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Abstract and Applied Analysis
VolumeΒ 2012, Article IDΒ 948050, 6 pages
http://dx.doi.org/10.1155/2012/948050
Research Article

On the Modified π‘ž-Bernoulli Numbers of Higher Order with Weight

T. Kim,1Β J. Choi,2Β Y.-H. Kim,2Β and S.-H. Rim3

1Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea
2Division of General Education, Kwangwoon University, Seoul 139-701, Republic of Korea
3Department of Mathematics Education, Kyungpook National University, Daegu 702-701, Republic of Korea

Received 11 August 2011; Revised 24 November 2011; Accepted 13 December 2011

Academic Editor: Ferhan M.Β Atici

Copyright Β© 2012 T. Kim 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.

Abstract

The purpose of this paper is to give some properties of the modified π‘ž-Bernoulli numbers and polynomials of higher order with weight. In particular, by using the bosonic 𝑝-adic π‘ž-integral on ℀𝑝, we derive new identities of π‘ž-Bernoulli numbers and polynomials with weight.

1. Introduction

Let 𝑝 be a fixed odd prime number. Throughout this paper ℀𝑝,β„šπ‘, and ℂ𝑝 will, respectively, denote the ring of 𝑝-adic rational integers, the field of 𝑝-adic rational numbers, and the completion of the algebraic closure of β„šπ‘. Let β„• be the set of natural numbers and β„€+=β„•βˆͺ{0}. The 𝑝-adic norm of ℂ𝑝 is defined by |𝑝|𝑝=1/𝑝. When one talks of a π‘ž-extension, π‘ž can be considered as an indeterminate, a complex number π‘žβˆˆβ„‚, or a 𝑝-adic number π‘žβˆˆβ„‚π‘. Throughout this paper we assume that π›Όβˆˆβ„š and π‘žβˆˆβ„‚π‘ with |1βˆ’π‘ž|𝑝<π‘βˆ’1/(π‘βˆ’1) so that π‘žπ‘₯=exp(π‘₯logπ‘ž).

Let π‘ˆπ·(℀𝑝) be the space of uniformly differentiable functions on ℀𝑝. For π‘“βˆˆπ‘ˆπ·(℀𝑝), the 𝑝-adic π‘ž-integral on ℀𝑝 is defined by Kim (see [1–3]) as follows:πΌπ‘ž(𝑓)=ξ€œβ„€π‘π‘“(π‘₯)π‘‘πœ‡π‘ž(π‘₯)=limπ‘β†’βˆž1ξ€Ίπ‘π‘ξ€»π‘žπ‘π‘βˆ’1π‘₯=0𝑓(π‘₯)π‘žπ‘₯,(1.1) where [π‘₯]π‘ž is the π‘ž-number of π‘₯ which is defined by [π‘₯]π‘ž=(1βˆ’π‘žπ‘₯)/(1βˆ’π‘ž).

From (1.1), we haveπ‘žπ‘›πΌπ‘žξ€·π‘“π‘›ξ€Έβˆ’πΌπ‘ž(𝑓)=(π‘žβˆ’1)π‘›βˆ’1𝑙=0π‘žπ‘™π‘“(𝑙)+π‘žβˆ’1logπ‘žπ‘›βˆ’1𝑙=0π‘žπ‘™π‘“ξ…ž(𝑙),(1.2) where 𝑓𝑛(π‘₯)=𝑓(π‘₯+𝑛) (see [2–4]).

As is well known, Bernoulli numbers are inductively defined by 𝐡0=1,(𝐡+1)π‘›βˆ’π΅π‘›=ξ‚»1if𝑛=1,0if𝑛>1,(1.3) with the usual convention about replacing 𝐡𝑛 by 𝐡𝑛 (see [3, 5]).

In [2, 5, 6], the π‘ž-Bernoulli numbers are defined by𝐡0,π‘ž=π‘žβˆ’1logπ‘ž,ξ€·π‘žπ΅π‘ž+1ξ€Έπ‘›βˆ’π΅π‘›,π‘ž=ξ‚»1if𝑛=1,0if𝑛>1,(1.4) with the usual convention about replacing π΅π‘›π‘ž by 𝐡𝑛,π‘ž. Note that limπ‘žβ†’1𝐡𝑛,π‘ž=𝐡𝑛. In the viewpoint of (1.4), we consider the modified π‘ž-Bernoulli numbers with weight.

In this paper we study families of the modified π‘ž-Bernoulli numbers and polynomials of higher order with weight. In particular, by using the multivariate 𝑝-adic π‘ž-integral on ℀𝑝, we give new identities of the higher-order π‘ž-Bernoulli numbers and polynomials with weight.

2. Modified π‘ž-Bernoulli Numbers with Weight of Higher Order

For π‘›βˆˆβ„€+, let us consider the following modified π‘ž-Bernoulli numbers with weight 𝛼 (see [1, 3]):𝐡(𝛼)𝑛,π‘ž=ξ€œβ„€π‘[π‘₯]π‘›π‘žπ›Όπ‘žβˆ’π‘₯π‘‘πœ‡π‘ž(π‘₯)=1(1βˆ’π‘ž)𝑛[𝛼]π‘›π‘žπ‘›ξ“π‘™=0βŽ›βŽœβŽœβŽπ‘›π‘™βŽžβŽŸβŽŸβŽ (βˆ’1)𝑙𝛼𝑙[𝛼𝑙]π‘ž,𝐡(𝛼)𝑛,π‘ž(π‘₯)=ξ€œβ„€π‘[π‘₯+𝑦]π‘›π‘žπ›Όπ‘žβˆ’π‘¦π‘‘πœ‡π‘ž(𝑦)=1(1βˆ’π‘ž)𝑛[𝛼]π‘›π‘žπ‘›ξ“π‘™=0βŽ›βŽœβŽœβŽπ‘›π‘™βŽžβŽŸβŽŸβŽ (βˆ’1)π‘™π‘žπ›Όπ‘™π‘₯𝛼𝑙[𝛼𝑙]π‘ž.(2.1) From (2.1), we note that𝐡(𝛼)𝑛,π‘ž(π‘₯)=𝑛𝑙=0βŽ›βŽœβŽœβŽπ‘›π‘™βŽžβŽŸβŽŸβŽ [π‘₯]π‘›βˆ’π‘™π‘žπ›Όπ‘žπ›Όπ‘™π‘₯𝐡(𝛼)𝑙,π‘ž(2.2) (see [1, 3]).

For π‘˜βˆˆβ„• and π‘›βˆˆβ„€+, by making use of the multivariate 𝑝-adic π‘ž-integral on ℀𝑝, we consider the following modified π‘ž-Bernoulli numbers with weight 𝛼 of order π‘˜, 𝐡(π‘˜,𝛼)𝑛,π‘ž:𝐡(π‘˜,𝛼)𝑛,π‘ž=ξ€œβ„€π‘β‹―ξ€œβ„€π‘ξ€Ίπ‘₯1+β‹―+π‘₯π‘˜ξ€»π‘›π‘žπ›Όπ‘žβˆ’π‘₯1βˆ’β‹―βˆ’π‘₯π‘˜π‘‘πœ‡π‘žξ€·π‘₯1ξ€Έβ‹―π‘‘πœ‡π‘žξ€·π‘₯π‘˜ξ€Έ.(2.3) Note that 𝐡(1,𝛼)𝑛,π‘ž=𝐡(𝛼)𝑛,π‘ž and limπ‘žβ†’1𝐡(π‘˜,𝛼)𝑛,π‘ž=𝐡(π‘˜)𝑛, where 𝐡(π‘˜)𝑛 are the 𝑛th ordinary Bernoulli numbers of order π‘˜.

For π‘˜,π‘βˆˆβ„•, we haveξ‚΅1βˆ’π‘ž1βˆ’π‘žπ‘π‘ξ‚Άπ‘˜π‘π‘βˆ’1𝑖1=0β‹―π‘π‘βˆ’1ξ“π‘–π‘˜=0𝑖1+β‹―+π‘–π‘˜ξ€»π‘›π‘žπ›Ό=ξ‚΅1βˆ’π‘ž1βˆ’π‘žπ‘π‘ξ‚Άπ‘˜ξ‚΅11βˆ’π‘žπ›Όξ‚Άπ‘›π‘π‘βˆ’1𝑖1,…,π‘–π‘˜=0𝑛𝑗=0βŽ›βŽœβŽœβŽπ‘›π‘—βŽžβŽŸβŽŸβŽ (βˆ’1)π‘—π‘žπ›Ό(𝑖1+β‹―+π‘–π‘˜)𝑗=1(1βˆ’π‘ž)𝑛[𝛼]π‘›π‘žπ‘›ξ“π‘—=0βŽ›βŽœβŽœβŽπ‘›π‘—βŽžβŽŸβŽŸβŽ (βˆ’1)𝑗(1βˆ’π‘ž)π‘˜ξ€·1βˆ’π‘žπ‘π‘ξ€Έπ‘˜βŽ›βŽœβŽ1βˆ’π‘žπ›Όπ‘π‘π‘—1βˆ’π‘žπ›Όπ‘—β‹―1βˆ’π‘žπ›Όπ‘π‘π‘—1βˆ’π‘žπ›Όπ‘—βŽžβŽŸβŽ ξ„Ώξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…ƒξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…Œπ‘˜βˆ’times.(2.4) By (1.1), (2.3), and (2.4), we get𝐡(π‘˜,𝛼)𝑛,π‘ž=1(1βˆ’π‘ž)𝑛[𝛼]π‘›π‘žπ‘›ξ“π‘—=0βŽ›βŽœβŽœβŽπ‘›π‘—βŽžβŽŸβŽŸβŽ (βˆ’1)𝑗(𝛼𝑗)π‘˜[𝛼𝑗]π‘˜π‘ž.(2.5) Therefore, by (2.5), we obtain the following theorem.

Theorem 2.1. For 𝑛β‰₯0, one has 𝐡(π‘˜,𝛼)𝑛,π‘ž=1(1βˆ’π‘ž)𝑛[𝛼]π‘›π‘žπ‘›ξ“π‘—=0βŽ›βŽœβŽœβŽπ‘›π‘—βŽžβŽŸβŽŸβŽ (βˆ’1)𝑗(𝛼𝑗)π‘˜[𝛼𝑗]π‘˜π‘ž.(2.6)

Let us consider the modified π‘ž-Bernoulli and polynomials with weight 𝛼 of order π‘˜ as follows:𝐡(π‘˜,𝛼)𝑛,π‘ž(π‘₯)=ξ€œβ„€π‘β‹―ξ€œβ„€π‘ξ€Ίπ‘₯+π‘₯1+β‹―+π‘₯π‘˜ξ€»π‘›π‘žπ›Όπ‘žβˆ’π‘₯1βˆ’β‹―βˆ’π‘₯π‘˜π‘‘πœ‡π‘žξ€·π‘₯1ξ€Έβ‹―π‘‘πœ‡π‘žξ€·π‘₯π‘˜ξ€Έ.(2.7) By the same method of (2.5), we obtain the following theorem.

Theorem 2.2. For π‘›βˆˆβ„€+, one has 𝐡(π‘˜,𝛼)𝑛,π‘ž(π‘₯)=1(1βˆ’π‘ž)𝑛[𝛼]π‘›π‘žπ‘›ξ“π‘—=0βŽ›βŽœβŽœβŽπ‘›π‘—βŽžβŽŸβŽŸβŽ (βˆ’1)π‘—π‘žπ›Όπ‘₯𝑗(𝛼𝑗)π‘˜[𝛼𝑗]π‘˜π‘ž.(2.8)

By Theorem 2.2, we get𝐡(π‘˜,𝛼)𝑛,π‘žβˆ’1(π‘˜βˆ’π‘₯)=1(1βˆ’π‘žβˆ’π›Ό)𝑛𝑛𝑗=0βŽ›βŽœβŽœβŽπ‘›π‘—βŽžβŽŸβŽŸβŽ (βˆ’1)𝑗(𝛼𝑗)π‘˜[𝛼𝑗]π‘˜π‘žβˆ’1π‘žβˆ’π›Όπ‘—(π‘˜βˆ’π‘₯)=(βˆ’1)π‘›π‘žπ›Όπ‘›(1βˆ’π‘žπ›Ό)𝑛𝑛𝑗=0βŽ›βŽœβŽœβŽπ‘›π‘—βŽžβŽŸβŽŸβŽ (βˆ’1)π‘—ξ‚΅π‘žβˆ’1(π‘žβˆ’1)𝛼𝑗(π‘žπ›Όπ‘—βˆ’1)π‘žβˆ’π›Όπ‘—ξ‚Άπ‘˜π‘žβˆ’π›Όπ‘—(π‘˜βˆ’π‘₯)=(βˆ’1)π‘›π‘žπ›Όπ‘›(1βˆ’π‘žπ›Ό)𝑛𝑛𝑗=0βŽ›βŽœβŽœβŽπ‘›π‘—βŽžβŽŸβŽŸβŽ (βˆ’1)π‘—π‘žπ›Όπ‘—π‘₯π‘žβˆ’π‘˜(𝛼𝑗)π‘˜[𝛼𝑗]π‘˜π‘ž=(βˆ’1)π‘›π‘žπ›Όπ‘›βˆ’π‘˜ξ‚π΅(π‘˜,𝛼)𝑛,π‘ž(π‘₯).(2.9) Therefore, by (2.9), we obtain the following theorem.

Theorem 2.3. For π‘›βˆˆβ„€+, one has 𝐡(π‘˜,𝛼)𝑛,π‘žβˆ’1(π‘˜βˆ’π‘₯)=(βˆ’1)π‘›π‘žπ›Όπ‘›βˆ’π‘˜ξ‚π΅(π‘˜,𝛼)𝑛,π‘ž(π‘₯),𝐡(π‘˜,𝛼)𝑛,π‘žβˆ’1(π‘˜)=(βˆ’1)π‘›π‘žπ›Όπ‘›βˆ’π‘˜ξ‚π΅(π‘˜,𝛼)𝑛,π‘ž.(2.10)

From Theorem 2.3, we note thatlimπ‘žβ†’1𝐡(π‘˜,𝛼)𝑛,π‘žβˆ’1(π‘˜βˆ’π‘₯)=𝐡(π‘˜)𝑛(π‘˜βˆ’π‘₯),limπ‘žβ†’1𝐡(π‘˜,𝛼)𝑛,π‘žβˆ’1(π‘˜)=(βˆ’1)𝑛𝐡(π‘˜)𝑛.(2.11) Thus, we have 𝐡(π‘˜)𝑛(π‘˜)=(βˆ’1)𝑛𝐡(π‘˜)𝑛, where 𝐡(π‘˜)𝑛 are the 𝑛th Bernoulli numbers of order π‘˜.

From (2.3) and (2.7), we can derive the following equations:𝐡(𝑙,𝛼)π‘˜,π‘ž(π‘₯)=limπ‘β†’βˆž1[π‘š]π‘™π‘žξ€Ίπ‘π‘ξ€»π‘™π‘žπ‘šπ‘šβˆ’1𝑖1,…,𝑖𝑙=0π‘π‘βˆ’1𝑛1,…,𝑛𝑙=0ξ€Ίπ‘₯+𝑖1+β‹―+𝑖𝑙+π‘šξ€·π‘›1+β‹―+π‘›π‘™ξ€Έξ€»π‘˜π‘žπ›Ό=[π‘š]π‘˜π‘žπ›Ό[π‘š]π‘™π‘žπ‘šβˆ’1𝑖1,…,𝑖𝑙=0ξ€œβ„€π‘β‹―ξ€œβ„€π‘ξ‚Έπ‘₯+𝑖1+β‹―+π‘–π‘™π‘š+π‘₯1+β‹―+π‘₯π‘™ξ‚Ήπ‘˜π‘žπ›Όπ‘šΓ—π‘žβˆ’π‘šπ‘₯1βˆ’β‹―βˆ’π‘šπ‘₯π‘™π‘‘πœ‡π‘žπ‘šξ€·π‘₯1ξ€Έβ‹―π‘‘πœ‡π‘žπ‘šξ€·π‘₯π‘˜ξ€Έ=[π‘š]π‘˜π‘žπ›Ό[π‘š]π‘™π‘žπ‘šβˆ’1𝑖1,…,𝑖𝑙=0𝐡(𝑙,𝛼)π‘˜,π‘žπ‘šξ‚΅π‘₯+𝑖1+β‹―+π‘–π‘™π‘šξ‚Ά.(2.12) Therefore, by (2.12), we obtain the following theorem.

Theorem 2.4. For π‘˜βˆˆβ„€+ and 𝑙,π‘šβˆˆβ„•, one has 𝐡(𝑙,𝛼)π‘˜,π‘ž(π‘₯)=[π‘š]π‘˜π‘žπ›Ό[π‘š]π‘™π‘žπ‘šβˆ’1𝑖1,…,𝑖𝑙=0𝐡(𝑙,𝛼)π‘˜,π‘žπ‘šξ‚΅π‘₯+𝑖1+β‹―+π‘–π‘™π‘šξ‚Ά.(2.13)

In particular,𝐡(𝑙,𝛼)π‘˜,π‘ž(π‘šπ‘₯)=[π‘š]π‘˜π‘žπ›Ό[π‘š]π‘™π‘žπ‘šβˆ’1𝑖1,…,𝑖𝑙=0𝐡(𝑙,𝛼)π‘˜,π‘žπ‘šξ‚΅π‘₯+𝑖1+β‹―+π‘–π‘™π‘šξ‚Ά.(2.14)

From (1.2), we can derive the following integral:ξ€œβ„€π‘π‘“(π‘₯+1)π‘žβˆ’π‘₯π‘‘πœ‡π‘ž(π‘₯)=ξ€œβ„€π‘π‘“(π‘₯)π‘žβˆ’π‘₯π‘‘πœ‡π‘ž(π‘₯)+π‘žβˆ’1logπ‘žπ‘“ξ…ž(0),ξ€œβ„€π‘π‘“(π‘₯+2)π‘žβˆ’π‘₯π‘‘πœ‡π‘ž(π‘₯)=ξ€œβ„€π‘π‘“1(π‘₯)π‘žβˆ’π‘₯π‘‘πœ‡π‘ž(π‘₯)+π‘žβˆ’1logπ‘žπ‘“ξ…ž(1)=ξ€œβ„€π‘π‘“(π‘₯)π‘žβˆ’π‘₯π‘‘πœ‡π‘ž(π‘₯)+π‘žβˆ’1logπ‘žξ€·π‘“ξ…ž(0)+π‘“ξ…ž(1)ξ€Έ.(2.15) Continuing this process, we obtainξ€œβ„€π‘π‘“(π‘₯+𝑛)π‘žβˆ’π‘₯π‘‘πœ‡π‘ž(π‘₯)=ξ€œβ„€π‘π‘“(π‘₯)π‘žβˆ’π‘₯π‘‘πœ‡π‘ž(π‘₯)+π‘žβˆ’1logπ‘žπ‘›βˆ’1𝑙=0π‘“ξ…ž(𝑙).(2.16) By (2.16), we getξ€œβ„€π‘[π‘₯+𝑛]π‘šπ‘žπ›Όπ‘žβˆ’π‘₯π‘‘πœ‡π‘ž(π‘₯)=ξ€œβ„€π‘[π‘₯]π‘šπ‘žπ›Όπ‘žβˆ’π‘₯π‘‘πœ‡π‘ž(π‘₯)+π‘šπ›Ό[𝛼]π‘žπ‘›βˆ’1𝑙=0[𝑙]π‘šβˆ’1π‘žπ›Όπ‘žπ›Όπ‘™.(2.17) Therefore, by (2.1) and (2.17), we obtain the following theorem.

Theorem 2.5. For π‘›βˆˆβ„• and π‘šβˆˆβ„€+, one has 𝐡(𝛼)π‘š,π‘ž(𝑛)βˆ’ξ‚π΅(𝛼)π‘š,π‘ž=π‘šπ›Ό[𝛼]π‘žπ‘›βˆ’1𝑙=0[𝑙]π‘šπ‘žπ›Όπ‘žπ›Όπ‘™.(2.18)

In an analogues manner as the previous investigation [7–10], we can define a further generalization of modified π‘ž-Bernoulli numbers with weight. Let πœ’ be the Dirichlet character with conductor π‘‘βˆˆβ„•. Then the generalized π‘ž-Bernoulli numbers with weight attached to πœ’ can be defined as follows:𝐡(𝛼)𝑛,πœ’,π‘ž=ξ€œπ‘‹πœ’(π‘₯)[π‘₯]π‘›π‘žπ›Όπ‘žβˆ’π‘₯π‘‘πœ‡π‘ž(π‘₯)=[𝑑]π‘›π‘žπ›Ό[𝑑]π‘žπ‘‘βˆ’1ξ“π‘Ž=0πœ’(π‘Ž)𝐡(𝛼)𝑛,π‘žπ‘‘ξ‚€π‘Žπ‘‘ξ‚.(2.19) We expect to investigate these objects in future papers. This definition 𝐡(𝛼)𝑛,π‘ž was also given in a previous paper (see [9]).

Acknowledgments

The authors express their sincere gratitude to the referees for their valuable suggestions and comments. This paper is supported in part by the Research Grant of Kwangwoon University in 2011.

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