International Scholarly Research Notices

International Scholarly Research Notices / 2011 / Article

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Volume 2011 |Article ID 271784 | https://doi.org/10.5402/2011/271784

Seog-Hoon Rim, Joohee Jeong, "Some New Identities on the π‘ž -Genocchi Numbers and Polynomials with Weight 𝜢 ", International Scholarly Research Notices, vol. 2011, Article ID 271784, 6 pages, 2011. https://doi.org/10.5402/2011/271784

Some New Identities on the π‘ž -Genocchi Numbers and Polynomials with Weight 𝜢

Academic Editor: A. Bellouquid
Received20 Sep 2011
Accepted15 Nov 2011
Published22 Dec 2011

Abstract

We construct a new type of π‘ž-Genocchi numbers and polynomials with weight 𝛼. From these π‘ž-Genocchi numbers and polynomials with weight 𝛼, we establish some interesting identities and relations.

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 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„€+=ℕ⋃{0}. Let 𝑣𝑝 be the normalized exponential valuation of ℂ𝑝 with |𝑝|𝑝=π‘βˆ’π‘£π‘(𝑝)=1/𝑝. When one talks of π‘ž-extension, π‘ž is variously considered as an indeterminate, a complex π‘žβˆˆβ„‚, or a 𝑝-adic number π‘ž. In this paper, we assume that π‘žβˆˆβ„‚π‘ with |1βˆ’π‘ž|𝑝<1. As a definition of π‘ž-numbers, we use the notation of π‘ž-number of [π‘₯]π‘ž=1βˆ’π‘žπ‘₯1βˆ’π‘ž,[π‘₯]βˆ’π‘ž=1βˆ’(βˆ’π‘ž)π‘₯1+π‘ž(1.1)(cf. [1–11]). Note that limπ‘žβ†’1[π‘₯]π‘ž=π‘₯. Let 𝐢(℀𝑝) be the space of continuous functions on ℀𝑝. For π‘“βˆˆπΆ(℀𝑝), the 𝑝-adic invariant integral on ℀𝑝 is defined by Kim [1, 3], πΌβˆ’π‘ž(𝑓)=ξ€œβ„€π‘π‘“(π‘₯)π‘‘πœ‡βˆ’π‘ž(π‘₯)=limπ‘β†’βˆž1ξ€Ίπ‘π‘ξ€»βˆ’π‘žπ‘π‘βˆ’1π‘₯=0𝑓(π‘₯)(βˆ’π‘ž)π‘₯.(1.2) From (1.2), we have the well-known integral equation π‘žπ‘›πΌβˆ’1𝑓𝑛+(βˆ’1)π‘›βˆ’1πΌπ‘ž(𝑓)=[2]π‘žπ‘›βˆ’1𝑙=0(βˆ’1)π‘™π‘žπ‘™π‘“(𝑙)(1.3)(see [1, 3]), where 𝑓𝑛(π‘₯)=𝑓(π‘₯+𝑛), (π‘›βˆˆβ„•).

For π›Όβˆˆβ„•, in [11], the π‘ž-Genocchi polynomials with weight 𝛼 are introduced by π‘‘ξ€œβ„€π‘π‘’[π‘₯+𝑦]π‘žπ›Όπ‘‘π‘‘πœ‡βˆ’π‘ž(𝑦)=βˆžξ“π‘›=0𝐺(𝛼)𝑛,π‘ž(π‘₯)𝑑𝑛𝑛!.(1.4) By comparing the coefficients of both sides of (1.4), we have 𝐺(𝛼)0,π‘ž(π‘₯)=0,𝐺(𝛼)𝑛+1,π‘ž(π‘₯)(𝑛+1)=ξ€œβ„€π‘[π‘₯+𝑦]π‘›π‘žπ›Όπ‘‘πœ‡βˆ’π‘ž(𝑦),forπ‘›βˆˆβ„•.(1.5)

In the special case, π‘₯=0, 𝐺(𝛼)𝑛,π‘ž(0)=𝐺(𝛼)𝑛,π‘ž are called the 𝑛th π‘ž-Genocchi numbers with weight 𝛼.

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

In this section, we show some new identities on the π‘ž-Genocchi numbers and polynomials with weight 𝛼. And we establish the distribution relation for π‘ž-Genocchi polynomials with weight 𝛼.

From (1.5), we can easily see that 𝐺(𝛼)𝑛+1,π‘ž(π‘₯)𝑛+1=[2]π‘ž[𝛼]π‘›π‘ž(1βˆ’π‘ž)𝑛𝑛𝑙=0βŽ›βŽœβŽœβŽπ‘›π‘™βŽžβŽŸβŽŸβŽ (βˆ’1)π‘™π‘žπ›Όπ‘™π‘₯1+π‘žπ›Όπ‘™+1.(2.1) From (1.5) and (2.1), we note that 𝐺(𝛼)𝑛+1,π‘ž(π‘₯)𝑛+1=ξ€œβ„€π‘[π‘₯+𝑦]π‘›π‘žπ›Όπ‘‘πœ‡βˆ’1(𝑦)=𝑛𝑙=0βŽ›βŽœβŽœβŽπ‘›π‘™βŽžβŽŸβŽŸβŽ [π‘₯]π‘›βˆ’π‘™π‘žπ›Όπ‘žπ›Όπ‘™π‘₯ξ€œβ„€π‘[𝑦]π‘™π‘žπ›Όπ‘‘πœ‡βˆ’π‘ž(𝑦)=𝑛𝑙=0βŽ›βŽœβŽœβŽπ‘›π‘™βŽžβŽŸβŽŸβŽ [π‘₯]π‘›βˆ’π‘™π‘žπ›Όπ‘žπ›Όπ‘™π‘₯𝐺(𝛼)𝑙+1,π‘žπ‘™+1.(2.2) Note that (1/(𝑙+1))(𝑛𝑙)=(1/(𝑛+1))𝑛+1𝑙+1ξ€Έ and from (2.2), we have the relation of polynomials and numbers, 𝐺(𝛼)𝑛+1,π‘ž(π‘₯)𝑛+1=1(𝑛+1)𝑛𝑙=0βŽ›βŽœβŽœβŽπ‘›+1𝑙+1⎞⎟⎟⎠[π‘₯]𝑛+1βˆ’π‘™π‘žπ›Όπ‘žπ›Όπ‘™π‘₯𝐺(𝛼)𝑙+1,π‘ž=1(𝑛+1)π‘žπ›Όπ‘₯𝑛+1𝑙=0βŽ›βŽœβŽœβŽπ‘›+1𝑙+1⎞⎟⎟⎠[π‘₯]𝑛+1βˆ’π‘™π‘žπ›Όπ‘žπ›Ό(𝑙+1)π‘₯𝐺(𝛼)𝑙+1,π‘ž=1(𝑛+1)π‘žπ›Όπ‘₯ξ‚€[π‘₯]π‘žπ›Ό+π‘žπ›Όπ‘™π‘₯𝐺(𝛼)π‘žξ‚π‘›+1,(2.3) with the usual convention of replacing (𝐺(𝛼)π‘ž)𝑛 by (𝐺(𝛼)𝑛,π‘ž).

Thus, by (2.3), we have a theorem.

Theorem 2.1. For π›Όβˆˆβ„• and π‘›βˆˆβ„€+, one has π‘žπ›Όπ‘₯𝐺(𝛼)𝑛+1,π‘ž(π‘₯)=(𝑛+1)π‘žπ›Όπ‘₯ξ‚€[π‘₯]π‘žπ›Ό+π‘žπ›Όπ‘₯𝐺(𝛼)π‘žξ‚π‘›+1=𝑛+1𝑙=0βŽ›βŽœβŽœβŽπ‘›+1π‘™βŽžβŽŸβŽŸβŽ [π‘₯]𝑛+1βˆ’π‘™π‘žπ›Όπ‘žπ›Όπ‘₯𝐺(𝛼)𝑙,π‘ž.(2.4)
In (1.3), if we take 𝑛=1, π‘žπΌβˆ’1𝑓1ξ€Έ+πΌβˆ’1(𝑓)=[2]π‘ž.(2.5)
We apply 𝑓(π‘₯)=𝑒[π‘₯]π‘žπ›Όπ‘‘ with (1.5), and we have the following: [2]π‘ž=βˆžξ“π‘›=0βŽ›βŽœβŽπ‘žξ€œβ„€π‘[π‘₯+1]π‘›π‘žπ›Όπ‘‘πœ‡βˆ’π‘ž(π‘₯)+ξ€œβ„€π‘[π‘₯]π‘›π‘žπ›Όπ‘‘πœ‡βˆ’π‘ž(π‘₯)βŽžβŽŸβŽ π‘‘π‘›π‘›!=βˆžξ“π‘›=0βŽ›βŽœβŽœβŽπ‘žξ‚πΊ(𝛼)𝑛+1,π‘ž(1)𝑛+1+𝐺(𝛼)𝑛+1,π‘žπ‘›+1βŽžβŽŸβŽŸβŽ π‘‘π‘›π‘›!.(2.6) By comparing the coefficients on both the sides in (2.6), we get π‘žξ‚πΊ(𝛼)𝑛+1,π‘ž(1)𝑛+1+𝐺(𝛼)𝑛+1,π‘žπ‘›+1=ξ‚»[2]π‘žif𝑛=0,0if𝑛>0.(2.7) From (2.2) and (2.7), we can derive the following: 𝐺(𝛼)1,π‘ž(1)=1,π‘ž1βˆ’π›Όξ‚€π‘žπ›Όξ‚πΊ(𝛼)π‘ž+1𝑛+𝐺(𝛼)𝑛,π‘ž=0ifπ‘›βˆˆβ„•,(2.8) with the usual convention of replacing (𝐺(𝛼)π‘ž)𝑛 by 𝐺(𝛼)𝑛,π‘ž.
For a fixed odd positive integer 𝑑 with (𝑝,𝑑)=1, we set 𝑋=𝑋𝑑=limβŸ΅π‘β„€π‘‘π‘π‘β„€,𝑋1=℀𝑝,π‘‹βˆ—=0<π‘Ž<𝑑𝑝,(π‘Ž,𝑝)=1ξ€·π‘Ž+𝑑𝑝℀𝑝,π‘Ž+𝑑𝑝𝑁℀𝑝=ξ€½π‘₯βˆˆπ‘‹βˆ£π‘₯β‰‘π‘Žξ€·mod𝑑𝑝𝑁,(2.9) where π‘Žβˆˆβ„€ satisfies the condition 0β‰€π‘Ž<𝑑𝑝𝑁. For the distribution relation for the π‘ž-Genocchi polynomials with weight 𝛼, we consider the following: ξ€œβ„€π‘[π‘₯+𝑦]π‘›π‘žπ›Όπ‘‘πœ‡βˆ’π‘ž(𝑦)=ξ€œβ„€π‘[𝑛+𝑦]π‘›π‘žπ›Όπ‘₯π‘‘πœ‡βˆ’π‘ž(𝑦)=[𝑑]π‘›π‘žπ›Ό[𝑑]βˆ’π‘žπ‘‘βˆ’1ξ“π‘Ž=0(βˆ’1)π‘Žπ‘žπ‘Žξ€œβ„€π‘ξ‚ƒπ‘₯+π‘Žπ‘‘+π‘¦ξ‚„π‘›π‘žπ›Όπ‘‘π‘‘πœ‡βˆ’π‘ž(𝑦).(2.10) By (1.5) and (2.10), we get a theorem.

Theorem 2.2. For π›Όβˆˆβ„• and π‘›βˆˆβ„€+, π‘‘βˆˆβ„• with 𝑑≑1(mod2), one has 𝐺(𝛼)𝑛+1,π‘ž(π‘₯)𝑛+1=[𝑑]π‘›π‘žπ›Ό[𝑑]βˆ’π‘žπ‘‘βˆ’1ξ“π‘Ž=0(βˆ’1)π‘Žπ‘žπ‘Žξ‚πΊ(𝛼)𝑛+1,π‘žπ‘‘ξ‚€π‘₯+π‘Žπ‘Žξ‚.(2.11)

3. Higher-Order π‘ž-Genocchi Numbers and Polynomials with Weight 𝛼

In this section, we define higher-order π‘ž-Genocchi polynomials 𝐺(𝛼)𝑛+1,π‘ž(β„Ž,π‘˜βˆ£π‘₯) and numbers 𝐺(𝛼)𝑛+1,π‘ž(β„Ž,π‘˜) with weight 𝛼. We find an integral equation for higher-order π‘ž-Genocchi numbers with weight 𝛼. And we establish a combination property.

Let π›Όβˆˆβ„€ and β„Ž,π‘˜βˆˆβ„€+, for π‘›βˆˆβ„€+, then we define higher-order π‘ž-Genocchi polynomials with weight 𝛼 as follows: 𝐺(𝛼)𝑛+1,π‘ž(β„Ž,π‘˜βˆ£π‘₯)(𝑛+1)=ξ€œπ‘π‘β‹―ξ€œπ‘π‘ξ„Ώξ…€ξ…€ξ…€ξ…€ξ…€ξ…ƒξ…€ξ…€ξ…€ξ…€ξ…€ξ…Œπ‘˜-timesξ€Ίπ‘₯1+π‘₯2+β‹―+π‘₯π‘˜+π‘₯ξ€»π‘›π‘žπ›Όπ‘ž(β„Žβˆ’1)π‘₯1+β‹―+(β„Žβˆ’π‘˜)π‘₯π‘˜π‘‘πœ‡βˆ’π‘žξ€·π‘₯1ξ€Έβ‹―π‘‘πœ‡βˆ’π‘žξ€·π‘₯π‘˜ξ€Έ=[2]π‘˜π‘ž[𝛼]π‘›π‘ž(1βˆ’π‘ž)𝑛𝑛𝑙=0βŽ›βŽœβŽœβŽπ‘›π‘™βŽžβŽŸβŽŸβŽ (βˆ’1)π‘™π‘žπ›Όπ‘™π‘₯ξ€·1+π‘žπ›Όπ‘™+β„Žξ€Έβ‹―ξ€·1+π‘žπ›Όπ‘™+β„Žβˆ’π‘˜+1ξ€Έ=[2]π‘˜π‘ž[𝛼]π‘›π‘ž(1βˆ’π‘ž)π‘šπ‘›ξ“π‘™=0βŽ›βŽœβŽœβŽπ‘›π‘™βŽžβŽŸβŽŸβŽ (βˆ’1)π‘™π‘žπ›Όπ‘™π‘₯ξ€·βˆ’π‘žπ›Όπ‘™+β„ŽβˆΆπ‘žβˆ’1ξ€Έπ‘˜,(3.1) where (π‘₯βˆΆπ‘ž)β„Ž=βˆβ„Žβˆ’1𝑖=0(1βˆ’π‘₯π‘žπ‘–).

In the special case, π‘₯=0, 𝐺(𝛼)𝑛+1,π‘ž(β„Ž,π‘˜|0)=𝐺(𝛼)𝑛+1,π‘ž(β„Ž,π‘˜) are called the (𝑛+1)th (β„Ž,π‘˜)-Genocchi numbers with weight 𝛼.

In (3.1), apply the following identity: ξ€Ίπ‘₯1+π‘₯2+β‹―+π‘₯π‘˜ξ€»π‘žπ›Ό(1βˆ’π‘žπ›Ό)+π‘žπ›Όξ€·π‘₯1+π‘₯2+β‹―+π‘₯π‘˜ξ€Έ=1,(3.2) and we have a theorem.

Theorem 3.1. For π›Όβˆˆβ„• and β„Ž,π‘˜βˆˆβ„€+, one has 𝐺(𝛼)𝑛+1,π‘ž(β„Ž,π‘˜)𝑛+1=(1βˆ’π‘žπ›Ό)𝐺(𝛼)𝑛+2,π‘ž(β„Ž,π‘˜)𝑛+2+𝐺(𝛼)𝑛+1,π‘ž(β„Ž+𝛼,π‘˜)𝑛+1.(3.3)

We consider, for π›Όβˆˆβ„• and β„Ž,π‘˜βˆˆβ„€+, 𝑖𝑗=0βŽ›βŽœβŽœβŽπ‘–π‘—βŽžβŽŸβŽŸβŽ (π‘žπ›Όβˆ’1)𝑗𝐺(𝛼)𝑛+π‘—βˆ’π‘–+1(β„Žβˆ’π›Ό,π‘˜)𝑛+π‘—βˆ’π‘–+1=𝑖𝑗=0βŽ›βŽœβŽœβŽπ‘–π‘—βŽžβŽŸβŽŸβŽ (π‘žπ›Όβˆ’1)π‘—ξ€œπ‘π‘β‹―ξ€œπ‘π‘ξ„Ώξ…€ξ…€ξ…€ξ…€ξ…€ξ…ƒξ…€ξ…€ξ…€ξ…€ξ…€ξ…Œπ‘˜-timesβŽ‘βŽ’βŽ£π‘˜ξ“β„“=1π‘₯β„“βŽ€βŽ₯βŽ¦π‘›βˆ’π‘–βˆ’π‘—π‘žπ›Όπ‘žβˆ‘π‘˜β„“=1(β„Žβˆ’π›Όβˆ’β„“)π‘₯β„“π‘‘πœ‡βˆ’π‘žξ€·π‘₯1ξ€Έβ‹―π‘‘πœ‡βˆ’π‘žξ€·π‘₯π‘˜ξ€Έ=𝑖𝑗=0βŽ›βŽœβŽœβŽπ‘–π‘—βŽžβŽŸβŽŸβŽ (π‘žπ›Όβˆ’1)π‘—ξ€œπ‘π‘β‹―ξ€œπ‘π‘ξ„Ώξ…€ξ…€ξ…€ξ…€ξ…€ξ…ƒξ…€ξ…€ξ…€ξ…€ξ…€ξ…Œπ‘˜-timesβŽ‘βŽ’βŽ£π‘˜ξ“β„“=1π‘₯β„“βŽ€βŽ₯βŽ¦π‘›βˆ’π‘–+π‘—π‘žπ›Όπ‘žβˆ‘π‘˜β„“=1(β„Žβˆ’β„“)π‘₯β„“π‘‘πœ‡βˆ’π‘žξ€·π‘₯1ξ€Έβ‹―π‘‘πœ‡βˆ’π‘žξ€·π‘₯π‘˜ξ€Έ=𝑖𝑗=0βŽ›βŽœβŽœβŽπ‘–π‘—βŽžβŽŸβŽŸβŽ (π‘žπ›Όβˆ’1)𝑗𝐺(𝛼)𝑛+π‘—βˆ’π‘–+1(β„Ž,π‘˜)𝑛+π‘—βˆ’π‘–+1.(3.4) Therefore, we obtain the following combinatorial property.

Theorem 3.2. For π›Όβˆˆβ„• and β„Ž,π‘˜βˆˆβ„€+, one has 𝑖𝑗=0βŽ›βŽœβŽœβŽπ‘–π‘—βŽžβŽŸβŽŸβŽ (π‘žπ›Όβˆ’1)𝑗𝐺(𝛼)𝑛+π‘—βˆ’π‘–+1(β„Žβˆ’π›Ό,π‘˜)𝑛+π‘—βˆ’π‘–+1=π‘–βˆ’1𝑗=0βŽ›βŽœβŽœβŽπ‘–βˆ’1π‘—βŽžβŽŸβŽŸβŽ (π‘žπ›Όβˆ’1)𝑗𝐺(𝛼)𝑛+π‘—βˆ’π‘–+1(β„Ž,π‘˜)𝑛+π‘—βˆ’π‘–+1.(3.5)

References

  1. A. Bayad and T. Kim, β€œIdentities involving values of Bernstein, q-Bernoulli, and q-Euler polynomials,” Russian Journal of Mathematical Physics, vol. 18, no. 2, pp. 133–143, 2011. View at: Publisher Site | Google Scholar
  2. L.-C. Jang, T. Kim, D.-H. Lee, and D.-W. Park, β€œAn application of polylogarithms in the analogs of Genocchi numbers,” Notes on Number Theory and Discrete Mathematics, vol. 7, no. 3, pp. 65–69, 2001. View at: Google Scholar
  3. T. Kim, β€œSome identities on the q-Euler polynomials of higher order and q-Stirling numbers by the fermionic p-adic integral on p,” Russian Journal of Mathematical Physics, vol. 16, no. 4, pp. 484–491, 2009. View at: Publisher Site | Google Scholar | MathSciNet
  4. T. Kim, β€œq-Volkenborn integration,” Russian Journal of Mathematical Physics, vol. 9, no. 3, pp. 288–299, 2002. View at: Google Scholar | Zentralblatt MATH
  5. T. Kim, β€œA note on p-adic q-integral on p associated with q-Euler numbers,” Advanced Studies in Contemporary Mathematics, vol. 15, no. 2, pp. 133–137, 2007. View at: Google Scholar
  6. T. Kim, β€œA note on q-Volkenborn integration,” Proceedings of the Jangjeon Mathematical Society, vol. 8, no. 1, pp. 13–17, 2005. View at: Google Scholar
  7. T. Kim, β€œOn the q-extension of Euler and Genocchi numbers,” Journal of Mathematical Analysis and Applications, vol. 326, no. 2, pp. 1458–1465, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  8. T. Kim, β€œA note on q-Bernstein polynomials,” Russian Journal of Mathematical Physics, vol. 18, no. 1, pp. 73–82, 2011. View at: Publisher Site | Google Scholar
  9. T. Kim, J. Choi, Y. H. Kim, and C. S. Ryoo, β€œOn the fermionic p-adic integral representation of Bernstein polynomials associated with Euler numbers and polynomials,” Journal of Inequalities and Applications, vol. 2010, Article ID 864247, 12 pages, 2010. View at: Publisher Site | Google Scholar
  10. T. Kim, J. Choi, Y. H. Kim, and C. S. Ryoo, β€œA note on the weighted p-adic q-Euler measure on p,” Advanced Studies in Contemporary Mathematics, vol. 12, pp. 35–40, 2011. View at: Google Scholar
  11. S.-H. Rim, K. H. Park, and E. J. Moon, β€œOn Genocchi numbers and polynomials,” Abstract and Applied Analysis, vol. 2008, Article ID 898471, 7 pages, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet

Copyright © 2011 Seog-Hoon Rim and Joohee Jeong. 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.


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