated with Gn,q,χ as follows:





(α) lp,q (s, χ) = lim ρ→∞

χ(a)(–)

a

a a –s α q

=

X∗

≤a≤f pρ ,(a,p)=

χ(x)x –s α dμ–q (x).

For k ≥ ,   klp,q  – k, χwk–  χ(a)(–)a qa [a]k– = k lim qα ρ→∞

 =k X

≤a≤f pρ



χ(x)[x]k– qα dμ–q (x) – k

 =k

X

χ(x)[x]k– qα dμ–q (x) –

(α) = Gx,q,χ –

pX

χ(x)[x]k– qα dμ–q (x)

k[]q χ(p) k– [p]qα []qp

[]q (α) χ(p)[p]k– qα Gk,qp ,χ . []qp

It is easy to show that   n a pα = exp pn loga α =  + pn loga α +   ≡  mod pn .

(pn logp a α ) !

+ ···

 X

χ(x)[x]k– qαp dμ–qp (x)

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(α) So, by the definition of lp,q ( – k, x), we get



(α) lp,q (–k, χ) = lim

ρ→∞

χ(a)(–)a qa a kα

≤a≤f pρ ,(a,p)=

≡ lim



ρ→∞



 χ(a)(–)a qa a kα mod pn ,

≤a≤f pρ ,(a,p)=

where k ≡ k (mod pn (p – )). Namely, we have   (α)  

  (α) lp,q –k, χwk ≡ lp,q –k , χwk mod pn . Theorem . For k ≡ k (mod pn (p – )), we have (α) (α) Gk(α) 

+,q,χ []q Gk+,qp ,χ []q Gk +,qp ,χ  – ≡ – mod pn . k+ []qp k +  k + []qp k + 

(α) Gk+,q,χ

Competing interests The authors declare that they have no competing interests. Authors? contributions All authors contributed equally to this work. All authors read and approved the final manuscript. Author details 1 Department of Mechanical System Engineering, Dongguk University, Gyeongju, 780-714, Korea. 2 General Education Institute, Konkuk University, Chungju, 138-701, Korea. Acknowledgements This paper was supported by Konkuk University in 2015. Received: 3 December 2014 Accepted: 22 December 2014 References 1. Araci, S, Acikgoz, M, Sen, E: On the von Staudt-Clausen?s theorem associated withq-Genocchi numbers. Appl. Math. Comput. 247, 780-785 (2014) 2. Kim, T: On the von Staudt-Clausen theorem for q-Euler numbers. Russ. J. Math. Phys. 20(1), 33-38 (2013) 3. Kim, T, Jang, L-C, Kim, Y-H: Some properties on the p-adic invariant integral on Zp associated with Genocchi and Bernoulli polynomials. J. Comput. Anal. Appl. 13(7), 1201-1207 (2011) 4. Kim, T: New approach to q-Euler polynomials of higher order. Russ. J. Math. Phys. 17(2), 218-225 (2010) 5. Kim, T: Identities on the weighted q-Euler numbers and q-Bernstein polynomials. Adv. Stud. Contemp. Math. (Kyungshang) 22(1), 7-12 (2012) 6. Kim, T, Kim, Y-H, Ryoo, CS: Some identities on the weighted q-Euler numbers and q-Bernstein polynomials. J. Inequal. Appl. 2011, 64 (2011) 7. Park, J-W: New approach to q-Bernoulli polynomials with weight or weak weight. Adv. Stud. Contemp. Math. 24(1), 39-44 (2014) 8. Ryoo, CS: A note on the weighted q-Euler numbers and polynomials. Adv. Stud. Contemp. Math. 21(1), 47-54 (2011) 9. Araci, S, Acikgoz, M, Jolany, H, Seo, J: A unified generating function of the q-Genocchi polynomials with their interpolation functions. Proc. Jangjeon Math. Soc. 15(2), 227-233 (2012) 10. Bayad, A, Kim, T: Identities for the Bernoulli, the Euler and the Genocchi numbers and polynomials. Adv. Stud. Contemp. Math. 20(2), 247-253 (2010) 11. Gaboury, S, Tremblay, R, Fugere, B-J: Some explicit formulas for certain new classes of Bernoulli, Euler and Genocchi polynomials. Proc. Jangjeon Math. Soc. 17(1), 115-123 (2014) 12. Jang, LC, Bell, ET: A study on the distribution of twisted q-Genocchi polynomials. Adv. Stud. Contemp. Math. 18(2), 181-189 (2009) 13. Jeong, J-H, Park, J-W, Rim, S-H: New approach to the analogue of Lebesgue-Radon-Nikodym theorem with respect to weighted p-adic q-measure on Zp . J. Comput. Anal. Appl. 15(7), 1310-1316 (2013) 14. Kim, DS, Kim, T: q-Bernoulli polynomials and q-umbral calculus. Sci. China Math. 57(9), 1867-1874 (2014) 15. Kim, T, Kim, DS, Dolgy, DV, Rim, SH: Some identities on the Euler numbers arising from Euler basis polynomials. Ars Comb. 109, 433-446 (2013) 16. Pak, HK, Rim, SH, Jeong, J: A note on the analogue of Lebesgue-Radon-Nikodym theorem with respect to weighted p-adic q-measure on Zp . J. Inequal. Appl. 2013, 15 (2013)