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Abstract and Applied Analysis
VolumeΒ 2011, Article IDΒ 392025, 14 pages
http://dx.doi.org/10.1155/2011/392025
Research Article

On the π‘ž-Bernoulli Numbers and Polynomials with Weight 𝜢

Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea

Received 13 May 2011; Accepted 26 July 2011

Academic Editor: ElenaΒ Litsyn

Copyright Β© 2011 T. Kim and J. Choi. 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

We present a systemic study of some families of higher-order π‘ž-Bernoulli numbers and polynomials with weight 𝛼. From these studies, we derive some interesting identities on the π‘ž-Bernoulli numbers and polynomials with weight 𝛼.

1. Introduction

Let 𝑝 be a fixed odd prime number. Throughout this paper, ℀𝑝, β„šπ‘, and ℂ𝑝 will denote the ring of 𝑝-adic rational integers, the field of 𝑝-adic rational numbers, and the completion of algebraic closure of β„šπ‘, respectively. The 𝑝-adic norm of ℂ𝑝 is defined as |π‘₯|𝑝=π‘βˆ’π‘Ÿ, where π‘₯=π‘π‘Ÿπ‘š/𝑛 with (𝑝,π‘š)=(𝑝,𝑛)=1, π‘Ÿβˆˆβ„š and π‘š,π‘›βˆˆβ„€. Let β„• and β„€ be the set of natural numbers and integers, respectively, β„€+=β„•βˆͺ{0}. Let π‘žβˆˆβ„‚π‘ with |1βˆ’π‘ž|𝑝<π‘βˆ’1/(π‘βˆ’1). The notation of π‘ž-number is defined by [π‘₯]𝑀=(1βˆ’π‘€π‘₯)/(1βˆ’π‘€) and [π‘₯]π‘ž=(1βˆ’π‘žπ‘₯)/(1βˆ’π‘ž), (see [1–13]).

As the well known definition, the Bernoulli polynomials are defined by π‘‘π‘’π‘‘π‘’βˆ’1π‘₯=βˆžξ“π‘›=0𝐡𝑛(𝑑π‘₯)𝑛.𝑛!(1.1) In the special case, π‘₯=0, 𝐡𝑛(0)=𝐡𝑛 are called the 𝑛th Bernoulli numbers. That is, the recurrence formula for the Bernoulli numbers is given by 𝐡0=1,(𝐡+1)π‘›βˆ’π΅π‘›=ξ‚»1if0𝑛=1,if𝑛>1,(1.2) with the usual convention about replacing 𝐡𝑖 with 𝐡𝑖.

In [1, 2], π‘ž-extension of Bernoulli numbers are defined by Carlitz as follows:𝛽0,π‘ž=1,π‘ž(π‘žπ›½+1)π‘›βˆ’π›½π‘›,π‘ž=ξ‚»1if0𝑛=1,if𝑛>1,(1.3) with the usual convention about replacing 𝛽𝑖 with 𝛽𝑖,π‘ž.

By (1.2) and (1.3), we get limπ‘žβ†’1𝛽𝑖,π‘ž=𝐡𝑖. In this paper, we assume that π›Όβˆˆβ„•.

In [7], the π‘ž-Bernoulli numbers with weight 𝛼 are defined by Kim as follows:̃𝛽(𝛼)0,π‘žξ€·π‘ž=1,π‘žπ›ΌΜƒπ›½π›Όξ€Έ+1π‘›βˆ’Μƒπ›½(𝛼)𝑛,π‘ž=𝛼[𝛼]π‘žif0𝑛=1,if𝑛>1,(1.4) with the usual convention about replacing (̃𝛽(𝛼))𝑖 with ̃𝛽(𝛼)𝑖,π‘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 as πΌπ‘ž(ξ€œπ‘“)=℀𝑝𝑓(π‘₯)π‘‘πœ‡π‘ž(π‘₯)=limπ‘β†’βˆž1ξ€Ίπ‘π‘ξ€»π‘žπ‘π‘βˆ’1π‘₯=0𝑓(π‘₯)π‘žπ‘₯,(1.5) (see[4, 5]). From (1.5), we note thatπ‘žπ‘›πΌπ‘žξ€·π‘“π‘›ξ€Έ=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),(1.6) where 𝑓𝑛(π‘₯)=𝑓(π‘₯+𝑛) and π‘“ξ…ž(𝑙)=(𝑑𝑓(π‘₯)/𝑑π‘₯)|π‘₯=𝑙.

By (1.4), (1.5), and (1.6), we set ̃𝛽(𝛼)𝑛,π‘ž=ξ€œβ„€π‘[π‘₯]π‘›π‘žπ›Όπ‘‘πœ‡π‘ž(π‘₯),whereπ‘›βˆˆβ„€+,(1.7) (see[7]). The π‘ž-Bernoulli polynomials are also given by ̃𝛽(𝛼)𝑛,π‘žξ€œ(π‘₯)=℀𝑝[]π‘₯+π‘¦π‘›π‘žπ›Όπ‘‘πœ‡π‘ž(π‘₯)=𝑛𝑙=0βŽ›βŽœβŽœβŽπ‘›π‘™βŽžβŽŸβŽŸβŽ [π‘₯]π‘žπ‘›βˆ’π‘™π›Όπ‘žπ›Όπ‘™π‘₯̃𝛽(𝛼)𝑙,π‘ž.(1.8)

The purpose of this paper is to derive a new concept of higher-order π‘ž-Bernoulli numbers and polynomials with weight 𝛼 from the fermionic 𝑝-adic π‘ž-integral on ℀𝑝. Finally, we present a systemic study of some families of higher-order π‘ž-Bernoulli numbers and polynomials with weight 𝛼.

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

Let π›½βˆˆβ„€ and π›Όβˆˆβ„• in this paper. For π‘˜βˆˆβ„• and π‘›βˆˆβ„€+, we consider the expansion of higher-order π‘ž-Bernoulli polynomials with weight 𝛼 as follows: ̃𝛽(𝛽,π‘˜βˆ£π›Ό)𝑛,π‘ž(ξ€œπ‘₯)=β„€π‘β‹―ξ€œβ„€π‘ξ€Ίπ‘₯1+β‹―+π‘₯π‘˜ξ€»+π‘₯π‘›π‘žπ›Όπ‘žπ‘₯1(π›½βˆ’1)+β‹―+π‘₯π‘˜(π›½βˆ’π‘˜)π‘‘πœ‡π‘žξ€·π‘₯1ξ€Έβ‹―π‘‘πœ‡π‘žξ€·π‘₯π‘˜ξ€Έ.(2.1)

From (2.1), we note that ̃𝛽(𝛽,π‘˜βˆ£π›Ό)𝑛,π‘ž(π‘₯)=(1βˆ’π‘ž)π‘˜βˆ’π‘›[𝛼]π‘›π‘žπ‘›ξ“π‘™=0βŽ›βŽœβŽœβŽπ‘›π‘™βŽžβŽŸβŽŸβŽ (βˆ’1)π‘™π‘žπ›Όπ‘™π‘₯βˆπ‘˜βˆ’1𝑖=0(𝛼𝑙+π›½βˆ’π‘–)βˆπ‘˜βˆ’1𝑖=0ξ€·1βˆ’π‘žπ›Όπ‘™+π›½βˆ’π‘–ξ€Έ=1(1βˆ’π‘ž)𝑛[𝛼]π‘›π‘žπ‘›ξ“π‘™=0βŽ›βŽœβŽœβŽπ‘›π‘™βŽžβŽŸβŽŸβŽ (βˆ’1)π‘™π‘žπ›Όπ‘™π‘₯ξ€·π‘˜π›Όπ‘™+π›½ξ€Έπ‘˜!ξ€·π‘˜π›Όπ‘™+π›½ξ€Έπ‘ž[π‘˜]π‘ž!,(2.2) where (𝛼𝑙)π‘ž=((1βˆ’π‘žπ›Ό)(1βˆ’π‘žπ›Όβˆ’1)β‹―(1βˆ’π‘žπ›Όβˆ’π‘™+1))/((1βˆ’π‘ž)(1βˆ’π‘ž2)β‹―(1βˆ’π‘žπ‘™)) and [π‘˜]π‘ž!=[π‘˜]π‘žβ‹―[2]π‘ž[1]π‘ž.

Therefore, we obtain the following theorem.

Theorem 2.1. For π‘›βˆˆβ„€+ and π‘˜βˆˆβ„•, we have ̃𝛽(𝛽,π‘˜βˆ£π›Ό)𝑛,π‘ž1(π‘₯)=(1βˆ’π‘ž)𝑛[𝛼]π‘›π‘žπ‘›ξ“π‘™=0βŽ›βŽœβŽœβŽπ‘›π‘™βŽžβŽŸβŽŸβŽ (βˆ’1)π‘™π‘žπ›Όπ‘™π‘₯ξ€·π‘˜π›Όπ‘™+π›½ξ€Έπ‘˜!𝛼𝑙+𝛽kξ€Έπ‘ž[π‘˜]π‘ž!.(2.3)

In the special case, π‘₯=0, ̃𝛽(𝛽,π‘˜βˆ£π›Ό)𝑛,π‘žΜƒπ›½(0)=(𝛽,π‘˜βˆ£π›Ό)𝑛,π‘ž are called the 𝑛th higher order π‘ž-Bernoulli numbers with weight 𝛼.

From (2.1) and (2.2), we can derive ̃𝛽(𝛽,π‘˜βˆ£π›Ό)𝑛,π‘ž=(π‘žπ›ΌΜƒπ›½βˆ’1)(π›½βˆ’π›Ό,π‘˜βˆ£π›Ό)𝑛+1,π‘ž+̃𝛽(π›½βˆ’π›Ό,π‘˜βˆ£π›Ό)𝑛,π‘ž.(2.4) By Theorem 2.1 and (2.4), we get ̃𝛽(π‘šπ›Ό,π‘˜βˆ£π›Ό)0,π‘ž=ξ€œβ„€π‘β‹―ξ€œβ„€π‘π‘žβˆ‘π‘˜π‘—=1(π›Όπ‘šβˆ’π‘—)π‘₯π‘—π‘‘πœ‡π‘žξ€·π‘₯1ξ€Έβ‹―π‘‘πœ‡π‘žξ€·π‘₯π‘˜ξ€Έ=π‘šξ“π‘™=0βŽ›βŽœβŽœβŽπ‘šπ‘™βŽžβŽŸβŽŸβŽ (π‘žπ›Όβˆ’1)π‘™ξ€œβ„€π‘β‹―ξ€œβ„€π‘ξ€Ίπ‘₯1+β‹―+π‘₯π‘˜ξ€»π‘™π‘žπ›Όπ‘žβˆ’βˆ‘π‘˜π‘—=1𝑗π‘₯π‘—π‘‘πœ‡π‘žξ€·π‘₯1ξ€Έβ‹―π‘‘πœ‡π‘žξ€·π‘₯π‘˜ξ€Έ=π‘šξ“π‘™=0βŽ›βŽœβŽœβŽπ‘šπ‘™βŽžβŽŸβŽŸβŽ (π‘žπ›Ό)βˆ’1𝑙̃𝛽(0,π‘˜βˆ£π›Ό)𝑙,π‘ž=(1βˆ’π‘ž)π‘˜βˆπ‘˜βˆ’1𝑖=0ξ€·1βˆ’π‘žπ›Όπ‘šβˆ’π‘˜+1+𝑖=(1βˆ’π‘ž)π‘˜π‘˜ξ“π‘™=0βŽ›βŽœβŽœβŽπ‘™βŽžβŽŸβŽŸβŽ π‘˜+π‘™βˆ’1π‘žπ‘ž(π›Όπ‘šβˆ’π‘˜+1)𝑙.(2.5) From (2.1), we have 𝑖𝑗=0βŽ›βŽœβŽœβŽπ‘–π‘—βŽžβŽŸβŽŸβŽ (π‘žπ›Όβˆ’1)π‘—ξ€œβ„€π‘β‹―ξ€œβ„€π‘ξ€Ίπ‘₯1+β‹―+π‘₯π‘˜ξ€»π‘žπ‘›βˆ’π‘–+π‘—π›Όπ‘ž(π›½βˆ’π›Όβˆ’1)π‘₯1+β‹―+(π›½βˆ’π›Όβˆ’π‘˜)π‘₯π‘˜π‘‘πœ‡π‘žξ€·π‘₯1ξ€Έβ‹―π‘‘πœ‡π‘žξ€·π‘₯π‘˜ξ€Έ=ξ€œβ„€π‘β‹―ξ€œβ„€π‘ξ€Ίπ‘₯1+β‹―+π‘₯π‘˜ξ€»π‘žπ‘›βˆ’π‘–π›Όπ‘ž(π›½βˆ’1)π‘₯1+β‹―+(π›½βˆ’π‘˜)π‘₯π‘˜π‘žπ›Ό(π‘₯1+β‹―+π‘₯π‘˜)(π‘–βˆ’1)π‘‘πœ‡π‘žξ€·π‘₯1ξ€Έβ‹―π‘‘πœ‡π‘žξ€·π‘₯π‘˜ξ€Έ=π‘–βˆ’1𝑗=0βŽ›βŽœβŽœβŽπ‘—βŽžβŽŸβŽŸβŽ π‘–βˆ’1(π‘žπ›Ό)βˆ’1𝑗̃𝛽(𝛽,π‘˜βˆ£π›Ό)π‘›βˆ’π‘–+𝑗,π‘ž.(2.6)

Thus, we obtain the following theorem.

Theorem 2.2. For π‘–βˆˆβ„•, we have 𝑖𝑗=0βŽ›βŽœβŽœβŽπ‘–π‘—βŽžβŽŸβŽŸβŽ (π‘žπ›Όβˆ’1)𝑗̃𝛽(π›½βˆ’π›Ό,π‘˜βˆ£π›Ό)π‘›βˆ’π‘–+𝑗,π‘ž=π‘–βˆ’1𝑗=0βŽ›βŽœβŽœβŽπ‘—βŽžβŽŸβŽŸβŽ π‘–βˆ’1(π‘žπ›Όβˆ’1)𝑗̃𝛽(𝛽,π‘˜βˆ£π›Ό)π‘›βˆ’π‘–+𝑗,π‘ž.(2.7)

It is easy to show that

π‘šβˆ‘π‘—=0ξ€·π‘šπ‘—ξ€Έ(π‘žπ›Όβˆ’1)𝑗̃𝛽(0,π‘˜βˆ£π›Ό)𝑗,π‘ž=(1βˆ’π‘ž)π‘˜π‘˜βˆ‘π‘™=0ξ€·π‘™π‘˜+π‘™βˆ’1ξ€Έπ‘žπ‘ž(π›Όπ‘šβˆ’π‘˜+1)𝑙=(1βˆ’π‘ž)π‘˜βˆπ‘˜βˆ’1𝑖=0(1βˆ’π‘žπ›Όπ‘šβˆ’π‘˜+1+𝑖).

3. Polynomials ̃𝛽(0,π‘˜βˆ£π›Ό)𝑛,π‘ž(π‘₯)

In this section, we consider the polynomials ̃𝛽(0,π‘˜|𝛼)𝑛,π‘ž(π‘₯) as follows: ̃𝛽(0,π‘˜βˆ£π›Ό)𝑛,π‘ž(ξ€œπ‘₯)=β„€π‘β‹―ξ€œβ„€π‘ξ€Ίπ‘₯1+β‹―+π‘₯π‘˜ξ€»+π‘₯π‘›π‘žπ›Όπ‘žβˆ’βˆ‘π‘˜π‘—=1𝑗π‘₯π‘—π‘‘πœ‡π‘žξ€·π‘₯1ξ€Έβ‹―π‘‘πœ‡π‘žξ€·π‘₯π‘˜ξ€Έ.(3.1)

From (3.1), we can easily derive the following equation: ̃𝛽(0,π‘˜|𝛼)𝑛,π‘ž(π‘₯)=(1βˆ’π‘ž)π‘˜(1βˆ’π‘žπ›Ό)𝑛𝑛𝑙=0βŽ›βŽœβŽœβŽπ‘›π‘™βŽžβŽŸβŽŸβŽ (βˆ’1)π‘™π‘žπ›Όπ‘™π‘₯βˆπ‘˜π‘–=0(π›Όπ‘™βˆ’π‘–)βˆπ‘˜βˆ’1𝑖=0ξ€·1βˆ’π‘žπ›Όπ‘™βˆ’π‘–ξ€Έ=1(1βˆ’π‘ž)𝑛[𝛼]π‘›π‘žπ‘›ξ“π‘™=0βŽ›βŽœβŽœβŽπ‘›π‘™βŽžβŽŸβŽŸβŽ (βˆ’1)π‘™π‘žπ›Όπ‘™π‘₯ξ€·π‘˜π›Όπ‘™ξ€Έπ‘˜!ξ€·π‘˜π›Όπ‘™ξ€Έπ‘ž[π‘˜]π‘ž!.(3.2) By (3.1) and (3.2), we get ξ€œβ„€π‘β‹―ξ€œβ„€π‘π‘žβˆ‘π‘˜π‘—=1(π›Όπ‘›βˆ’π‘—)π‘₯𝑗+𝛼𝑛π‘₯π‘‘πœ‡π‘žξ€·π‘₯1ξ€Έβ‹―π‘‘πœ‡π‘žξ€·π‘₯π‘˜ξ€Έ=𝑛𝑙=0βŽ›βŽœβŽœβŽπ‘›π‘™βŽžβŽŸβŽŸβŽ [𝛼]π‘™π‘ž(π‘žβˆ’1)𝑙̃𝛽(0,π‘˜βˆ£π›Ό)𝑙,π‘ž(π‘₯),(3.3)ξ€œβ„€π‘β‹―ξ€œβ„€π‘π‘žβˆ‘π‘˜π‘™=1(π›Όπ‘›βˆ’π‘™)π‘₯𝑙+𝛼𝑛π‘₯π‘‘πœ‡π‘žξ€·π‘₯1ξ€Έβ‹―π‘‘πœ‡π‘žξ€·π‘₯π‘˜ξ€Έ=π‘žπ›Όπ‘›π‘₯(1βˆ’π‘ž)π‘˜ξ‚€βˆπ‘˜βˆ’1𝑗=0(π›Όπ‘›βˆ’π‘—)βˆπ‘˜βˆ’1𝑗=0(1βˆ’π‘žπ›Όπ‘›βˆ’π‘—)=π‘žπ›Όπ‘›π‘₯(π‘˜π›Όπ‘›)π‘˜!(π‘˜π›Όπ‘›)π‘ž[π‘˜]π‘ž!.(3.4) Therefore, by (3.3) and (3.4), we obtain the following theorem.

Theorem 3.1. For π‘›βˆˆβ„€+, we have (1βˆ’π‘ž)𝑛̃𝛽(0,π‘˜βˆ£π›Ό)𝑛,π‘ž1(π‘₯)=[𝛼]π‘›π‘žπ‘›ξ“π‘™=0βŽ›βŽœβŽœβŽπ‘›π‘™βŽžβŽŸβŽŸβŽ (βˆ’1)π‘™π‘žπ›Όπ‘™π‘₯ξ€·π‘˜π›Όπ‘™ξ€Έπ‘˜!ξ€·π‘˜π›Όπ‘™ξ€Έπ‘ž[π‘˜]π‘ž!.(3.5) Moreover, 𝑛𝑙=0βŽ›βŽœβŽœβŽπ‘›π‘™βŽžβŽŸβŽŸβŽ [𝛼]π‘™π‘ž(π‘žβˆ’1)𝑙̃𝛽(0,π‘˜βˆ£π›Ό)𝑙,π‘žπ‘ž(π‘₯)=𝛼𝑛π‘₯(π‘˜π›Όπ‘›)π‘˜!(π‘˜π›Όπ‘›)π‘ž[π‘˜]π‘ž!.(3.6)

Let π‘‘βˆˆβ„•. Then, we have ξ€œβ„€π‘β‹―ξ€œβ„€π‘ξ€Ίπ‘₯+π‘₯1+β‹―+π‘₯π‘˜ξ€»π‘›π‘žπ›Όπ‘žβˆ’βˆ‘π‘˜π‘—=1𝑗π‘₯π‘—π‘‘πœ‡π‘žξ€·π‘₯1ξ€Έβ‹―π‘‘πœ‡π‘žξ€·π‘₯π‘˜ξ€Έ=[𝑑]π‘›π‘žπ›Ό[𝑑]π‘˜π‘žπ‘‘βˆ’1ξ“π‘Ž1,β‹―,π‘Žπ‘˜=0π‘žβˆ’βˆ‘π‘˜π‘—=2(π‘—βˆ’1)π‘Žπ‘—Γ—ξ€œβ„€π‘β‹―ξ€œβ„€π‘ξƒ¬βˆ‘π‘₯+π‘˜π‘—=1π‘Žπ‘—π‘‘+π‘˜ξ“π‘–=1π‘₯π‘–ξƒ­π‘›π‘žπ›Όπ‘‘π‘žβˆ‘βˆ’π‘‘π‘˜π‘—=1𝑗π‘₯π‘—π‘‘πœ‡π‘žπ‘‘ξ€·π‘₯1ξ€Έβ‹―π‘‘πœ‡π‘žπ‘‘ξ€·π‘₯π‘˜ξ€Έ.(3.7) Thus, by (3.1) and (3.7), we obtain the following theorem.

Theorem 3.2. For 𝑑,π‘˜βˆˆβ„•, and π‘›βˆˆβ„€+, we have ̃𝛽(0,π‘˜βˆ£π›Ό)𝑛,π‘ž[𝑑](π‘₯)=π‘›π‘žπ›Ό[𝑑]π‘˜π‘žπ‘‘βˆ’1ξ“π‘Ž1,…,π‘Žπ‘˜=0π‘žβˆ’βˆ‘π‘˜π‘—=2(π‘—βˆ’1)π‘Žπ‘—Μƒπ›½(0,π‘˜βˆ£π›Ό)𝑛,π‘žπ›Όξ‚΅π‘₯+π‘Ž1+β‹―+π‘Žπ‘˜π‘‘ξ‚Ά.(3.8)

From (3.1), we note that ̃𝛽(0,π‘˜|𝛼)𝑛,π‘ž(π‘₯)=𝑛𝑙=0βŽ›βŽœβŽœβŽπ‘›π‘™βŽžβŽŸβŽŸβŽ [π‘₯]π‘žπ‘›βˆ’π‘™π›Όπ‘žπ›Όπ‘™π‘₯̃𝛽(0,π‘˜βˆ£π›Ό)𝑙,π‘ž,̃𝛽(0,π‘˜βˆ£π›Ό)𝑛,π‘ž(π‘₯+𝑦)=𝑛𝑙=0βŽ›βŽœβŽœβŽπ‘›π‘™βŽžβŽŸβŽŸβŽ [𝑦]π‘žπ‘›βˆ’π‘™π›Όπ‘žπ›Όπ‘™π‘₯̃𝛽(0,π‘˜βˆ£π›Ό)𝑙,π‘ž(π‘₯).(3.9)

4. Polynomials ̃𝛽(β„Ž,1|𝛼)𝑛,π‘ž(π‘₯)

For β„Žβˆˆβ„€, let us define weighted (β„Ž,π‘ž)-Bernoulli polynomials ̃𝛽(β„Ž,1βˆ£π›Ό)𝑛,π‘ž(π‘₯) as follows: ̃𝛽(β„Ž,1βˆ£π›Ό)𝑛,π‘ž(ξ€œπ‘₯)=℀𝑝π‘₯+π‘₯1ξ€»π‘›π‘žπ›Όπ‘žπ‘₯1(β„Žβˆ’1)π‘‘πœ‡π‘žξ€·π‘₯1ξ€Έ.(4.1) By (4.1), we easily see that ̃𝛽(β„Ž,1βˆ£π›Ό)𝑛,π‘ž1(π‘₯)=[𝛼]π‘›π‘ž(1βˆ’π‘ž)𝑛𝑛𝑙=0βŽ›βŽœβŽœβŽπ‘›π‘™βŽžβŽŸβŽŸβŽ (βˆ’1)π‘™π‘žπ›Όπ‘™π‘₯𝛼𝑙+β„Ž[]𝛼𝑙+β„Žπ‘ž.(4.2) Therefore, by (4.2), we obtain the following theorem.

Theorem 4.1. For β„Žβˆˆβ„€ and π‘›βˆˆβ„€+, we have ̃𝛽(β„Ž,1|𝛼)𝑛,π‘ž1(π‘₯)=[𝛼]π‘›π‘ž(1βˆ’π‘ž)𝑛𝑛𝑙=0βŽ›βŽœβŽœβŽπ‘›π‘™βŽžβŽŸβŽŸβŽ (βˆ’1)π‘™π‘žπ›Όπ‘™π‘₯𝛼𝑙+β„Ž[]𝛼𝑙+β„Žπ‘ž.(4.3)

From (4.1), we can derive the following equation: π‘žπ›Όπ‘₯ξ€œβ„€π‘ξ€Ίπ‘₯+π‘₯1ξ€»π‘›π‘žπ›Όπ‘žπ‘₯1(β„Žβˆ’1)π‘‘πœ‡π‘žξ€·π‘₯1ξ€Έ=(π‘žπ›Ό)ξ€œβˆ’1℀𝑝π‘₯+π‘₯1ξ€»π‘žπ‘›+1π›Όπ‘žπ‘₯1(β„Žβˆ’π›Όβˆ’1)π‘‘πœ‡π‘žξ€·π‘₯1ξ€Έ+ξ€œβ„€π‘ξ€Ίπ‘₯+π‘₯1ξ€»π‘›π‘žπ›Όπ‘žπ‘₯1(β„Žβˆ’π›Όβˆ’1)π‘‘πœ‡π‘žξ€·π‘₯1ξ€Έ.(4.4) By (4.4), we easily get π‘žπ›Όπ‘₯̃𝛽(β„Ž,1βˆ£π›Ό)𝑛,π‘ž(π‘₯)=(π‘žπ›ΌΜƒπ›½βˆ’1)(β„Žβˆ’π›Όβˆ’1,1βˆ£π›Ό)𝑛+1,π‘žΜƒπ›½(π‘₯)+(β„Žβˆ’π›Όβˆ’1,1βˆ£π›Ό)𝑛,π‘ž(π‘₯).(4.5) From (4.1), we have ̃𝛽(β„Ž,1|𝛼)𝑛,π‘žξ€œ(π‘₯)=℀𝑝π‘₯+π‘₯1ξ€»π‘›π‘žπ›Όπ‘žπ‘₯1(β„Žβˆ’1)π‘‘πœ‡π‘žξ€·π‘₯1ξ€Έ=𝑛𝑙=0βŽ›βŽœβŽœβŽπ‘›π‘™βŽžβŽŸβŽŸβŽ [π‘₯]π‘žπ‘›βˆ’π‘™π›Όπ‘žπ›Όπ‘™π‘₯̃𝛽(β„Ž,1βˆ£π›Ό)𝑙,π‘ž,(4.6) where ̃𝛽(β„Ž,1βˆ£π›Ό)𝑙,π‘žΜƒπ›½(0)=(β„Ž,1βˆ£π›Ό)𝑙,π‘ž.

By (4.6), we get the following recurrence formula: ̃𝛽(β„Ž,1βˆ£π›Ό)𝑛,π‘žξ‚€π‘ž(π‘₯)=𝛼π‘₯Μƒπ›½π‘ž(β„Ž,1βˆ£π›Ό)+[π‘₯]π‘žπ›Όξ‚π‘›,for𝑛β‰₯1,(4.7) with the usual convention about replacing (Μƒπ›½π‘ž(β„Ž,1βˆ£π›Ό))𝑛 with ̃𝛽(β„Ž,1βˆ£π›Ό)𝑛,π‘ž.

From (1.6), we note that π‘žπΌπ‘žξ€·π‘“1ξ€Έ=πΌπ‘ž(𝑓)+(π‘žβˆ’1)𝑓(0)+π‘žβˆ’1𝑓logπ‘žξ…ž(0).(4.8) For β„Žβˆˆβ„€+, by (4.8), we have π‘žβ„Žξ€œβ„€π‘π‘“(π‘₯+1)π‘ž(β„Žβˆ’1)π‘₯π‘‘πœ‡π‘ž(ξ€œπ‘₯)=℀𝑝𝑓(π‘₯)π‘‘πœ‡π‘ž(π‘₯)+(π‘žβˆ’1)β„Žπ‘“(0)+π‘žβˆ’1𝑓logπ‘žξ…ž(0).(4.9) If β„Žβˆˆ{βˆ’1,βˆ’2,βˆ’3,…}, then we get π‘žβ„Žξ€œβ„€π‘π‘“(π‘₯+1)π‘ž(β„Žβˆ’1)π‘₯π‘‘πœ‡π‘ž(ξ€œπ‘₯)=℀𝑝𝑓(π‘₯)π‘‘πœ‡π‘ž(π‘₯)+(1βˆ’π‘ž)β„Žπ‘“(0)+π‘žβˆ’1𝑓logπ‘žξ…ž(0).(4.10) Let β„Žβˆˆβ„€+. By (4.9), we get π‘žβ„Žξ€œβ„€π‘ξ€Ίπ‘₯+π‘₯1ξ€»+1π‘›π‘žπ›Όπ‘ž(β„Žβˆ’1)π‘₯1π‘‘πœ‡π‘žξ€·π‘₯1ξ€Έβˆ’ξ€œβ„€π‘ξ€Ίπ‘₯+π‘₯1ξ€»π‘›π‘žπ›Όπ‘ž(β„Žβˆ’1)π‘₯1π‘‘πœ‡π‘žξ€·π‘₯1ξ€Έ[π‘₯]=(π‘žβˆ’1)β„Žπ‘›π‘žπ›Ό+𝛼[𝛼]π‘ž[π‘₯]π‘žπ‘›βˆ’1π›Όπ‘žπ›Όπ‘₯.(4.11) From (4.6) and (4.11), we note that π‘žβ„ŽΜƒπ›½(β„Ž,1βˆ£π›Ό)𝑛,π‘žΜƒπ›½(π‘₯+1)βˆ’(β„Ž,1|𝛼)𝑛,π‘ž[π‘₯](π‘₯)=(π‘žβˆ’1)β„Žπ‘›π‘žπ›Όπ›Ό+𝑛[𝛼]π‘ž[π‘₯]π‘žπ‘›βˆ’1π›Όπ‘žπ›Όπ‘₯.(4.12) If we take π‘₯=0 in (4.12), then we have ̃𝛽(β„Ž,1|𝛼)0,π‘ž=β„Ž[β„Ž]π‘ž,π‘žβ„ŽΜƒπ›½(β„Ž,1βˆ£π›Ό)𝑛,π‘ž(̃𝛽1)βˆ’(β„Ž,1βˆ£π›Ό)𝑛,π‘ž=𝛼[𝛼]π‘žif0𝑛=1,if𝑛>1.(4.13) Therefore, by (4.12) and (4.13), we obtain the following theorem.

Theorem 4.2. For β„Žβˆˆβ„€+, we have ̃𝛽(β„Ž,1βˆ£π›Ό)0,π‘ž=β„Ž[β„Ž]π‘ž,π‘žβ„ŽΜƒπ›½(β„Ž,1βˆ£π›Ό)𝑛,π‘ž(̃𝛽1)βˆ’(β„Ž,1βˆ£π›Ό)𝑛,π‘ž=𝛼[𝛼]π‘žif0𝑛=1,if𝑛>1.(4.14)

By (4.7) and Theorem 4.2, we obtain the following corollary.

Corollary 4.3. For β„Žβˆˆβ„€+, we have ̃𝛽(β„Ž,1βˆ£π›Ό)0,π‘ž=β„Ž[β„Ž]π‘ž,π‘žβ„Žξ‚€π‘žπ›ΌΜƒπ›½π‘ž(β„Ž,1βˆ£π›Ό)+1π‘›βˆ’Μƒπ›½(β„Ž,1βˆ£π›Ό)𝑛,π‘ž=𝛼[𝛼]π‘žif0𝑛=1,if𝑛>1,(4.15) with the usual convention about replacing (Μƒπ›½π‘ž(β„Ž,1βˆ£π›Ό))𝑛 with ̃𝛽(β„Ž,1βˆ£π›Ό)𝑛,π‘ž.

From (4.1), we have ̃𝛽(β„Ž,1βˆ£π›Ό)0,π‘ž=ξ€œβ„€π‘π‘žπ‘₯1(β„Žβˆ’1)π‘‘πœ‡π‘žξ€·π‘₯1ξ€Έ=β„Ž[β„Ž]π‘ž,ifβ„Žβˆˆβ„€+.(4.16) It is not difficult to show that ̃𝛽(β„Ž,1βˆ£π›Ό)𝑛,π‘žβˆ’1(ξ€œ1βˆ’π‘₯)=℀𝑝1βˆ’π‘₯+π‘₯1ξ€»π‘›π‘žβˆ’π›Όπ‘žβˆ’π‘₯1(β„Žβˆ’1)π‘‘πœ‡π‘žβˆ’1ξ€·π‘₯1ξ€Έ=(βˆ’1)π‘›π‘žπ›Όπ‘›+β„Žβˆ’1ξ€œβ„€π‘ξ€Ίπ‘₯+π‘₯1ξ€»π‘žπ›Όπ‘žπ‘₯1(β„Žβˆ’1)π‘‘πœ‡π‘žξ€·π‘₯1ξ€Έ=(βˆ’1)π‘›π‘žπ›Όπ‘›+β„Žβˆ’1̃𝛽(β„Ž,1βˆ£π›Ό)𝑛,π‘ž(π‘₯).(4.17) Therefore, by (4.17), we obtain the following theorem.

Theorem 4.4. For β„Ž,π‘›βˆˆβ„€+, we have ̃𝛽(β„Ž,1βˆ£π›Ό)𝑛,π‘žβˆ’1(1βˆ’π‘₯)=(βˆ’1)π‘›π‘žπ›Όπ‘›+β„Žβˆ’1̃𝛽(β„Ž,1βˆ£π›Ό)𝑛,π‘ž(π‘₯).(4.18)

For π‘₯=1 in Theorem 4.4, we get ̃𝛽(β„Ž,1βˆ£π›Ό)𝑛,π‘žβˆ’1=(βˆ’1)π‘›π‘žπ›Όπ‘›+β„Žβˆ’1̃𝛽(β„Ž,1βˆ£π›Ό)𝑛,π‘ž(1)=(βˆ’1)π‘›π‘žπ›Όπ‘›βˆ’1̃𝛽(β„Ž,1βˆ£π›Ό)𝑛,π‘žif𝑛>1.(4.19) Therefore, by (4.19), we obtain the following corollary.

Corollary 4.5. For β„Žβˆˆβ„€+ and π‘›βˆˆβ„• with 𝑛>1, we have ̃𝛽(β„Ž,1βˆ£π›Ό)𝑛,π‘žβˆ’1=(βˆ’1)π‘›π‘žπ›Όπ‘›βˆ’1̃𝛽(β„Ž,1βˆ£π›Ό)𝑛,π‘ž.(4.20)

Let π‘‘βˆˆβ„•. By (4.1), we see that ξ€œβ„€π‘π‘ž(β„Žβˆ’1)π‘₯1ξ€Ίπ‘₯+π‘₯1ξ€»π‘›π‘žπ›Όπ‘‘πœ‡π‘žξ€·π‘₯1ξ€Έ=[𝑑]π‘›π‘žπ›Ό[𝑑]π‘žπ‘‘βˆ’1ξ“π‘Ž=0π‘žβ„Žπ‘Žξ€œβ„€π‘ξ‚ƒπ‘₯+π‘Žπ‘‘+π‘₯1ξ‚„π‘›π‘žπ›Όπ‘‘π‘žπ‘₯1(β„Žβˆ’1)π‘‘π‘‘πœ‡π‘žπ›Όξ€·π‘₯1ξ€Έ.(4.21) By (4.1) and (4.21), we obtain the following equation: ̃𝛽(β„Ž,1βˆ£π›Ό)𝑛,π‘ž[𝑑](π‘₯)=π‘›π‘žπ›Ό[𝑑]π‘žπ‘‘βˆ’1ξ“π‘Ž=0π‘žβ„Žπ‘ŽΜƒπ›½(β„Ž,1βˆ£π›Ό)𝑛,π‘žξ‚€π‘₯+π‘Žπ‘‘ξ‚,(4.22) where π‘‘βˆˆβ„• and β„Žβˆˆβ„€+.

5. Polynomials ̃𝛽(β„Ž,π‘˜|𝛼)𝑛,π‘ž(π‘₯) and β„Ž=π‘˜

From (2.1), we note that ̃𝛽(β„Ž,π‘˜|𝛼)𝑛,π‘ž(ξ€œπ‘₯)=β„€π‘β‹―ξ€œβ„€π‘ξ€Ίπ‘₯+π‘₯1+β‹―+π‘₯π‘˜ξ€»π‘›π‘žπ›Όπ‘ž(β„Žβˆ’1)π‘₯1+β‹―+(β„Žβˆ’π‘˜)π‘₯π‘˜π‘‘πœ‡π‘žξ€·π‘₯1ξ€Έβ‹―π‘‘πœ‡π‘žξ€·π‘₯π‘˜ξ€Έ=1(1βˆ’π‘ž)𝑛[𝛼]π‘›π‘žπ‘›ξ“π‘™=0βŽ›βŽœβŽœβŽπ‘›π‘™βŽžβŽŸβŽŸβŽ (βˆ’1)π‘™π‘žπ›Όπ‘™π‘₯(𝛼𝑙+β„Ž)β‹―(𝛼𝑙+β„Žβˆ’π‘˜+1)[]𝛼𝑙+β„Žπ‘žβ‹―[]𝛼𝑙+β„Žβˆ’π‘˜+1π‘ž,(5.1)π‘žβ„Žξ€œβ„€π‘β‹―ξ€œβ„€π‘ξ€Ίπ‘₯+1+π‘₯1+β‹―+π‘₯π‘˜ξ€»π‘›π‘žπ›Όπ‘ž(β„Žβˆ’1)π‘₯1+β‹―+(β„Žβˆ’π‘˜)π‘₯π‘˜π‘‘πœ‡π‘žξ€·π‘₯1ξ€Έβ‹―π‘‘πœ‡π‘žξ€·π‘₯π‘˜ξ€Έ=ξ€œβ„€π‘β‹―ξ€œβ„€π‘ξ€Ίπ‘₯+π‘₯1+β‹―+π‘₯π‘˜ξ€»π‘›π‘žπ›Όπ‘ž(β„Žβˆ’1)π‘₯1+β‹―+(β„Žβˆ’π‘˜)π‘₯π‘˜π‘‘πœ‡π‘žξ€·π‘₯1ξ€Έβ‹―π‘‘πœ‡π‘žξ€·π‘₯π‘˜ξ€Έ+ξ€œ(π‘žβˆ’1)β„Žβ„€π‘β‹―ξ€œβ„€π‘ξ€Ίπ‘₯+π‘₯2+β‹―+π‘₯π‘˜ξ€»π‘›π‘žπ›Όπ‘ž(β„Žβˆ’1)π‘₯2+β‹―+(β„Žβˆ’π‘˜)π‘₯π‘˜π‘‘πœ‡π‘žξ€·π‘₯2ξ€Έβ‹―π‘‘πœ‡π‘žξ€·π‘₯π‘˜ξ€Έπ›Ό+𝑛[𝛼]π‘žπ‘žπ›Όπ‘₯ξ€œβ„€π‘β‹―ξ€œβ„€π‘ξ€Ίπ‘₯+π‘₯2+β‹―+π‘₯π‘˜ξ€»π‘žπ‘›βˆ’1π›ΌΓ—π‘žπ›Ό(β„Žβˆ’1)π‘₯2+β‹―+(β„Žβˆ’π‘˜+1)π‘₯π‘˜π‘‘πœ‡π‘žξ€·π‘₯2ξ€Έβ‹―π‘‘πœ‡π‘žξ€·π‘₯π‘˜ξ€Έ.(5.2) From (5.1), we have π‘žβ„ŽΜƒπ›½(β„Ž,π‘˜βˆ£π›Ό)𝑛,π‘žΜƒπ›½(π‘₯+1)=(β„Ž,π‘˜βˆ£π›Ό)𝑛,π‘žΜƒπ›½(π‘₯)+(π‘žβˆ’1)β„Ž(β„Žβˆ’1,π‘˜βˆ’1βˆ£π›Ό)𝑛,π‘ž(π‘₯)+π‘žπ›Όπ‘₯𝑛𝛼[𝛼]π‘žΜƒπ›½(β„Ž,π‘˜βˆ’1βˆ£π›Ό)𝑛,π‘ž(π‘₯).(5.3) Therefore, by (5.3), we obtain the following theorem.

Theorem 5.1. For β„Ž,π‘›βˆˆβ„€+ and π‘˜βˆˆβ„•, we have π‘žβ„ŽΜƒπ›½(β„Ž,π‘˜βˆ£π›Ό)𝑛,π‘žΜƒπ›½(π‘₯+1)βˆ’(β„Ž,π‘˜βˆ£π›Ό)𝑛,π‘žΜƒπ›½(π‘₯)=(π‘žβˆ’1)β„Ž(β„Žβˆ’1,π‘˜βˆ’1βˆ£π›Ό)𝑛,π‘ž(π‘₯)+π‘›π‘žπ›Όπ‘₯𝛼[𝛼]π‘žΜƒπ›½(β„Ž,π‘˜βˆ’1βˆ£π›Ό)𝑛,π‘ž(π‘₯).(5.4)

It is easy to show that π‘žπ›Όπ‘₯ξ€œβ„€π‘β‹―ξ€œβ„€π‘ξ€Ίπ‘₯+π‘₯1+β‹―+π‘₯π‘˜ξ€»π‘›π‘žπ›Όπ‘žβ„Žπ‘₯1+(β„Žβˆ’1)π‘₯2+β‹―+(β„Ž+1βˆ’π‘˜)π‘₯π‘˜π‘‘πœ‡π‘žξ€·π‘₯1ξ€Έβ‹―π‘‘πœ‡π‘žξ€·π‘₯π‘˜ξ€Έ=(π‘žπ›Ό)ξ€œβˆ’1β„€π‘β‹―ξ€œβ„€π‘ξ€Ίπ‘₯+π‘₯1+β‹―+π‘₯π‘˜ξ€»π‘žπ‘›+1π›Όπ‘ž(β„Žβˆ’π›Ό)π‘₯1+β‹―+(β„Žβˆ’π›Ό+1βˆ’π‘˜)π‘₯π‘˜π‘‘πœ‡π‘žξ€·π‘₯1ξ€Έβ‹―π‘‘πœ‡π‘žξ€·π‘₯π‘˜ξ€Έ+ξ€œβ„€π‘β‹―ξ€œβ„€π‘ξ€Ίπ‘₯+π‘₯1+β‹―+π‘₯π‘˜ξ€»π‘›π‘žπ›Όπ‘ž(β„Žβˆ’π›Ό)π‘₯1+β‹―+(β„Žβˆ’π›Ό+1βˆ’π‘˜)π‘₯π‘˜π‘‘πœ‡π‘žξ€·π‘₯1ξ€Έβ‹―π‘‘πœ‡π‘žξ€·π‘₯π‘˜ξ€Έ=(π‘žπ›ΌΜƒπ›½βˆ’1)(β„Ž+1βˆ’π›Ό,π‘˜βˆ£π›Ό)𝑛+1,π‘žΜƒπ›½(π‘₯)+(β„Ž+1βˆ’π›Ό,π‘˜βˆ£π›Ό)𝑛,π‘ž(π‘₯).(5.5) Thus, by (5.5), we obtain the following proposition.

Proposition 5.2. For β„Ž,π‘›βˆˆβ„€+, we have π‘žπ›Όπ‘₯̃𝛽(β„Ž+1,π‘˜βˆ£π›Ό)𝑛,π‘ž(π‘₯)=(π‘žπ›ΌΜƒπ›½βˆ’1)(β„Ž+1βˆ’π›Ό,π‘˜βˆ£π›Ό)𝑛+1,π‘žΜƒπ›½(π‘₯)+(β„Ž+1βˆ’π›Ό,π‘˜βˆ£π›Ό)𝑛,π‘ž(π‘₯).(5.6)

For π‘‘βˆˆβ„•, we get ξ€œβ„€π‘β‹―ξ€œβ„€π‘ξƒ¬π‘₯+π‘˜ξ“π‘—=1π‘₯π‘—ξƒ­π‘›π‘žπ›Όπ‘žβˆ‘π‘˜π‘—=1(β„Žβˆ’π‘—)π‘₯π‘—π‘‘πœ‡π‘žξ€·π‘₯1ξ€Έβ‹―π‘‘πœ‡π‘žξ€·π‘₯π‘˜ξ€Έ=[𝑑]π‘›π‘žπ›Ό[𝑑]π‘˜π‘žπ‘‘βˆ’1ξ“π‘Ž1,…,π‘Žπ‘˜=0π‘žβ„Žβˆ‘π‘˜π‘—=1π‘Žπ‘—βˆ’βˆ‘π‘˜π‘—=2(π‘—βˆ’1)π‘Žπ‘—Γ—ξ€œβ„€π‘β‹―ξ€œβ„€π‘ξƒ¬βˆ‘π‘₯+π‘˜π‘—=1π‘Žπ‘—π‘‘+π‘˜ξ“π‘—=1π‘₯π‘—ξƒ­π‘›π‘žπ›Όπ‘‘π‘žπ‘‘βˆ‘π‘˜π‘—=1(β„Žβˆ’π‘—)π‘₯π‘—π‘‘πœ‡π‘žπ‘‘ξ€·π‘₯1ξ€Έβ‹―π‘‘πœ‡π‘žπ‘‘ξ€·π‘₯π‘˜ξ€Έ.(5.7) Thus, we obtain ̃𝛽(β„Ž,π‘˜βˆ£π›Ό)𝑛,π‘ž[𝑑](π‘₯)=π‘›π‘žπ›Ό[𝑑]π‘˜π‘žπ‘‘βˆ’1ξ“π‘Ž1,…,π‘Žπ‘˜=0π‘žβ„Žβˆ‘π‘˜π‘—=1π‘Žπ‘—βˆ’βˆ‘π‘˜π‘—=2(π‘—βˆ’1)π‘Žπ‘—Μƒπ›½(β„Ž,π‘˜βˆ£π›Ό)𝑛,π‘žπ‘‘ξ‚΅π‘₯+π‘Ž1+β‹―+π‘Žπ‘˜π‘‘ξ‚Ά.(5.8) Equation (5.8) is multiplication formula for the π‘ž-Bernoulli polynomials of order (β„Ž,π‘˜) with weight 𝛼.

Let us define ̃𝛽(π‘˜,π‘˜βˆ£π›Ό)𝑛,π‘žΜƒπ›½(π‘₯)=(π‘˜βˆ£π›Ό)𝑛,π‘ž(π‘₯). Then we see that ̃𝛽(π‘˜βˆ£π›Ό)𝑛,π‘ž1(π‘₯)=(1βˆ’π‘ž)𝑛[𝛼]π‘›π‘žπ‘›ξ“π‘™=0βŽ›βŽœβŽœβŽπ‘›π‘™βŽžβŽŸβŽŸβŽ (βˆ’1)π‘™π‘žπ›Όπ‘™π‘₯(𝛼𝑙+π‘˜)β‹―(𝛼𝑙+1)[]𝛼𝑙+π‘˜π‘žβ‹―[]𝛼𝑙+1π‘ž,(5.9)ξ€œβ„€π‘β‹―ξ€œβ„€π‘ξ€Ίπ‘˜βˆ’π‘₯+π‘₯1+β‹―π‘₯π‘˜ξ€»π‘›π‘žβˆ’π›Όπ‘žβˆ’(π‘˜βˆ’1)π‘₯1βˆ’β‹―βˆ’(π‘˜βˆ’π‘˜)π‘₯π‘˜π‘‘πœ‡π‘žβˆ’1ξ€·π‘₯1ξ€Έβ‹―π‘‘πœ‡π‘žβˆ’1ξ€·π‘₯π‘˜ξ€Έ=(βˆ’1)π‘›π‘žξ‚€2ξ‚π›Όπ‘›βˆ’π‘˜+π‘˜+1(1βˆ’π‘ž)𝑛[𝛼]π‘›π‘žπ‘›ξ“π‘™=0βŽ›βŽœβŽœβŽπ‘›π‘™βŽžβŽŸβŽŸβŽ (βˆ’1)π‘™π‘žπ›Όπ‘™π‘₯(𝛼𝑙+π‘˜)β‹―(𝛼𝑙+1)[]𝛼𝑙+π‘˜π‘žβ‹―[]𝛼𝑙+1π‘ž=(βˆ’1)π‘›π‘žξ‚€2ξ‚π›Όπ‘›βˆ’π‘˜+π‘˜+1̃𝛽(π‘˜βˆ£π›Ό)𝑛,π‘ž(π‘₯).(5.10) Therefore, by (5.9) and (5.10), we obtain the following theorem.

Theorem 5.3. For π‘›βˆˆβ„€+ and π‘˜βˆˆβ„•, we have ̃𝛽(π‘˜βˆ£π›Ό)𝑛,π‘žβˆ’1(π‘˜βˆ’π‘₯)=(βˆ’1)π‘›π‘žξ‚€2ξ‚π›Όπ‘›βˆ’π‘˜+π‘˜+1̃𝛽(π‘˜βˆ£π›Ό)𝑛,π‘ž(π‘₯).(5.11)

Let π‘₯=π‘˜ in Theorem 5.3. Then we see that ̃𝛽(π‘˜βˆ£π›Ό)𝑛,π‘žβˆ’1=(βˆ’1)π‘›π‘žξ‚€2ξ‚π›Όπ‘›βˆ’π‘˜+π‘˜+1̃𝛽(π‘˜βˆ£π›Ό)𝑛,π‘ž(π‘˜).(5.12) From Theorem 5.1, we can derive the following equation: π‘žπ‘˜Μƒπ›½(π‘˜βˆ£π›Ό)𝑛,π‘žΜƒπ›½(π‘₯+1)βˆ’(π‘˜βˆ£π›Ό)𝑛,π‘žΜƒπ›½(π‘₯)=π‘˜(π‘žβˆ’1)(π‘˜βˆ’1βˆ£π›Ό)𝑛,π‘ž(π‘₯)+π‘›π‘žπ›Όπ‘₯𝛼[𝛼]π‘žΜƒπ›½(π‘˜,π‘˜βˆ’1βˆ£π›Ό)𝑛,π‘ž(π‘₯).(5.13) By (5.9), we easily get ̃𝛽(π‘˜βˆ£π›Ό)𝑛,π‘ž=1(1βˆ’π‘ž)𝑛[𝛼]π‘›π‘žπ‘›ξ“π‘™=0βŽ›βŽœβŽœβŽπ‘›π‘™βŽžβŽŸβŽŸβŽ (βˆ’1)𝑙(𝛼𝑙+π‘˜)β‹―(𝛼𝑙+1)[]𝛼𝑙+π‘˜π‘žβ‹―[]𝛼𝑙+1π‘ž.(5.14) From the definition of 𝑝-adic π‘ž-integral on ℀𝑝, we note that 𝑛𝑙=0βŽ›βŽœβŽœβŽπ‘›π‘™βŽžβŽŸβŽŸβŽ (π‘žπ›Όβˆ’1)π‘™ξ€œβ„€π‘β‹―ξ€œβ„€π‘ξ€Ίπ‘₯1+β‹―+π‘₯π‘˜ξ€»π‘™π‘žπ›Όπ‘žβˆ‘π‘˜π‘™=0(π‘˜βˆ’π‘™)π‘₯π‘™π‘‘πœ‡π‘žξ€·π‘₯1ξ€Έβ‹―π‘‘πœ‡π‘žξ€·π‘₯π‘˜ξ€Έ=(𝛼𝑛+π‘˜)β‹―(𝛼𝑛+1)[]𝛼𝑛+π‘˜π‘žβ‹―[]𝛼𝑛+1π‘ž=(π‘˜π›Όπ‘›)π‘˜!(π‘˜π›Όπ‘›)π‘ž[π‘˜]π‘ž!.(5.15) Thus, by (5.15), we get 𝑛𝑙=0βŽ›βŽœβŽœβŽπ‘›π‘™βŽžβŽŸβŽŸβŽ (π‘žπ›Όβˆ’1)𝑙̃𝛽(π‘˜βˆ£π›Ό)𝑛,π‘ž=(π‘˜π›Όπ‘›)π‘˜!(π‘˜π›Όπ‘›)π‘ž[π‘˜]π‘ž!.(5.16) By the definition of polynomial ̃𝛽(π‘˜βˆ£π›Ό)𝑛,π‘ž(π‘₯), we see that ̃𝛽(π‘˜βˆ£π›Ό)𝑛,π‘ž(ξ€œπ‘₯)=β„€π‘β‹―ξ€œβ„€π‘ξ€Ίπ‘₯+π‘₯1+β‹―π‘₯π‘˜ξ€»π‘›π‘žπ›Όπ‘ž(π‘˜βˆ’1)π‘₯1+β‹―+(π‘˜βˆ’π‘˜)π‘₯π‘˜π‘‘πœ‡π‘žξ€·π‘₯1ξ€Έβ‹―π‘‘πœ‡π‘žξ€·π‘₯π‘˜ξ€Έ=𝑛𝑙=0βŽ›βŽœβŽœβŽπ‘›π‘™βŽžβŽŸβŽŸβŽ π‘žπ›Όπ‘™π‘₯̃𝛽(π‘˜βˆ£π›Ό)𝑙,π‘ž[π‘₯]π‘žπ‘›βˆ’π‘™=ξ‚€π‘žπ›Όπ‘₯Μƒπ›½π‘ž(π‘˜βˆ£π›Ό)+[π‘₯]π‘žπ›Όξ‚π‘›,whereπ‘›βˆˆβ„€+,(5.17) with the usual convention about replacing (Μƒπ›½π‘ž(π‘˜βˆ£π›Ό))𝑛 with ̃𝛽(π‘˜βˆ£π›Ό)𝑛,π‘ž.

Let π‘₯=0 in (5.13). Then we have π‘žπ‘˜Μƒπ›½(π‘˜βˆ£π›Ό)𝑛,π‘žΜƒπ›½(1)βˆ’(π‘˜βˆ£π›Ό)𝑛,π‘žΜƒπ›½=π‘˜(π‘žβˆ’1)(π‘˜βˆ’1βˆ£π›Ό)𝑛,π‘ž+𝑛𝛼[𝛼]π‘žΜƒπ›½(π‘˜,π‘˜βˆ’1βˆ£π›Ό)𝑛,π‘ž.(5.18)

Acknowledgments

The present research has been conducted by the Research Grant of Kwangwoon University in 2011. The authors would like to express their gratitude to referees for their valuable suggestions and comments.

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