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Journal of Function Spaces and Applications
VolumeΒ 2012Β (2012), Article IDΒ 214961, 13 pages
http://dx.doi.org/10.1155/2012/214961
Research Article

Novel Identities for π‘ž-Genocchi Numbers and Polynomials

Department of Mathematics, Faculty of Science and Arts, University of Gaziantep, 27310 Gaziantep, Turkey

Received 26 February 2012; Revised 25 April 2012; Accepted 9 May 2012

Academic Editor: GesturΒ Γ“lafsson

Copyright Β© 2012 Serkan Araci. 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 essential aim of this paper is to introduce novel identities for q-Genocchi numbers and polynomials by using the method by T. Kim et al. (article in press). We show that these polynomials are related to p-adic analogue of Bernstein polynomials. Also, we derive relations between q-Genocchi and q-Bernoulli numbers.

1. Preliminaries

Imagine that 𝑝 be a fixed odd prime number. We now start with definition of the following notations. Let β„šπ‘ be the field 𝑝-adic rational numbers and let ℂ𝑝 be the completion of algebraic closure of β„šπ‘.

Thus, β„šπ‘=ξƒ―π‘₯=βˆžξ“π‘›=βˆ’π‘˜π‘Žπ‘›π‘π‘›βˆΆ0β‰€π‘Žπ‘›ξƒ°.<𝑝(1.1)

Then ℀𝑝 is integral domain, which is defined by ℀𝑝=ξƒ―π‘₯=βˆžξ“π‘›=0π‘Žπ‘›π‘π‘›βˆΆ0β‰€π‘Žπ‘›ξƒ°,<𝑝(1.2) or ℀𝑝=ξ€½π‘₯βˆˆβ„šπ‘βˆΆ|π‘₯|𝑝.≀1(1.3)

We assume that π‘žβˆˆβ„‚π‘ with |1βˆ’π‘ž|𝑝<1 as an indeterminate. The 𝑝-adic absolute value |β‹…|𝑝, is normally defined by |π‘₯|𝑝=π‘βˆ’π‘Ÿ,(1.4) where π‘₯=π‘π‘Ÿ(𝑠/𝑑) with (𝑝,𝑠)=(𝑝,𝑑)=(𝑠,𝑑)=1, and π‘Ÿβˆˆβ„š.

[π‘₯]π‘ž is a π‘ž-extension of π‘₯, which is defined by [π‘₯]π‘ž=1βˆ’π‘žπ‘₯,1βˆ’π‘ž(1.5) we note that limπ‘žβ†’1[π‘₯]π‘ž=π‘₯ (see [1–17]).

Throughout this paper, we use notation of β„•βˆ—βˆΆ=β„•βˆͺ{0}, where β„• denotes set of Natural numbers.

We say that 𝑓 is a uniformly differentiable function at a point π‘Žβˆˆβ„€π‘, if the difference quotient, 𝐹𝑓(π‘₯,𝑦)=𝑓(π‘₯)βˆ’π‘“(𝑦),π‘₯βˆ’π‘¦(1.6) has a limit π‘“ξ…ž(π‘Ž) as (π‘₯,𝑦)β†’(π‘Ž,π‘Ž), and we denote this by π‘“βˆˆπ‘ˆπ·(℀𝑝). Then, for π‘“βˆˆπ‘ˆπ·(℀𝑝), we can start with the following expression: 1ξ€Ίπ‘π‘ξ€»π‘žξ“0β‰€πœ‰<𝑝𝑁𝑓(πœ‰)π‘žπœ‰=0β‰€πœ‰<𝑝𝑁𝑓(πœ‰)πœ‡π‘žξ€·πœ‰+𝑝𝑁℀𝑝,(1.7) which represents a 𝑝-adic π‘ž-analogue of Riemann sums for 𝑓. The integral of 𝑓 on ℀𝑝 will be defined as the limit (π‘β†’βˆž) of these sums, when it exists. The 𝑝-adic π‘ž-integral of function π‘“βˆˆπ‘ˆπ·(℀𝑝) is defined by Kim in [7, 12] as πΌπ‘ž(ξ€œπ‘“)=℀𝑝𝑓(πœ‰)π‘‘πœ‡π‘ž(πœ‰)=limπ‘β†’βˆž1ξ€Ίπ‘π‘ξ€»π‘žπ‘π‘βˆ’1ξ“πœ‰=0𝑓(πœ‰)π‘žπœ‰.(1.8)

The bosonic integral is considered as a bosonic limit π‘žβ†’1, 𝐼1(𝑓)=limπ‘žβ†’1πΌπ‘ž(𝑓). Similarly, the fermionic 𝑝-adic integral on ℀𝑝 is introduced by Kim as follows: πΌβˆ’π‘ž(ξ€œπ‘“)=℀𝑝𝑓(πœ‰)π‘‘πœ‡βˆ’π‘ž(πœ‰)(1.9) (for more details, see [13–16]).

From (1.9), it is well-known equality that π‘žπΌβˆ’π‘žξ€·π‘“1ξ€Έ+πΌβˆ’π‘ž[2](𝑓)=π‘žπ‘“(0),(1.10) where 𝑓1(π‘₯)=𝑓(π‘₯+1) (for details, see [2, 3, 8, 9, 12, 13, 15–17]).

The π‘ž-Genocchi polynomials with wegiht 0 are introduced as 𝐺𝑛+1,π‘ž(π‘₯)=ξ€œπ‘›+1℀𝑝(π‘₯+πœ‰)π‘›π‘‘πœ‡βˆ’π‘ž(πœ‰).(1.11)

From (1.11), we have 𝐺𝑛,π‘ž(π‘₯)=𝑛𝑙=0βŽ›βŽœβŽœβŽπ‘›π‘™βŽžβŽŸβŽŸβŽ π‘₯π‘™ξ‚πΊπ‘›βˆ’π‘™,π‘ž,(1.12) where 𝐺𝑛,π‘žξ‚πΊ(0)∢=𝑛,π‘ž are called π‘ž-Genocchi numbers with weight 0. Then, π‘ž-Genocchi numbers are defined as 𝐺0,π‘žξ‚€ξ‚πΊ=0,π‘žπ‘žξ‚+1𝑛+𝐺𝑛,π‘ž=ξ‚»[2]π‘ž,if𝑛=1,0,if𝑛≠1,(1.13) with the usual convention about replacing (ξ‚πΊπ‘ž)𝑛 by 𝐺𝑛,π‘ž is used (for details, see [3]).

Let π‘ˆπ·(℀𝑝) be the space of continuous functions on ℀𝑝. For π‘“βˆˆπ‘ˆπ·(℀𝑝), 𝑝-adic analogue of Bernstein operator for 𝑓 is defined by 𝐡𝑛(𝑓,π‘₯)=π‘›ξ“π‘˜=0π‘“ξ‚€π‘˜π‘›ξ‚π΅π‘˜,𝑛(π‘₯)=π‘›ξ“π‘˜=0π‘“ξ‚€π‘˜π‘›ξ‚βŽ›βŽœβŽœβŽπ‘›π‘˜βŽžβŽŸβŽŸβŽ π‘₯π‘˜(1βˆ’π‘₯)π‘›βˆ’π‘˜,(1.14) where 𝑛,π‘˜βˆˆβ„•βˆ—. Here, π΅π‘˜,𝑛(π‘₯) is called 𝑝-adic Bernstein polynomials, which are defined by π΅π‘˜,π‘›βŽ›βŽœβŽœβŽπ‘›π‘˜βŽžβŽŸβŽŸβŽ π‘₯(π‘₯)=π‘˜(1βˆ’π‘₯)π‘›βˆ’π‘˜[],π‘₯∈0,1(1.15) (for details, see [1, 4, 5, 7]).

The π‘ž-Bernoulli polynomials and numbers with weight 0 are defined by Kim et al., respectively, 𝐡𝑛,π‘ž(π‘₯)=limπ‘›β†’βˆž1[𝑝𝑛]π‘žπ‘π‘›βˆ’1𝑦=0(π‘₯+𝑦)π‘›π‘žπ‘¦=ξ€œβ„€π‘(π‘₯+πœ‰)π‘›π‘‘πœ‡π‘žξ‚π΅(πœ‰),𝑛,π‘ž=ξ€œβ„€π‘πœ‰π‘›π‘‘πœ‡π‘ž(πœ‰)(1.16) (for more information, see [10]).

The author, by using derivative operator, will investigate some interesting identities on the π‘ž-Genocchi numbers and polynomials arising from their generating function. Also, the author derives some relations between π‘ž-Genocchi numbers and π‘ž-Bernoulli numbers by using Kim’s π‘ž-Volkenborn integral and fermionic 𝑝-adic π‘ž-integral on ℀𝑝.

2. Novel Properties of π‘ž-Genocchi Numbers and Polynomials with Weight 0

Let 𝑓(π‘₯)=𝑒𝑑(π‘₯+𝑦). Then, by using (1.10), we easily procure the following: ξ€œβ„€π‘π‘’π‘‘(π‘₯+πœ‰)π‘‘πœ‡βˆ’π‘ž[2](πœ‰)=π‘žπ‘žπ‘’π‘‘π‘’+1π‘₯𝑑.(2.1)

From the last equality, by (1.11), we get Araci, Acikgoz, and Qi’s π‘ž-Genocchi polynomials with weight 0 in [3] as follows: [2]π‘žπ‘‘π‘žπ‘’π‘‘π‘’+1π‘₯𝑑=βˆžξ“π‘›=0𝐺𝑛,π‘ž(𝑑π‘₯)𝑛,||||𝑛!logπ‘ž+𝑑<πœ‹.(2.2)

Here, we assume that π‘₯ is a fixed parameter. Let ξ‚πΉπ‘ž([2]π‘₯,𝑑)=π‘žπ‘žπ‘’π‘‘π‘’+1π‘₯𝑑=βˆžξ“π‘›=0𝐺𝑛,π‘ž(𝑑π‘₯)π‘›βˆ’1.𝑛!(2.3) Thus, by expression of (2.3), we can readily see the following: π‘žπ‘’π‘‘ξ‚πΉπ‘žξ‚πΉ(π‘₯,𝑑)+π‘ž[2](π‘₯,𝑑)=π‘žπ‘’π‘₯𝑑.(2.4)

Last from equality, taking derivative operator 𝐷 as 𝐷=𝑑/𝑑𝑑 on the both sides of (2.4), then, we easily see that π‘žπ‘’π‘‘(𝐷+𝐼)π‘˜ξ‚πΉπ‘ž(π‘₯,𝑑)+π·π‘˜ξ‚πΉπ‘ž[2](π‘₯,𝑑)=π‘žπ‘₯π‘˜π‘’π‘₯𝑑,(2.5) where π‘˜βˆˆβ„•βˆ— and 𝐼 is identity operator. By multiplying π‘’βˆ’π‘‘ on both sides of (2.5), we get π‘ž(𝐷+𝐼)π‘˜ξ‚πΉπ‘ž(π‘₯,𝑑)+π‘’βˆ’π‘‘π·π‘˜ξ‚πΉπ‘ž[2](π‘₯,𝑑)=π‘žπ‘₯π‘˜π‘’(π‘₯βˆ’1)𝑑.(2.6)

Let us take derivative operator π·π‘š(π‘šβˆˆβ„•) on the both sides of (2.6). Then, we get π‘žπ‘’π‘‘π·π‘š(𝐷+𝐼)π‘˜ξ‚πΉπ‘ž(π‘₯,𝑑)+π·π‘˜(π·βˆ’πΌ)π‘šξ‚πΉπ‘ž[2](π‘₯,𝑑)=π‘žπ‘₯π‘˜(π‘₯βˆ’1)π‘šπ‘’π‘₯𝑑.(2.7)

Let 𝐺[0] (not 𝐺(0)) be the constant term in a Laurent series of 𝐺(𝑑) in (2.3). Then, we get π‘˜ξ“π‘—=0βŽ›βŽœβŽœβŽπ‘˜π‘—βŽžβŽŸβŽŸβŽ ξ‚€π‘žπ‘’π‘‘π·π‘˜+π‘šβˆ’π‘—ξ‚πΉπ‘žξ‚[0]+(π‘₯,𝑑)π‘šξ“π‘—=0βŽ›βŽœβŽœβŽπ‘šπ‘—βŽžβŽŸβŽŸβŽ (βˆ’1)π‘—ξ‚€π·π‘˜+π‘šβˆ’π‘—ξ‚πΉπ‘žξ‚[0]=[2](π‘₯,𝑑)π‘žπ‘₯π‘˜(π‘₯βˆ’1)π‘š.(2.8)

By (2.3), we easily see ξ‚€π·π‘ξ‚πΉπ‘žξ‚[0]=𝐺(π‘₯,𝑑)𝑁+1,π‘ž(π‘₯),𝑒𝑁+1π‘‘π·π‘ξ‚πΉπ‘žξ‚[0]=𝐺(π‘₯,𝑑)𝑁+1,π‘ž(π‘₯).𝑁+1(2.9)

We see that the members of (2.11) are proportional to the Bernstein polynomials with the following theorem.

Theorem 2.1. For π‘˜,π‘šβˆˆβ„•, one has [2]π‘ž(βˆ’1)π‘šπ΅π‘˜,π‘š+π‘˜(π‘₯)ξ€·π‘˜π‘š+π‘˜ξ€Έ=max{π‘˜,π‘š}𝑗=0π‘žξ€·π‘˜π‘—ξ€Έ+(βˆ’1)π‘—ξ€·π‘šπ‘—ξ€Έξ‚πΊπ‘˜+π‘šβˆ’π‘—+1π‘˜+π‘šβˆ’π‘—+1,π‘ž(π‘₯).(2.10)

Proof. By expressions of (2.8) and (2.9), we see that max{π‘˜,π‘š}𝑗=0π‘žξ€·π‘˜π‘—ξ€Έ+(βˆ’1)π‘—ξ€·π‘šπ‘—ξ€Έξ‚πΊπ‘˜+π‘šβˆ’π‘—+1π‘˜+π‘šβˆ’π‘—+1,π‘ž[2](π‘₯)=π‘žπ‘₯π‘˜(π‘₯βˆ’1)π‘š.(2.11)
By applying basic operations to above equality, we can easily reach to the desired result.

As a special case, we derive the following.

Corollary 2.2. For π‘˜βˆˆβ„•, one has [2]π‘ž(βˆ’1)π‘˜π΅π‘˜,2π‘˜(π‘₯)ξ€·π‘˜2π‘˜ξ€Έ=[2]π‘ž[π‘˜/2]𝑗=0ξ€·π‘˜2𝑗𝐺2π‘˜βˆ’2𝑗+12π‘˜βˆ’2𝑗+1,π‘žβŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ (π‘₯)+2π‘˜(π‘žβˆ’1)[(π‘˜βˆ’1)/2]𝑗=0ξ€·π‘˜2𝑗+1𝐺2π‘˜βˆ’2𝑗2π‘˜βˆ’2𝑗,π‘ž(π‘₯).(2.12)

Proof. When π‘˜=π‘š into (2.10), we derive the following identity: (βˆ’1)π‘˜π΅π‘˜,2π‘˜ξ€·(π‘₯)=π‘˜2π‘˜ξ€Έ1+π‘žπ‘˜ξ“π‘—=0βŽ‘βŽ’βŽ’βŽ£π‘žβŽ›βŽœβŽœβŽπ‘˜π‘—βŽžβŽŸβŽŸβŽ +(βˆ’1)π‘—βŽ›βŽœβŽœβŽπ‘˜π‘—βŽžβŽŸβŽŸβŽ βŽ€βŽ₯βŽ₯βŽ¦ξ‚πΊ2π‘˜βˆ’π‘—+1,π‘ž(π‘₯)=βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ 2π‘˜βˆ’π‘—+12π‘˜[π‘˜/2]𝑗=0ξ€·π‘˜2𝑗𝐺2π‘˜βˆ’2𝑗+12π‘˜βˆ’2𝑗+1,π‘ž(+βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ π‘₯)2π‘˜π‘žβˆ’1π‘ž+1[(π‘˜βˆ’1)/2]𝑗=0ξ€·π‘˜2𝑗+1𝐺2π‘˜βˆ’2𝑗2π‘˜βˆ’2𝑗,π‘ž(π‘₯).(2.13) Here, [π‘₯] is greatest integer ≀π‘₯. Then, we complete the proof of Theorem.

From (2.2), we note that 𝑑𝐺𝑑π‘₯𝑛,π‘žξ‚(π‘₯)=π‘›π‘›βˆ’1𝑙=0βŽ›βŽœβŽœβŽπ‘™βŽžβŽŸβŽŸβŽ ξ‚πΊπ‘›βˆ’1𝑙,π‘žπ‘₯π‘›βˆ’1βˆ’π‘™ξ‚πΊ=π‘›π‘›βˆ’1,π‘ž(π‘₯).(2.14)

By (2.14) and (1.11), we easily see that ξ€œ10𝐺𝑛,π‘žξ‚πΊ(π‘₯)𝑑π‘₯=𝑛+1,π‘žξ‚πΊ(1)βˆ’π‘›+1,π‘ž[2]𝑛+1=βˆ’π‘žβˆ’1𝐺𝑛+1,π‘ž[2]𝑛+1=βˆ’π‘žβˆ’1ξ€œβ„€π‘πœ‰π‘›π‘‘πœ‡βˆ’π‘ž(πœ‰).(2.15)

Now, let us consider definition of integral from 0 to 1 in (2.11), then we have βˆ’[2]π‘žβˆ’1max{π‘˜,π‘š}𝑗=0π‘žξ€·π‘˜π‘—ξ€Έ+(βˆ’1)π‘—ξ€·π‘šπ‘—ξ€Έξ‚πΊπ‘˜+π‘šβˆ’π‘—+1π‘˜+π‘šβˆ’π‘—+2,π‘ž=[2]π‘˜+π‘šβˆ’π‘—+2π‘ž(βˆ’1)π‘š=[2]𝐡(π‘˜+1,π‘š+1)π‘ž(βˆ’1)π‘šΞ“(π‘˜+1)Ξ“(π‘š+1),Ξ“(π‘˜+π‘š+2)(2.16) where 𝐡(π‘˜+1,π‘š+1) is beta function which is defined by ξ€œπ΅(π‘˜+1,π‘š+1)=10π‘₯π‘˜(1βˆ’π‘₯)π‘š1𝑑π‘₯=ξ€·(π‘˜+π‘š+1)π‘šπ‘˜+π‘šξ€Έ,π‘˜>0,π‘š>0.(2.17)

As a result, we obtain the following theorem.

Theorem 2.3. For π‘š,π‘˜βˆˆβ„•, one has max{π‘˜,π‘š}𝑗=1π‘žξ€·π‘˜π‘—ξ€Έ+(βˆ’1)π‘—ξ€·π‘šπ‘—ξ€Έξ‚πΊπ‘˜+π‘šβˆ’π‘—+1π‘˜+π‘šβˆ’π‘—+2,π‘žπ‘˜+π‘šβˆ’π‘—+2=π‘ž(βˆ’1)π‘š+1ξ€·(π‘˜+π‘š+1)π‘˜π‘˜+π‘šξ€Έβˆ’[2]π‘žξ‚πΊπ‘˜+π‘š+1π‘˜+π‘š+2,π‘ž.π‘˜+π‘š+2(2.18)

Proof. By taking integral from 0 to 1 in (2.11), we easily reach to desired result.

Substituting π‘š=π‘˜+1 into Theorem 2.1, we readily get π‘˜+1𝑗=1π‘žξ€·π‘˜π‘—ξ€Έ+(βˆ’1)π‘—ξ€·π‘—π‘˜+1𝐺2π‘˜βˆ’π‘—+32π‘˜βˆ’π‘—+3,π‘ž=π‘ž(βˆ’1)π‘˜(ξ€·2π‘˜+2)π‘˜2π‘˜+1ξ€Έβˆ’[2]π‘žξ‚πΊ2π‘˜+22π‘˜+3,π‘ž.2π‘˜+3(2.19)

By differentiating both sides of (2.11) with respect to 𝑑, we have the following: max{π‘˜,π‘š}𝑗=0⎧βŽͺ⎨βŽͺβŽ©π‘žβŽ›βŽœβŽœβŽπ‘˜π‘—βŽžβŽŸβŽŸβŽ +(βˆ’1)π‘—βŽ›βŽœβŽœβŽπ‘šπ‘—βŽžβŽŸβŽŸβŽ βŽ«βŽͺ⎬βŽͺβŽ­ξ‚πΊπ‘˜+π‘šβˆ’π‘—,π‘ž[2](π‘₯)=π‘žπ‘₯π‘˜βˆ’1(π‘₯βˆ’1)π‘šβˆ’1((π‘˜+π‘š)π‘₯βˆ’π‘˜).(2.20)

We now give interesting theorem for π‘ž-Genocchi numbers with weight 0 as follows.

Theorem 2.4. For π‘˜βˆˆβ„•, one has [2]π‘ž[π‘˜/2]𝑗=0ξ€·π‘˜2𝑗𝐺2π‘˜βˆ’2𝑗+12π‘˜βˆ’2𝑗+2,π‘ž2π‘˜βˆ’2𝑗+2+(π‘žβˆ’1)[(π‘˜βˆ’1)/2]𝑗=0ξ€·π‘˜2𝑗+1𝐺2π‘˜βˆ’2𝑗2π‘˜βˆ’2𝑗+1,π‘ž=2π‘˜βˆ’2𝑗+1π‘ž(βˆ’1)π‘˜+1(ξ€·2π‘˜+1)π‘˜2π‘˜ξ€Έ.(2.21)

Proof. It is proved by using definition of integral on the both sides in the following equality, that is, π‘˜ξ“π‘—=0π‘žξ€·π‘˜π‘—ξ€Έ+(βˆ’1)π‘—ξ€·π‘˜π‘—ξ€Έξ‚»ξ€œ2π‘˜βˆ’π‘—+110𝐺2π‘˜βˆ’π‘—+1,π‘žξ‚Ό=[2](π‘₯)𝑑π‘₯π‘žξ‚»ξ€œ10π‘₯π‘˜(π‘₯βˆ’1)π‘˜ξ‚Ό.𝑑π‘₯(2.22) Last from equality, we discover the following: [2]π‘ž[π‘˜/2]𝑗=0βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ ξ‚†βˆ«2𝑗10𝐺2π‘˜βˆ’2𝑗+1,π‘žξ‚‡(π‘₯)𝑑π‘₯2π‘˜βˆ’2𝑗+1+(π‘žβˆ’1)[(π‘˜βˆ’1)/2]𝑗=0βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ ξ‚†βˆ«2𝑗+110𝐺2π‘˜βˆ’2𝑗,π‘žξ‚‡(π‘₯)𝑑π‘₯=[2]2π‘˜βˆ’2π‘—π‘ž(βˆ’1)π‘˜ξ‚»ξ€œ10π‘₯π‘˜(1βˆ’π‘₯)π‘˜ξ‚Ό.𝑑π‘₯(2.23) Then, taking integral from 0 to 1 both sides of last equality, we get βˆ’[2]π‘žβˆ’1[2]π‘ž[π‘˜/2]𝑗=0ξ€·π‘˜2𝑗𝐺2π‘˜βˆ’2𝑗+12π‘˜βˆ’2𝑗+2,π‘ž+[2]2π‘˜βˆ’2𝑗+2π‘žβˆ’1(1βˆ’π‘ž)[(π‘˜βˆ’1)/2]𝑗=0ξ€·π‘˜2𝑗+1𝐺2π‘˜βˆ’2𝑗2π‘˜βˆ’2𝑗+1,π‘ž=[2]2π‘˜βˆ’2𝑗+1π‘ž(βˆ’1)π‘˜[2]𝐡(π‘˜+1,π‘˜+1)=π‘ž(βˆ’1)π‘˜ξ€·(2π‘˜+1)π‘˜2π‘˜ξ€Έ.(2.24)
Thus, we complete the proof of the theorem.

Theorem 2.5. For π‘˜βˆˆβ„•, one has [2]π‘ž[(π‘˜+1)/2]𝑗=0ξ€·π‘˜2𝑗𝐺2π‘˜βˆ’2𝑗+12π‘˜βˆ’2𝑗+1,π‘ž(π‘₯)+[π‘˜/2]𝑗=1ξ€·π‘˜2π‘—βˆ’1𝐺2π‘˜βˆ’2𝑗+12π‘˜βˆ’2𝑗+1,π‘ž(π‘₯)βˆ’[π‘˜/2]𝑗=0ξ€·π‘˜2𝑗𝐺2π‘˜βˆ’2𝑗2π‘˜βˆ’2𝑗,π‘ž[2](π‘₯)+(π‘žβˆ’1)π‘ž[π‘˜/2]𝑗=0βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ ξƒ―ξ‚πΊ2𝑗+12π‘˜βˆ’2𝑗,π‘ž(π‘₯)[2]π‘ž+𝐺(2π‘˜βˆ’2𝑗)2π‘˜βˆ’2𝑗+1,π‘ž(π‘₯)[2]2π‘žξƒ°(2π‘˜βˆ’2𝑗+1)=π‘₯π‘˜(π‘₯βˆ’1)π‘˜ξ€·[2]π‘žξ€Έ.π‘₯βˆ’π‘ž(2.25)

Proof. In view of (2.2) and (2.23), we discover the following applications: π‘˜+1𝑗=0βŽ‘βŽ’βŽ’βŽ£π‘žβŽ›βŽœβŽœβŽπ‘˜π‘—βŽžβŽŸβŽŸβŽ +(βˆ’1)π‘—βŽ›βŽœβŽœβŽπ‘—βŽžβŽŸβŽŸβŽ βŽ€βŽ₯βŽ₯βŽ¦ξ‚πΊπ‘˜+12π‘˜βˆ’π‘—+1,π‘ž(π‘₯)=[2]2π‘˜βˆ’π‘—+1π‘žξ‚πΊ2π‘˜+1,π‘ž(π‘₯)+2π‘˜+1[(π‘˜+1)/2]𝑗=1βŽ‘βŽ’βŽ’βŽ£π‘žβŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ +βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ +βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ βŽ€βŽ₯βŽ₯βŽ¦ξ‚πΊ2𝑗2𝑗2π‘—βˆ’12π‘˜βˆ’2𝑗+1,π‘ž(+2π‘˜βˆ’2𝑗+1π‘₯)[π‘˜/2]𝑗=0βŽ‘βŽ’βŽ’βŽ£π‘žβŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ βˆ’βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ βˆ’βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ βŽ€βŽ₯βŽ₯βŽ¦ξ‚πΊ2𝑗+12𝑗+12𝑗2π‘˜βˆ’2𝑗,π‘ž(π‘₯)⎧βŽͺ⎨βŽͺ⎩2π‘˜βˆ’2𝑗=βˆ’[(π‘˜+1)/2]𝑗=0βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ ξ‚πΊ2𝑗2π‘˜βˆ’2𝑗,π‘ž(π‘₯)+2π‘˜βˆ’2π‘—π‘žβˆ’11+π‘ž[π‘˜/2]𝑗=0βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ ξ‚πΊ2𝑗+12π‘˜βˆ’2𝑗+1,π‘ž(π‘₯)⎫βŽͺ⎬βŽͺ⎭+[2]2π‘˜βˆ’2𝑗+1π‘ž[(π‘˜+1)/2]𝑗=0βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ Γ—ξ‚πΊ2𝑗2π‘˜βˆ’2𝑗+1,π‘ž(π‘₯)+2π‘˜βˆ’2𝑗+1[(π‘˜+1)/2]𝑗=1βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ ξ‚πΊ2π‘—βˆ’12π‘˜βˆ’2𝑗+1,π‘ž(π‘₯)βˆ’2π‘˜βˆ’2𝑗+1[π‘˜/2]𝑗=0βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ ξ‚πΊ2𝑗2π‘˜βˆ’2𝑗,π‘ž(π‘₯)+2π‘˜βˆ’2𝑗(π‘žβˆ’1)[π‘˜/2]𝑗=0βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ ξ‚πΊ2𝑗+12π‘˜βˆ’2𝑗,π‘ž(π‘₯)+2π‘˜βˆ’2π‘—π‘žβˆ’11+π‘ž[π‘˜/2]𝑗=0βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ ξ‚πΊ2𝑗+12π‘˜βˆ’2𝑗+1,π‘ž(π‘₯).2π‘˜βˆ’2𝑗+1(2.26)
Thus, we give evidence of the theorem.

As π‘žβ†’1 into Theorem 2.5, it leads to the following interesting property.

Corollary 2.6. For π‘˜βˆˆβ„•, one has [(π‘˜+1)/2]𝑗=0βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ πΊ2𝑗2π‘˜βˆ’2𝑗+1(π‘₯)+2π‘˜+1βˆ’2𝑗[π‘˜/2]𝑗=1βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ πΊ2π‘—βˆ’12π‘˜βˆ’2𝑗+1(π‘₯)βˆ’4π‘˜βˆ’4𝑗+2[π‘˜/2]𝑗=0βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ πΊ2𝑗2π‘˜βˆ’2𝑗,π‘ž(π‘₯)4π‘˜βˆ’4𝑗=π‘₯π‘˜(π‘₯βˆ’1)π‘˜ξ‚€1π‘₯βˆ’2,(2.27) where 𝐺𝑛(π‘₯) is ordinary Genocchi polynomials, which is defined by the means of the following generating function [9]: βˆžξ“π‘›=0𝐺𝑛(𝑑π‘₯)𝑛=𝑛!2𝑑𝑒𝑑𝑒+1π‘₯𝑑,|𝑑|<πœ‹.(2.28)

3. Some Identities π‘ž-Genocchi Numbers and π‘ž-Bernoulli Numbers by Using Kim’s 𝑝-Adic π‘ž-Integrals on ℀𝑝

In this section, we consider π‘ž-Genocchi numbers and π‘ž-Bernoulli numbers by means of 𝑝-adic π‘ž-integral on ℀𝑝. Now, we start with the following theorem.

Theorem 3.1. For π‘š,π‘˜βˆˆβ„•, one has max{π‘˜,π‘š}𝑗=0π‘žξ€·π‘˜π‘—ξ€Έ+(βˆ’1)π‘—ξ€·π‘šπ‘—ξ€Έπ‘˜+π‘šβˆ’π‘—+1π‘˜+π‘šβˆ’π‘—+1𝑙=0ξ€·π‘™π‘˜+π‘šβˆ’π‘—+1𝐺𝑙+1π‘˜+π‘šβˆ’π‘—βˆ’π‘™+1,π‘žξ‚πΊπ‘™+1,π‘ž=[2]π‘žπ‘šξ“π‘™=0βŽ›βŽœβŽœβŽπ‘šπ‘™βŽžβŽŸβŽŸβŽ (βˆ’1)π‘šβˆ’π‘™ξ‚πΊπ‘™+π‘˜+1,π‘žπ‘™+π‘˜+1.(3.1)

Proof. For π‘š,π‘˜βˆˆβ„•, then by (2.11), 𝐼1=[2]π‘žξ€œβ„€π‘π‘₯π‘˜(π‘₯βˆ’1)π‘šπ‘‘πœ‡βˆ’π‘ž[2](π‘₯)=π‘žπ‘šξ“π‘™=0βŽ›βŽœβŽœβŽπ‘šπ‘™βŽžβŽŸβŽŸβŽ (βˆ’1)π‘šβˆ’π‘™ξ€œβ„€π‘π‘₯𝑙+π‘˜π‘‘πœ‡βˆ’π‘ž=[2](π‘₯)π‘žπ‘šξ“π‘™=0βŽ›βŽœβŽœβŽπ‘šπ‘™βŽžβŽŸβŽŸβŽ (βˆ’1)π‘šβˆ’π‘™ξ‚πΊπ‘™+π‘˜+1,π‘ž.𝑙+π‘˜+1(3.2) On the other hand, the right hand side of (2.11), 𝐼2=max{π‘˜,π‘š}𝑗=0π‘žξ€·π‘˜π‘—ξ€Έ+(βˆ’1)π‘—ξ€·π‘šπ‘—ξ€Έπ‘˜+π‘šβˆ’π‘—+1π‘˜+π‘šβˆ’π‘—+1𝑙=0βŽ›βŽœβŽœβŽπ‘™βŽžβŽŸβŽŸβŽ ξ‚πΊπ‘˜+π‘šβˆ’π‘—+1π‘˜+π‘šβˆ’π‘—βˆ’π‘™+1,π‘žξ€œβ„€π‘π‘₯π‘™π‘‘πœ‡βˆ’π‘ž=(π‘₯)max{π‘˜,π‘š}𝑗=0π‘žξ€·π‘˜π‘—ξ€Έ+(βˆ’1)π‘—ξ€·π‘šπ‘—ξ€Έπ‘˜+π‘šβˆ’π‘—+1π‘˜+π‘šβˆ’π‘—+1𝑙=0ξ€·π‘™π‘˜+π‘šβˆ’π‘—+1𝐺𝑙+1π‘˜+π‘šβˆ’π‘—βˆ’π‘™+1,π‘žξ‚πΊπ‘™+1,π‘ž.(3.3) Combining 𝐼1 and 𝐼2, we arrive to the proof of the theorem.

Theorem 3.2. For π‘˜βˆˆβ„•, one has π‘˜ξ“π‘™=0βŽ›βŽœβŽœβŽπ‘˜π‘™βŽžβŽŸβŽŸβŽ (βˆ’1)π‘˜βˆ’π‘™ξƒ―[2]π‘žξ‚πΊπ‘˜+𝑙+2,π‘žξ‚πΊπ‘˜+𝑙+2βˆ’π‘žπ‘˜+𝑙+1,π‘žξƒ°=[2]π‘˜+𝑙+1π‘ž[π‘˜/2]𝑗=0ξ€·π‘˜2𝑗2π‘˜βˆ’2𝑗+12π‘˜βˆ’2𝑗+1𝑙=0𝑙2π‘˜βˆ’2𝑗+1𝐺𝑙+12π‘˜+1βˆ’2π‘—βˆ’π‘™,π‘žξ‚πΊπ‘™+1,π‘ž+[π‘˜/2]𝑗=1ξ€·π‘˜2π‘—βˆ’1ξ€Έ2π‘˜βˆ’2𝑗+12π‘˜βˆ’2𝑗+1𝑙=0𝑙2π‘˜βˆ’2𝑗+1𝐺𝑙+12π‘˜+1βˆ’2π‘—βˆ’π‘™,π‘žξ‚πΊπ‘™+1,π‘ž+π‘žβˆ’11+π‘ž[π‘˜/2]𝑗=0βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ π‘‡2𝑗+1π‘žπ‘˜,𝑗,(3.4) here π‘‡π‘žπ‘˜,𝑗=[2]π‘žβˆ‘2π‘˜βˆ’2𝑗𝑙=0(𝑙2π‘˜βˆ’2𝑗𝐺/(2π‘˜βˆ’2𝑗))(𝑙+1,π‘žξ‚πΊ2π‘˜βˆ’2π‘—βˆ’π‘™,π‘žβˆ‘/(𝑙+1))+2π‘˜βˆ’2𝑗+1𝑙=0(𝑙2π‘˜βˆ’2𝑗+1𝐺/(2π‘˜βˆ’2𝑗+1))(𝑙+1,π‘žξ‚πΊ2π‘˜βˆ’2π‘—βˆ’π‘™+1,π‘ž/(𝑙+1)).

Proof. Let us take fermionic 𝑝-adic π‘ž-inetgral on ℀𝑝 left-hand side of Theorem 2.5, we get 𝐼3=ξ€œβ„€π‘π‘₯π‘˜(π‘₯βˆ’1)π‘˜ξ€·[2]π‘žξ€Έπ‘₯βˆ’π‘žπ‘‘πœ‡βˆ’π‘ž(=[2]π‘₯)π‘žπ‘˜ξ“π‘™=0βŽ›βŽœβŽœβŽπ‘˜π‘™βŽžβŽŸβŽŸβŽ (βˆ’1)π‘˜βˆ’π‘™ξ€œβ„€π‘π‘₯π‘˜+𝑙+1π‘‘πœ‡βˆ’π‘ž(π‘₯)βˆ’π‘žπ‘˜ξ“π‘™=0βŽ›βŽœβŽœβŽπ‘˜π‘™βŽžβŽŸβŽŸβŽ (βˆ’1)π‘˜βˆ’π‘™ξ€œβ„€π‘π‘₯π‘˜+π‘™π‘‘πœ‡βˆ’π‘ž=[2](π‘₯)π‘žπ‘˜ξ“π‘™=0βŽ›βŽœβŽœβŽπ‘˜π‘™βŽžβŽŸβŽŸβŽ (βˆ’1)π‘˜βˆ’π‘™ξ‚πΊπ‘˜+𝑙+2,π‘žπ‘˜+𝑙+2βˆ’π‘žπ‘˜ξ“π‘™=0βŽ›βŽœβŽœβŽπ‘˜π‘™βŽžβŽŸβŽŸβŽ (βˆ’1)π‘˜βˆ’π‘™ξ‚πΊπ‘˜+𝑙+1,π‘ž.π‘˜+𝑙+1(3.5) In other word, we consider the right-hand side of Theorem 2.5 as follows: 𝐼4=[2]π‘ž[π‘˜/2]𝑗=0ξ€·π‘˜2𝑗2π‘˜βˆ’2𝑗+12π‘˜βˆ’2𝑗+1𝑙=0βŽ›βŽœβŽœβŽπ‘™βŽžβŽŸβŽŸβŽ ξ‚πΊ2π‘˜βˆ’2𝑗+12π‘˜+1βˆ’2π‘—βˆ’π‘™,π‘žξ€œβ„€π‘π‘₯π‘™π‘‘πœ‡βˆ’π‘ž+(π‘₯)[π‘˜/2]𝑗=1ξ€·π‘˜2π‘—βˆ’1ξ€Έ2π‘˜βˆ’2𝑗+12π‘˜βˆ’2𝑗+1𝑙=0βŽ›βŽœβŽœβŽπ‘™βŽžβŽŸβŽŸβŽ ξ‚πΊ2π‘˜βˆ’2𝑗+12π‘˜+1βˆ’2π‘—βˆ’π‘™,π‘žξ€œβ„€π‘π‘₯π‘™π‘‘πœ‡βˆ’π‘ž(+π‘₯)[π‘˜/2]𝑗=0βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ βŽ§βŽͺβŽͺ⎨βŽͺβŽͺ⎩2𝑗+1(π‘žβˆ’1)2π‘˜βˆ’2π‘—βˆ‘π‘—=0𝑙2π‘˜βˆ’2π‘—βˆ’π‘™ξ€Έξ‚πΊ2π‘˜βˆ’2𝑗2π‘˜βˆ’2π‘—βˆ’π‘™,π‘žβˆ«β„€π‘π‘₯π‘™π‘‘πœ‡βˆ’π‘ž+(π‘₯)π‘žβˆ’11+π‘ž2π‘˜βˆ’2𝑗+1βˆ‘π‘™=0𝑙2π‘˜βˆ’2𝑗+1𝐺2π‘˜βˆ’2𝑗+12π‘˜βˆ’2π‘—βˆ’π‘™+1βˆ«β„€π‘π‘₯π‘™π‘‘πœ‡βˆ’π‘ž(⎫βŽͺβŽͺ⎬βŽͺβŽͺ⎭=[2]π‘₯)π‘ž[π‘˜/2]𝑗=0ξ€·π‘˜2𝑗2π‘˜βˆ’2𝑗+12π‘˜βˆ’2𝑗+1𝑙=0𝑙2π‘˜βˆ’2π‘—βˆ’π‘™+1𝐺𝑙+12π‘˜+1βˆ’2π‘—βˆ’π‘™,π‘žξ‚πΊπ‘™+1,π‘ž+[π‘˜/2]𝑗=1ξ€·π‘˜2π‘—βˆ’1ξ€Έ2π‘˜βˆ’2𝑗+12π‘˜βˆ’2𝑗+1𝑙=0𝑙2π‘˜βˆ’2π‘—βˆ’π‘™+1𝐺𝑙+12π‘˜+1βˆ’2π‘—βˆ’π‘™,π‘žξ‚πΊπ‘™+1,π‘ž+[π‘˜/2]𝑗=0βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ βŽ§βŽͺβŽͺ⎨βŽͺβŽͺ⎩(2𝑗+1π‘žβˆ’1)2π‘˜βˆ’2π‘—βˆ‘π‘—=0𝑙2π‘˜βˆ’2π‘—βˆ’π‘™ξ€Έξ‚πΊ2π‘˜βˆ’2𝑗2π‘˜βˆ’2π‘—βˆ’π‘™,π‘žξ‚πΊπ‘™+1,π‘ž+𝑙+1π‘žβˆ’11+π‘ž2π‘˜βˆ’2𝑗+1βˆ‘π‘™=0𝑙2π‘˜βˆ’2π‘—βˆ’π‘™+1𝐺2π‘˜βˆ’2𝑗+12π‘˜βˆ’2π‘—βˆ’π‘™+1𝐺𝑙+1,π‘žβŽ«βŽͺβŽͺ⎬βŽͺβŽͺ⎭.𝑙+1(3.6)
Equating 𝐼3 and 𝐼4, we complete the proof of the theorem.

As π‘žβ†’1 in the above theorem, we reach interesting property in Analytic Numbers Theory concerning ordinary Genocchi polynomials.

Corollary 3.3. For π‘˜βˆˆβ„•, one has π‘˜ξ“π‘™=0βŽ›βŽœβŽœβŽπ‘˜π‘™βŽžβŽŸβŽŸβŽ (βˆ’1)π‘˜βˆ’π‘™ξ‚»2πΊπ‘˜+𝑙+2βˆ’πΊπ‘˜+𝑙+2π‘˜+𝑙+1ξ‚Όπ‘˜+𝑙+1=2[π‘˜/2]𝑗=0ξ€·π‘˜2𝑗2π‘˜βˆ’2𝑗+12π‘˜βˆ’2𝑗+1𝑙=0𝑙2π‘˜βˆ’2𝑗+1𝐺𝑙+12π‘˜+1βˆ’2π‘—βˆ’π‘™πΊπ‘™+1+[π‘˜/2]𝑗=1ξ€·π‘˜2π‘—βˆ’1ξ€Έ2π‘˜βˆ’2𝑗+12π‘˜βˆ’2𝑗+1𝑙=0𝑙2π‘˜βˆ’2𝑗+1𝐺𝑙+12π‘˜+1βˆ’2π‘—βˆ’π‘™πΊπ‘™+1.(3.7)

Theorem 3.4. For π‘š,π‘˜βˆˆβ„•, one has [2]π‘žπ‘šξ“π‘™=0βŽ›βŽœβŽœβŽπ‘šπ‘™βŽžβŽŸβŽŸβŽ (βˆ’1)π‘šβˆ’π‘™ξ‚π΅π‘™+π‘˜,π‘ž=max{π‘˜,π‘š}𝑗=0π‘žξ€·π‘˜π‘—ξ€Έ+(βˆ’1)π‘—ξ€·π‘šπ‘—ξ€Έπ‘˜+π‘šβˆ’π‘—+1π‘˜+π‘šβˆ’π‘—+1𝑙=0βŽ›βŽœβŽœβŽπ‘™βŽžβŽŸβŽŸβŽ ξ‚πΊπ‘˜+π‘šβˆ’π‘—+1π‘˜+π‘š+1βˆ’π‘—βˆ’π‘™,π‘žξ‚π΅π‘™,π‘ž.(3.8)

Proof. We consider (2.11) and (2.2) by means of π‘ž-Volkenborn integral. Then, by (2.11), we see [2]π‘žξ€œβ„€π‘π‘₯π‘˜(π‘₯βˆ’1)π‘šπ‘‘πœ‡π‘ž[2](π‘₯)=π‘žπ‘šξ“π‘™=0βŽ›βŽœβŽœβŽπ‘šπ‘™βŽžβŽŸβŽŸβŽ (βˆ’1)π‘šβˆ’π‘™ξ€œβ„€π‘π‘₯𝑙+π‘˜π‘‘πœ‡π‘ž[2](π‘₯)=π‘žπ‘šξ“π‘™=0βŽ›βŽœβŽœβŽπ‘šπ‘™βŽžβŽŸβŽŸβŽ (βˆ’1)π‘šβˆ’π‘™ξ‚π΅π‘™+π‘˜,π‘ž.(3.9) On the other hand, max{π‘˜,π‘š}𝑗=0π‘žξ€·π‘˜π‘—ξ€Έ+(βˆ’1)π‘—ξ€·π‘šπ‘—ξ€Έπ‘˜+π‘šβˆ’π‘—+1π‘˜+π‘šβˆ’π‘—+1𝑙=0βŽ›βŽœβŽœβŽπ‘™βŽžβŽŸβŽŸβŽ ξ‚πΊπ‘˜+π‘šβˆ’π‘—+1π‘˜+π‘š+1βˆ’π‘—βˆ’π‘™,π‘žξ€œβ„€π‘π‘₯π‘™π‘‘πœ‡π‘ž=(π‘₯)max{π‘˜,π‘š}𝑗=0π‘žξ€·π‘˜π‘—ξ€Έ+(βˆ’1)π‘—ξ€·π‘šπ‘—ξ€Έπ‘˜+π‘šβˆ’π‘—+1π‘˜+π‘šβˆ’π‘—+1𝑙=0βŽ›βŽœβŽœβŽπ‘™βŽžβŽŸβŽŸβŽ ξ‚πΊπ‘˜+π‘šβˆ’π‘—+1π‘˜+π‘š+1βˆ’π‘—βˆ’π‘™,π‘žξ‚π΅π‘™,π‘ž.(3.10) Therefore, we get the proof of theorem.

Corollary 3.5. For π‘˜βˆˆβ„•, one gets π‘˜ξ“π‘™=0βŽ›βŽœβŽœβŽπ‘˜π‘™βŽžβŽŸβŽŸβŽ (βˆ’1)π‘˜βˆ’π‘™ξ‚†[2]π‘žξ‚π΅π‘˜+𝑙+1,π‘žξ‚π΅βˆ’π‘žπ‘˜+𝑙,π‘žξ‚‡=[2]π‘ž[π‘˜/2]𝑗=0ξ€·π‘˜2𝑗2π‘˜βˆ’2𝑗+12π‘˜βˆ’2𝑗+1𝑙=0βŽ›βŽœβŽœβŽπ‘™βŽžβŽŸβŽŸβŽ ξ‚πΊ2π‘˜βˆ’2𝑗+12π‘˜+1βˆ’2π‘—βˆ’π‘™,π‘žξ‚π΅π‘™,π‘ž+[π‘˜/2]𝑗=1ξ€·π‘˜2π‘—βˆ’1ξ€ΈΓ—2π‘˜βˆ’2𝑗+12π‘˜βˆ’2𝑗+1𝑙=0βŽ›βŽœβŽœβŽπ‘™βŽžβŽŸβŽŸβŽ ξ‚πΊ2kβˆ’2𝑗+12π‘˜+1βˆ’2π‘—βˆ’π‘™,π‘žξ‚π΅π‘™,π‘ž+ξ‚΅π‘žβˆ’1ξ‚Άπ‘ž+1[π‘˜/2]𝑗=0βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ π‘†2𝑗+1π‘žπ‘˜,𝑗,(3.11) where π‘†π‘žπ‘˜,𝑗=[2]π‘žβˆ‘2π‘˜βˆ’2𝑗𝑗=0ξ€·(1/(2π‘˜βˆ’2𝑗))𝑙2π‘˜βˆ’2𝑗𝐺2π‘˜βˆ’2π‘—βˆ’π‘™,π‘žξ‚π΅π‘™,π‘ž+βˆ‘2π‘˜βˆ’2𝑗+1𝑙=0ξ€·(1/(2π‘˜βˆ’2𝑗+1))𝑙2π‘˜βˆ’2𝑗+1𝐺2π‘˜βˆ’2π‘—βˆ’π‘™+1𝐡𝑙,π‘ž.

Proof. By using 𝑝-adic π‘ž-integral on ℀𝑝 left-hand side of Theorem 2.5, we get 𝐼5=[2]π‘žξ€œβ„€π‘π‘₯π‘˜(π‘₯βˆ’1)π‘˜([2]π‘₯βˆ’π‘ž)π‘‘πœ‡π‘ž(=[2]π‘₯)π‘žπ‘˜ξ“π‘™=0βŽ›βŽœβŽœβŽπ‘˜π‘™βŽžβŽŸβŽŸβŽ (βˆ’1)π‘˜βˆ’π‘™ξ€œβ„€π‘π‘₯π‘˜+𝑙+1π‘‘πœ‡π‘ž(π‘₯)βˆ’π‘žπ‘˜ξ“π‘™=0βŽ›βŽœβŽœβŽπ‘˜π‘™βŽžβŽŸβŽŸβŽ (βˆ’1)π‘˜βˆ’π‘™ξ€œβ„€π‘π‘₯π‘˜+π‘™π‘‘πœ‡π‘ž=[2](π‘₯)π‘žπ‘˜ξ“π‘™=0βŽ›βŽœβŽœβŽπ‘˜π‘™βŽžβŽŸβŽŸβŽ (βˆ’1)π‘˜βˆ’π‘™ξ‚π΅π‘˜+𝑙+1,π‘žβˆ’π‘žπ‘˜ξ“π‘™=0βŽ›βŽœβŽœβŽπ‘˜π‘™βŽžβŽŸβŽŸβŽ (βˆ’1)π‘˜βˆ’π‘™ξ‚π΅π‘˜+𝑙,π‘ž.(3.12) Also, we compute the right-hand side of Theorem 2.5 as follows: 𝐼6=[2]π‘ž[π‘˜/2]𝑗=01βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ 2π‘˜βˆ’2𝑗+12𝑗2π‘˜βˆ’2𝑗+1𝑙=0βŽ›βŽœβŽœβŽπ‘™βŽžβŽŸβŽŸβŽ ξ‚πΊ2π‘˜βˆ’2𝑗+12π‘˜+1βˆ’2π‘—βˆ’π‘™,π‘žξ€œβ„€π‘π‘₯π‘™π‘‘πœ‡π‘ž+(π‘₯)[π‘˜/2]𝑗=11βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ 2π‘˜βˆ’2𝑗+12π‘—βˆ’12π‘˜βˆ’2𝑗+1𝑙=0βŽ›βŽœβŽœβŽπ‘™βŽžβŽŸβŽŸβŽ ξ‚πΊ2π‘˜βˆ’2𝑗+12π‘˜+1βˆ’2π‘—βˆ’π‘™,π‘žξ€œβ„€π‘π‘₯π‘™π‘‘πœ‡π‘ž(+π‘₯)[π‘˜/2]𝑗=0βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ βŽ§βŽͺβŽͺβŽͺ⎨βŽͺβŽͺβŽͺ⎩2𝑗+1(π‘žβˆ’1)2π‘˜βˆ’2π‘—βˆ‘π‘—=01βŽ›βŽœβŽœβŽπ‘™βŽžβŽŸβŽŸβŽ ξ‚πΊ2π‘˜βˆ’2𝑗2π‘˜βˆ’2𝑗2π‘˜βˆ’2π‘—βˆ’π‘™,π‘žβˆ«β„€π‘π‘₯π‘™π‘‘πœ‡π‘ž+(π‘₯)π‘žβˆ’11+π‘ž2π‘˜βˆ’2𝑗+1βˆ‘π‘™=01βŽ›βŽœβŽœβŽπ‘™βŽžβŽŸβŽŸβŽ ξ‚πΊ2π‘˜βˆ’2𝑗+12π‘˜βˆ’2𝑗+12π‘˜βˆ’2π‘—βˆ’π‘™+1βˆ«β„€π‘π‘₯π‘™π‘‘πœ‡π‘žβŽ«βŽͺβŽͺβŽͺ⎬βŽͺβŽͺβŽͺ⎭=[2](π‘₯)π‘ž[π‘˜/2]𝑗=01βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ 2π‘˜βˆ’2𝑗+12𝑗2π‘˜βˆ’2𝑗+1𝑙=0βŽ›βŽœβŽœβŽπ‘™βŽžβŽŸβŽŸβŽ ξ‚πΊ2π‘˜βˆ’2𝑗+12π‘˜+1βˆ’2π‘—βˆ’π‘™,π‘žξ‚π΅π‘™,π‘ž+[π‘˜/2]𝑗=11βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ 2π‘˜βˆ’2𝑗+12π‘—βˆ’12π‘˜βˆ’2𝑗+1𝑙=0βŽ›βŽœβŽœβŽπ‘™βŽžβŽŸβŽŸβŽ ξ‚πΊ2π‘˜βˆ’2𝑗+12π‘˜+1βˆ’2π‘—βˆ’π‘™,π‘žξ‚π΅π‘™,π‘ž+[π‘˜/2]𝑗=0βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ βŽ§βŽͺβŽͺβŽͺ⎨βŽͺβŽͺβŽͺ⎩2𝑗+1(π‘žβˆ’1)2π‘˜βˆ’2π‘—βˆ‘π‘—=01βŽ›βŽœβŽœβŽπ‘™βŽžβŽŸβŽŸβŽ ξ‚πΊ2π‘˜βˆ’2𝑗2π‘˜βˆ’2𝑗2π‘˜βˆ’2π‘—βˆ’π‘™,π‘žξ‚π΅π‘™,π‘ž+π‘žβˆ’11+π‘ž2π‘˜βˆ’2𝑗+1βˆ‘π‘™=01βŽ›βŽœβŽœβŽπ‘™βŽžβŽŸβŽŸβŽ ξ‚πΊ2π‘˜βˆ’2𝑗+12π‘˜βˆ’2𝑗+12π‘˜βˆ’2π‘—βˆ’π‘™+1𝐡𝑙,π‘žβŽ«βŽͺβŽͺβŽͺ⎬βŽͺβŽͺβŽͺ⎭.(3.13)
Equating 𝐼5 and 𝐼6, we get the proof of Corollary.

As π‘žβ†’1 in the above theorem, we easily derive the following corollary.

Corollary 3.6. For π‘˜βˆˆβ„•, one has π‘˜ξ“π‘™=0βŽ›βŽœβŽœβŽπ‘˜π‘™βŽžβŽŸβŽŸβŽ (βˆ’1)π‘˜βˆ’π‘™ξ€½2π΅π‘˜+𝑙+1βˆ’π΅π‘˜+𝑙=2[π‘˜/2]𝑗=0ξ€·π‘˜2𝑗2π‘˜βˆ’2𝑗+12π‘˜βˆ’2𝑗+1𝑙=0βŽ›βŽœβŽœβŽπ‘™βŽžβŽŸβŽŸβŽ πΊ2π‘˜βˆ’2𝑗+12π‘˜+1βˆ’2π‘—βˆ’π‘™π΅π‘™+[π‘˜/2]𝑗=1ξ€·π‘˜2π‘—βˆ’1ξ€Έ2π‘˜βˆ’2𝑗+12π‘˜βˆ’2𝑗+1𝑙=0βŽ›βŽœβŽœβŽπ‘™βŽžβŽŸβŽŸβŽ πΊ2π‘˜βˆ’2𝑗+12π‘˜+1βˆ’2π‘—βˆ’π‘™π΅π‘™.(3.14)

Acknowledgment

The author would like to express his sincere gratitude to the referee for his/her valuable comments and suggestions which have improved the presentation of the paper.

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