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Journal of Applied Mathematics
VolumeΒ 2012Β (2012), Article IDΒ 743939, 8 pages
http://dx.doi.org/10.1155/2012/743939
Research Article

On the π‘ž-Genocchi Numbers and Polynomials with Weight 𝛼 and Weak Weight 𝛽

Department of Mathematics, Hannam University, Daejeon 306-791, Republic of Korea

Received 20 November 2011; Accepted 19 February 2012

Academic Editor: Francis T. K.Β Au

Copyright Β© 2012 J. Y. Kang 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

We construct a new type of π‘ž-Genocchi numbers and polynomials with weight 𝛼 and weak weight π›½βˆΆπΊ(𝛼,𝛽)𝑛,π‘ž,𝐺(𝛼,𝛽)𝑛,π‘ž(π‘₯), respectively. Some interesting results and relationships are obtained.

1. Introduction

The Genocchi numbers and polynomials possess many interesting properties and are arising in many areas of mathematics and physics. Recently, many mathematicians have studied in the area of the π‘ž-Genocchi numbers and polynomials (see [1–13]). In this paper, we construct a new type of π‘ž-Genocchi numbers 𝐺(𝛼,𝛽)𝑛,π‘ž and polynomials 𝐺(𝛼,𝛽)𝑛,π‘ž(π‘₯) with weight 𝛼 and weak weight 𝛽.

Throughout this paper, we use the following notations. By ℀𝑝, we denote the ring of 𝑝-adic rational integers, β„šπ‘ denotes the field of 𝑝-adic rational numbers, ℂ𝑝 denotes the completion of algebraic closure of β„šπ‘, β„• denotes the set of natural numbers, β„€ denotes the ring of rational integers, β„š denotes the field of rational numbers, β„‚ denotes the set of complex numbers, and β„€+=β„•βˆͺ{0}. Let πœˆπ‘ be the normalized exponential valuation of ℂ𝑝 with |𝑝|𝑝=π‘βˆ’πœˆπ‘(𝑝)=π‘βˆ’1. When one talks of π‘ž-extension, π‘ž is considered in many ways such as an indeterminate, a complex number π‘žβˆˆβ„‚, or 𝑝-adic number π‘žβˆˆβ„‚π‘. If π‘žβˆˆβ„‚, one normally assume that |π‘ž|<1. If π‘žβˆˆβ„‚π‘, we normally assumes that |π‘žβˆ’1|𝑝<π‘βˆ’(1/π‘βˆ’1) so that π‘žπ‘₯=exp(π‘₯logπ‘ž) for |π‘₯|𝑝≀1. Throughout this paper, we use the notation [π‘₯]π‘ž=1βˆ’π‘žπ‘₯,[π‘₯]1βˆ’π‘žβˆ’π‘ž=1βˆ’(βˆ’π‘ž)π‘₯,1+π‘ž(1.1) cf. [1–13].

Hence, limπ‘žβ†’1[π‘₯]=π‘₯ for any π‘₯ with |π‘₯|𝑝≀1 in the present 𝑝-adic case. For ξ€·β„€π‘“βˆˆπ‘ˆπ·π‘ξ€Έ=ξ€½π‘“βˆ£π‘“βˆΆβ„€π‘βŸΆβ„‚π‘isuniformlydifferentiablefunctionξ€Ύ,(1.2) the fermionic 𝑝-adic π‘ž-integral on ℀𝑝 is defined by Kim as follows: πΌβˆ’π‘ž(ξ€œπ‘“)=℀𝑝𝑓(π‘₯)π‘‘πœ‡βˆ’π‘ž(π‘₯)=limπ‘β†’βˆž1ξ€Ίπ‘π‘ξ€»π‘βˆ’π‘žπ‘βˆ’1π‘₯=0𝑓(π‘₯)(βˆ’π‘ž)π‘₯,(1.3) cf. [3–6].

If we take 𝑓1(π‘₯)=𝑓(π‘₯+1) in (1.1), then we easily see that π‘žπΌβˆ’π‘žξ€·π‘“1ξ€Έ+πΌβˆ’π‘ž[2](𝑓)=π‘žπ‘“(0).(1.4) From (1.4), we obtain π‘žπ‘›πΌβˆ’π‘žξ€·π‘“π‘›ξ€Έ+(βˆ’1)π‘›βˆ’1πΌβˆ’π‘ž[2](𝑓)=π‘žπ‘›βˆ’1𝑙=0(βˆ’1)π‘›βˆ’1βˆ’π‘™π‘žπ‘™π‘“(𝑙),(1.5) where 𝑓𝑛(π‘₯)=𝑓(π‘₯+𝑛) (cf. [3–6]).

As-well-known definition, the Genocchi polynomials are defined by 𝐹(𝑑)=2𝑑𝑒𝑑+1=𝑒𝐺𝑑=βˆžξ“π‘›=0𝐺𝑛𝑑𝑛,𝐹𝑛!(𝑑,π‘₯)=2𝑑𝑒𝑑𝑒+1π‘₯𝑑=𝑒𝐺(π‘₯)𝑑=βˆžξ“π‘›=0𝐺𝑛𝑑(π‘₯)𝑛.𝑛!(1.6) with the usual convention of replacing 𝐺𝑛(π‘₯) by 𝐺𝑛(π‘₯). In the special case, π‘₯=0, 𝐺𝑛(0)=𝐺𝑛 are called the 𝑛-th Genocchi numbers (cf. [1–11]).

These numbers and polynomials are interpolated by the Genocchi zeta function and Hurwitz-type Genocchi zeta function, respectively. 𝜁𝐺(𝑠)=2βˆžξ“π‘›=1(βˆ’1)𝑛𝑛𝑠,𝜁𝐺(𝑠,π‘₯)=2βˆžξ“π‘›=0(βˆ’1)𝑛(𝑛+π‘₯)𝑠.(1.7) Our aim in this paper is to define π‘ž-Genocchi numbers 𝐺(𝛼,𝛽)𝑛,π‘ž and polynomials 𝐺(𝛼,𝛽)𝑛,π‘ž(π‘₯) with weight 𝛼 and weak weight 𝛽. We investigate some properties which are related to π‘ž-Genocchi numbers 𝐺(𝛼,𝛽)𝑛,π‘ž and polynomials 𝐺(𝛼,𝛽)𝑛,π‘ž(π‘₯) with weight 𝛼 and weak weight 𝛽. We also derive the existence of a specific interpolation function which interpolates π‘ž-Genocchi numbers 𝐺(𝛼,𝛽)𝑛,π‘ž and polynomials 𝐺(𝛼,𝛽)𝑛,π‘ž(π‘₯) with weight 𝛼 and weak weight 𝛽 at negative integers.

2. π‘ž-Genocchi Numbers and Polynomials with Weight 𝛼 and Weak Weight 𝛽

Our primary goal of this section is to define π‘ž-Genocchi numbers 𝐺(𝛼,𝛽)𝑛,π‘ž and polynomials 𝐺(𝛼,𝛽)𝑛,π‘ž(π‘₯) with weight 𝛼 and weak weight 𝛽. We also find generating functions of π‘ž-Genocchi numbers 𝐺(𝛼,𝛽)𝑛,π‘ž and polynomials 𝐺(𝛼,𝛽)𝑛,π‘ž(π‘₯) with weight 𝛼 and weak weight 𝛽.

For π›Όβˆˆβ„€ and π‘žβˆˆβ„‚π‘ with |1βˆ’π‘ž|𝑝≀1, π‘ž-Genocchi numbers 𝐺(𝛼,𝛽)𝑛,π‘ž are defined by 𝐺(𝛼,𝛽)𝑛,π‘žξ€œ=𝑛℀𝑝[π‘₯]π‘žπ‘›βˆ’1π›Όπ‘‘πœ‡βˆ’π‘žπ›½(π‘₯).(2.1) By using 𝑝-adic π‘ž-integral on ℀𝑝, we obtain π‘›ξ€œβ„€π‘[π‘₯]π‘žπ‘›βˆ’1π›Όπ‘‘πœ‡βˆ’π‘žπ›½(π‘₯)=𝑛limπ‘β†’βˆž1ξ€Ίπ‘π‘ξ€»βˆ’π‘žπ›½π‘π‘βˆ’1π‘₯=0[π‘₯]π‘žπ‘›βˆ’1π›Όξ€·βˆ’π‘žπ›½ξ€Έπ‘₯[2]=π‘›π‘žπ›½ξ‚΅11βˆ’π‘žπ›Όξ‚Άπ‘›βˆ’1π‘›βˆ’1𝑙=0βŽ›βŽœβŽœβŽπ‘™βŽžβŽŸβŽŸβŽ π‘›βˆ’1(βˆ’1)𝑙11+π‘žπ›Όπ‘™+𝛽[2]=π‘›π‘žπ›½βˆžξ“π‘š=0(βˆ’1)π‘šπ‘žπ›½π‘š[π‘š]π‘žπ‘›βˆ’1𝛼.(2.2) By (2.1), we have 𝐺(𝛼,𝛽)𝑛,π‘ž[2]=π‘›π‘žπ›½βˆžξ“π‘š=0(βˆ’1)π‘šπ‘žπ›½π‘š[π‘š]π‘žπ‘›βˆ’1𝛼.(2.3) From the above, we can easily obtain that πΉπ‘ž(𝛼,𝛽)(𝑑)=βˆžξ“π‘›=0𝐺(𝛼,𝛽)𝑛,π‘žπ‘‘π‘›[2]𝑛!=π‘‘π‘žπ›½βˆžξ“π‘š=0(βˆ’1)π‘šπ‘žπ›½π‘šπ‘’[π‘š]π‘žπ›Όπ‘‘.(2.4) Thus, π‘ž-Genocchi numbers 𝐺(𝛼,𝛽)𝑛,π‘ž with weight 𝛼 and weak weight 𝛽 are defined by means of the generating function πΉπ‘ž(𝛼,𝛽)([2]𝑑)=π‘‘π‘žπ›½βˆžξ“π‘š=0(βˆ’1)π‘šπ‘žπ›½π‘šπ‘’[π‘š]π‘žπ›Όπ‘‘.(2.5)

Using similar method as above, we introduce π‘ž-Genocchi polynomials 𝐺(𝛼,𝛽)𝑛,π‘ž(π‘₯) with weight 𝛼 and weak weight 𝛽.

𝐺(𝛼,𝛽)𝑛,π‘ž(π‘₯) are defined by 𝐺(𝛼,𝛽)𝑛,π‘ž(ξ€œπ‘₯)=𝑛℀𝑝[]π‘₯+π‘¦π‘žπ‘›βˆ’1π›Όπ‘‘πœ‡βˆ’π‘žπ›½(𝑦).(2.6) By using 𝑝-adic π‘ž-integral, we have 𝐺(𝛼,𝛽)𝑛,π‘ž[2](π‘₯)=π‘›π‘žπ›½ξ‚΅11βˆ’π‘žπ›Όξ‚Άπ‘›βˆ’1π‘›βˆ’1𝑙=0βŽ›βŽœβŽœβŽπ‘™βŽžβŽŸβŽŸβŽ π‘›βˆ’1(βˆ’1)π‘™π‘žπ›Όπ‘₯𝑙11+π‘žπ›Όπ‘™+𝛽.(2.7) By using (2.6) and (2.7), we obtain πΉπ‘ž(𝛼,𝛽)(𝑑,π‘₯)=βˆžξ“π‘›=0𝐺(𝛼,𝛽)𝑛,π‘ž(𝑑π‘₯)𝑛[2]𝑛!=π‘‘π‘žπ›½βˆžξ“π‘š=0(βˆ’1)π‘šπ‘žπ›½π‘šπ‘’[π‘š+π‘₯]π‘žπ›Όπ‘‘.(2.8)

Remark 2.1. In (2.8), we simply see that limπ‘žβ†’1πΉπ‘ž(𝛼,𝛽)(𝑑,π‘₯)=2π‘‘βˆžξ“π‘š=0(βˆ’1)π‘šπ‘’(π‘š+π‘₯)𝑑=2𝑑1+𝑒𝑑𝑒π‘₯𝑑=𝐹(𝑑,π‘₯).(2.9) Since [π‘₯+𝑦]π‘žπ›Ό=[π‘₯]π‘žπ›Ό+π‘žπ›Όπ‘₯[𝑦]π‘žπ›Ό, we easily obtain that 𝐺(𝛼,𝛽)𝑛+1,π‘ž(ξ€œπ‘₯)=(𝑛+1)℀𝑝[]π‘₯+π‘¦π‘›π‘žπ›Όπ‘‘πœ‡βˆ’π‘žπ›½(𝑦)=π‘žβˆ’π›Όπ‘₯𝑛+1ξ“π‘˜=0βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ [π‘₯]𝑛+1π‘žπ‘›+1βˆ’π‘˜π›Όπ‘žπ›Όπ‘₯π‘˜πΊ(𝛼,𝛽)π‘˜,π‘ž=π‘žβˆ’π›Όπ‘₯ξ‚€[π‘₯]π‘žπ›Ό+π‘žπ›Όπ‘₯πΊπ‘ž(𝛼,𝛽)𝑛+1[2]=(𝑛+1)π‘žπ›½βˆžξ“π‘š=0(βˆ’1)π‘šπ‘žπ›½π‘š[]π‘₯+π‘šπ‘›π‘žπ›Ό.(2.10) Observe that, if π‘žβ†’1, then 𝐺(𝛼,𝛽)𝑛,π‘žβ†’πΊπ‘› and 𝐺(𝛼,𝛽)𝑛,π‘ž(π‘₯)→𝐺𝑛(π‘₯).
By (2.7), we have the following complement relation.

Theorem 2.2. Property of complement 𝐺(𝛼,𝛽)𝑛,π‘žβˆ’1(1βˆ’π‘₯)=(βˆ’1)π‘›βˆ’1π‘žπ›Ό(π‘›βˆ’1)𝐺(𝛼,𝛽)𝑛,π‘ž(π‘₯).(2.11)

By (2.7), we have the following distribution relation.

Theorem 2.3. For any positive integer π‘š (=odd), one has 𝐺(𝛼,𝛽)𝑛,π‘ž[2](π‘₯)=π‘žπ›½[2]π‘žπ›½π‘š[π‘š]π‘žπ‘›βˆ’1π›Όπ‘šβˆ’1𝑖=0(βˆ’1)π‘–π‘žπ›½π‘–πΊ(𝛼,𝛽)𝑛,π‘žπ‘šξ‚€π‘–+π‘₯π‘šξ‚,π‘›βˆˆβ„€+.(2.12) By (1.5), (2.1), and (2.6), one easily sees that π‘š[2]π‘žπ›½π‘›βˆ’1𝑙=0(βˆ’1)π‘›βˆ’1βˆ’π‘™π‘žπ›½π‘™[𝑙]π‘žπ‘šβˆ’1𝛼=π‘žπ›½π‘›πΊ(𝛼,𝛽)π‘š,π‘ž(𝑛)+(βˆ’1)π‘›βˆ’1𝐺(𝛼,𝛽)π‘š,π‘ž.(2.13)

Hence, we have the following theorem.

Theorem 2.4. Let π‘šβˆˆβ„€+.
If 𝑛≑0(mod2), then π‘žπ›½π‘›πΊ(𝛼,𝛽)π‘š,π‘ž(𝑛)βˆ’πΊ(𝛼,𝛽)π‘š,π‘ž[2]=π‘šπ‘žπ›½π‘›βˆ’1𝑙=0(βˆ’1)𝑙+1π‘žπ›½π‘™[𝑙]π‘žπ‘šβˆ’1𝛼.(2.14) If 𝑛≑1(mod2), then π‘žπ›½π‘›πΊ(𝛼,𝛽)π‘š,π‘ž(𝑛)+𝐺(𝛼,𝛽)π‘š,π‘ž[2]=π‘šπ‘žπ›½π‘›βˆ’1𝑙=0(βˆ’1)π‘™π‘žπ›½π‘™[𝑙]π‘žπ‘šβˆ’1𝛼.(2.15) From (1.4), one notes that [2]π‘žπ›½π‘‘=π‘žπ›½ξ€œβ„€π‘π‘‘π‘’[π‘₯+1]π‘žπ›Όπ‘‘π‘‘πœ‡βˆ’π‘žπ›½(ξ€œπ‘₯)+℀𝑝𝑑𝑒[π‘₯]π‘žπ›Όπ‘‘π‘‘πœ‡βˆ’π‘žπ›½(=π‘₯)βˆžξ“π‘›=0ξƒ©π‘žπ›½ξ€œβ„€π‘π‘›[]π‘₯+1π‘žπ‘›βˆ’1π›Όπ‘‘πœ‡βˆ’π‘žπ›½ξ€œ(π‘₯)+℀𝑝𝑛[π‘₯]π‘žπ‘›βˆ’1π›Όπ‘‘πœ‡βˆ’π‘žπ›½ξƒͺ𝑑(π‘₯)𝑛=𝑛!βˆžξ“π‘›=0ξ‚€π‘žπ›½πΊ(𝛼,𝛽)𝑛,π‘ž(1)+𝐺(𝛼,𝛽)𝑛,π‘žξ‚π‘‘π‘›.𝑛!(2.16)

Therefore, we obtain the following theorem.

Theorem 2.5. For π‘›βˆˆβ„€+, one has π‘žπ›½πΊ(𝛼,𝛽)𝑛,π‘ž(1)+𝐺(𝛼,𝛽)𝑛,π‘ž=ξ‚»[2]π‘žπ›½,if𝑛=1,0,if𝑛≠1.(2.17)

By Theorem 2.4 and (2.10), we have the following corollary.

Corollary 2.6. For π‘›βˆˆβ„€+, one has π‘žπ›½βˆ’π›Όξ‚€π‘žπ›ΌπΊπ‘ž(𝛼,𝛽)+1𝑛+𝐺(𝛼,𝛽)𝑛,π‘ž=ξ‚»[2]π‘žπ›½,if𝑛=1,0,if𝑛≠1.(2.18) with the usual convention of replacing (πΊπ‘ž(𝛼,𝛽))𝑛 by 𝐺(𝛼,𝛽)𝑛,π‘ž.

3. The Analogue of the Genocchi Zeta Function

By using π‘ž-Genocchi numbers and polynomials with weight 𝛼 and weak weight 𝛽, π‘ž-Genocchi zeta function and Hurwitz π‘ž-Genocchi zeta functions are defined. These functions interpolate the π‘ž-Genocchi numbers and π‘ž-Genocchi polynomials with weight 𝛼 and weak weight 𝛽, respectively. In this section, we assume that π‘žβˆˆβ„‚ with |π‘ž|<1. From (2.4), we note that π‘‘π‘˜+1π‘‘π‘‘π‘˜+1πΉπ‘ž(𝛼,𝛽)(||||𝑑)𝑑=0[2]=(π‘˜+1)π‘žπ›½βˆžξ“π‘š=0(βˆ’1)π‘šπ‘žπ›½π‘š[π‘š]π‘˜π‘žπ›Ό=𝐺(𝛼,𝛽)π‘˜+1,π‘ž,(π‘˜βˆˆβ„•).(3.1) By using the above equation, we are now ready to define π‘ž-Genocchi zeta functions.

Definition 3.1. Let π‘ βˆˆβ„‚. We define πœπ‘ž(𝛼,𝛽)([2]𝑠)=π‘žπ›½βˆžξ“π‘›=1(βˆ’1)π‘›π‘žπ›½π‘›[𝑛]π‘ π‘žπ›Ό.(3.2)

Note that πœπ‘ž(𝛼,𝛽)(𝑠) is a meromorphic function on β„‚. Note that, if π‘žβ†’1, then πœπ‘ž(𝛼,𝛽)(𝑠)=𝜁(𝑠) which is the Genocchi zeta functions. Relation between πœπ‘ž(𝛼,𝛽)(𝑠) and 𝐺(𝛼,𝛽)π‘˜,π‘ž is given by the following theorem.

Theorem 3.2. For π‘˜βˆˆβ„•, we have πœπ‘ž(𝛼,𝛽)(πΊβˆ’π‘˜)=(𝛼,𝛽)π‘˜+1,π‘ž.π‘˜+1(3.3)Observe that πœπ‘ž(𝛼,𝛽)(𝑠) function interpolates 𝐺(𝛼,𝛽)π‘˜,π‘ž numbers at nonnegative integers. By using (2.3), one notes that π‘‘π‘˜+1π‘‘π‘‘π‘˜+1πΉπ‘ž(𝛼,𝛽)(||||𝑑,π‘₯)𝑑=0[2]=(π‘˜+1)π‘žπ›½βˆžξ“π‘š=0(βˆ’1)π‘šπ‘žπ›½π‘š[]π‘₯+π‘šπ‘˜π‘žπ›Ό=𝐺(𝛼,𝛽)π‘˜+1,π‘ž(π‘₯),(π‘˜βˆˆβ„•),(3.4)ξ‚€π‘‘ξ‚π‘‘π‘‘π‘˜+1ξƒ©βˆžξ“π‘›=0𝐺(𝛼,𝛽)𝑛,π‘žπ‘‘(π‘₯)𝑛ξƒͺ|||||𝑛!𝑑=0=𝐺(𝛼,𝛽)π‘˜+1,π‘ž(π‘₯),forπ‘˜βˆˆβ„•.(3.5)

By (3.2) and (3.5), we are now ready to define the Hurwitz π‘ž-Genocchi zeta functions.

Definition 3.3. Let π‘ βˆˆβ„‚. We define πœπ‘ž(𝛼,𝛽)([2]𝑠,π‘₯)=π‘žπ›½βˆžξ“π‘›=0(βˆ’1)π‘›π‘žπ›½π‘›[]𝑛+π‘₯π‘ π‘žπ›Ό.(3.6) Note that πœπ‘ž(𝛼,𝛽)(𝑠,π‘₯) is a meromorphic function on β„‚.

Remark 3.4. It holds that limπ‘žβ†’1πœπ‘ž(𝛼,𝛽)(𝑠,π‘₯)=2βˆžξ“π‘›=0(βˆ’1)𝑛(𝑛+π‘₯)𝑠.(3.7)

Relation between πœπ‘ž(𝛼)(𝑠,π‘₯) and 𝐺(𝛼)π‘˜,π‘ž(π‘₯) is given by the following theorem.

Theorem 3.5. For π‘˜βˆˆβ„•, one has πœπ‘ž(𝛼,𝛽)(πΊβˆ’π‘˜,π‘₯)=(𝛼,𝛽)π‘˜+1,π‘ž(π‘₯).π‘˜+1(3.8) Observe that πœπ‘ž(𝛼,𝛽)(βˆ’π‘˜,π‘₯) function interpolates 𝐺(𝛼,𝛽)π‘˜,π‘ž(π‘₯) numbers at nonnegative integers.

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