Table of Contents Author Guidelines Submit a Manuscript
Discrete Dynamics in Nature and Society
VolumeΒ 2011, Article IDΒ 176296, 11 pages
http://dx.doi.org/10.1155/2011/176296
Research Article

Some Relations between Twisted (β„Ž,π‘ž)-Euler Numbers with Weight Ξ± and π‘ž-Bernstein Polynomials with Weight Ξ±

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

Received 19 July 2011; Accepted 26 August 2011

Academic Editor: JohnΒ Rassias

Copyright Β© 2011 N. S. Jung 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

By using fermionic p-adic q-integral on ℀𝑝, we give some interesting relationship between the twisted (h, q)-Euler numbers with weight Ξ± and the q-Bernstein polynomials.

1. Introduction

Let 𝑝 be a fixed odd prime number. Throughout this paper, we always make use of the following notations: β„€ denotes the ring of rational integers, ℀𝑝 denotes the ring of 𝑝-adic rational integers, β„šπ‘ denotes the field of 𝑝-adic rational numbers, and ℂ𝑝 denotes the completion of algebraic closure of β„šπ‘, respectively. Let β„• be the set of natural numbers and β„€+=β„•βˆͺ{0}. Let 𝐢𝑝𝑛={π‘€βˆ£π‘€π‘π‘›=1} be the cyclic group of order 𝑝𝑛, and let 𝑇𝑝=limπ‘›β†’βˆžπΆπ‘π‘›=πΆπ‘βˆž=βˆͺ𝑛β‰₯0𝐢𝑝𝑛,(1.1) (see [1–22]), be the locally constant space. For π‘€βˆˆπ‘‡π‘, we denote by πœ™π‘€βˆΆβ„€π‘β†’β„‚π‘ the locally constant function π‘₯↦𝑀π‘₯. The 𝑝-adic absolute value is defined by |π‘₯|𝑝=1/π‘π‘Ÿ, where π‘₯=π‘π‘Ÿπ‘ /𝑑(π‘Ÿβˆˆβ„šand𝑠,π‘‘βˆˆβ„€with(𝑠,𝑑)=(𝑝,𝑠)=(𝑝,𝑑)=1). In this paper, we assume that π‘žβˆˆβ„‚π‘ with |π‘žβˆ’1|𝑝<1 as an indeterminate. The π‘ž-number is defined by [π‘₯]π‘ž=1βˆ’π‘žπ‘₯1βˆ’π‘ž,(1.2) (see [1–22]). Note that limπ‘žβ†’1[π‘₯]π‘ž=π‘₯. For π‘“βˆˆπ‘ˆπ·ξ€·β„€π‘ξ€Έ=ξ€½π‘“βˆ£π‘“βˆΆβ„€π‘βŸΆβ„‚π‘isuniformlydifferentiablefunctionξ€Ύ,(1.3) the fermionic 𝑝-adic π‘ž-integral on ℀𝑝 is defined by Kim as follows: πΌβˆ’π‘ž(𝑓)=ξ€œβ„€π‘π‘“(π‘₯)π‘‘πœ‡βˆ’π‘ž(π‘₯)=limπ‘β†’βˆž1+π‘ž1+π‘žπ‘π‘π‘π‘βˆ’1π‘₯=0𝑓(π‘₯)(βˆ’π‘ž)π‘₯,(1.4) (see [1–7]). From (1.4), we note that π‘žπ‘›πΌβˆ’π‘žξ€·π‘“π‘›ξ€Έ=(βˆ’1)π‘›πΌβˆ’π‘ž(𝑓)+[2]π‘žπ‘›βˆ’1𝑙=0(βˆ’1)π‘›βˆ’1βˆ’π‘™π‘žπ‘™π‘“(𝑙),(1.5) where 𝑓𝑛(π‘₯)=𝑓(π‘₯+𝑛) for π‘›βˆˆβ„•.

For π‘˜,π‘›βˆˆβ„€+ and π‘₯∈[0,1], Kim defined π‘ž-Bernstein polynomials, which are different π‘ž-Bernstein polynomials of Phillips, as follows: π΅π‘˜,𝑛(π‘₯,π‘ž)=βŽ›βŽœβŽœβŽπ‘›π‘˜βŽžβŽŸβŽŸβŽ [π‘₯]π‘˜π‘ž[1βˆ’π‘₯]π‘›βˆ’π‘˜π‘žβˆ’1,(1.6) (see [5]). In [9], the 𝑝-adic extension of (1.6) is given by π΅π‘˜,𝑛(π‘₯,π‘ž)=βŽ›βŽœβŽœβŽπ‘›π‘˜βŽžβŽŸβŽŸβŽ [π‘₯]π‘˜π‘ž[1βˆ’π‘₯]π‘›βˆ’π‘˜π‘žβˆ’1,whereπ‘₯βˆˆβ„€π‘,𝑛,π‘˜βˆˆβ„€+.(1.7) For π›Όβˆˆβ„€, β„Žβˆˆβ„€, π‘€βˆˆπ‘‡π‘, and π‘žβˆˆβ„‚π‘ with |1βˆ’π‘ž|𝑝≀1, twisted (β„Ž,π‘ž)-Euler numbers 𝐸(β„Ž,𝛼)𝑛,π‘ž,𝑀 with weight 𝛼 are defined by 𝐸(β„Ž,𝛼)𝑛,π‘ž,𝑀=ξ€œβ„€π‘πœ™π‘€(π‘₯)π‘žπ‘₯(β„Žβˆ’1)[π‘₯]π‘›π‘žπ›Όπ‘‘πœ‡βˆ’π‘ž(π‘₯).(1.8) In the special case, π‘₯=0, 𝐸(β„Ž,𝛼)𝑛,π‘ž,𝑀(0)=𝐸(β„Ž,𝛼)𝑛,π‘ž,𝑀 are called the 𝑛-th twisted (β„Ž,π‘ž)-Euler numbers with weight 𝛼.

In this paper, we investigate some relations between the π‘ž-Bernstein polynomials and the twisted (β„Ž,π‘ž)-Euler numbers with weight 𝛼. From these relations, we derive some interesting identities on the twisted (β„Ž,π‘ž)-Euler numbers and polynomials with weight 𝛼.

2. Twisted (β„Ž,π‘ž)-Euler Numbers and Polynomials with Weight 𝛼

By using 𝑝-adic π‘ž-integral and (1.8), we obtain ξ€œβ„€π‘πœ™π‘€(π‘₯)π‘žπ‘₯(β„Žβˆ’1)[π‘₯]π‘›π‘žπ›Όπ‘‘πœ‡βˆ’π‘ž(π‘₯)=limπ‘β†’βˆž1ξ€Ίπ‘π‘ξ€»βˆ’π‘žπ‘π‘βˆ’1π‘₯=0[π‘₯]π‘›π‘žπ›Όπ‘€π‘₯π‘žπ‘₯(β„Žβˆ’1)(βˆ’π‘ž)π‘₯=[2]π‘žξ‚΅11βˆ’π‘žπ›Όξ‚Άπ‘›π‘›ξ“π‘™=0βŽ›βŽœβŽœβŽπ‘›π‘™βŽžβŽŸβŽŸβŽ (βˆ’1)𝑙11+π‘€π‘žπ›Όπ‘™+β„Ž.(2.1) We set 𝐹(β„Ž,𝛼)π‘ž,𝑀(𝑑)=βˆžξ“π‘›=0𝐸(β„Ž,𝛼)𝑛,π‘ž,𝑀𝑑𝑛𝑛!.(2.2) By (2.1) and (2.2), we have 𝐹(β„Ž,𝛼)π‘ž,𝑀(𝑑)=βˆžξ“π‘›=0𝐸(β„Ž,𝛼)𝑛,π‘ž,𝑀𝑑𝑛𝑛!=[2]π‘žβˆžξ“π‘›=0βŽ›βŽœβŽœβŽξ‚΅11βˆ’π‘žπ›Όξ‚Άπ‘›π‘›ξ“π‘™=0βŽ›βŽœβŽœβŽπ‘›π‘™βŽžβŽŸβŽŸβŽ (βˆ’1)𝑙11+π‘€π‘žπ›Όπ‘™+β„ŽβŽžβŽŸβŽŸβŽ π‘‘π‘›π‘›!=[2]π‘žβˆžξ“π‘š=0(βˆ’1)π‘šπ‘€π‘šπ‘žβ„Žπ‘šπ‘’[π‘š]π‘žπ›Όπ‘‘.(2.3) Since [π‘₯+𝑦]π‘žπ›Ό=[π‘₯]π‘žπ›Ό+π‘žπ›Όπ‘₯[𝑦]π‘žπ›Ό, we obtain 𝐸(β„Ž,𝛼)𝑛,π‘ž,𝑀(π‘₯)=ξ€œβ„€π‘πœ™π‘€(𝑦)π‘žπ‘¦(β„Žβˆ’1)[𝑦+π‘₯]π‘žπ›Όπ‘‘π‘‘πœ‡βˆ’π‘ž(𝑦)=𝑛𝑙=0βŽ›βŽœβŽœβŽπ‘›π‘™βŽžβŽŸβŽŸβŽ π‘žπ›Όπ‘₯𝑙[π‘₯]π‘›βˆ’π‘™π‘žπ›Όξ€œβ„€π‘πœ™π‘€(𝑦)π‘žπ‘¦(β„Žβˆ’1)[𝑦]π‘™π‘žπ›Όπ‘‘πœ‡βˆ’π‘ž(𝑦)=𝑛𝑙=0βŽ›βŽœβŽœβŽπ‘›π‘™βŽžβŽŸβŽŸβŽ π‘žπ›Όπ‘₯𝑙[π‘₯]π‘›βˆ’π‘™π‘žπ›ΌπΈ(β„Ž,𝛼)𝑙,π‘ž,𝑀.(2.4)

Therefore, we obtain the following theorem.

Theorem 2.1. For π‘›βˆˆβ„€+ and π‘€βˆˆπ‘‡π‘, we have 𝐸(β„Ž,𝛼)𝑛,π‘ž,𝑀(π‘₯)=[2]π‘žβˆžξ“π‘š=0(βˆ’1)π‘šπ‘€π‘šπ‘žβ„Žπ‘š[π‘₯+π‘š]π‘›π‘žπ›Ό.(2.5) Furthermore, 𝐸(β„Ž,𝛼)𝑛,π‘ž,𝑀(π‘₯)=𝑛𝑙=0βŽ›βŽœβŽœβŽπ‘›π‘™βŽžβŽŸβŽŸβŽ π‘žπ›Όπ‘₯𝑙[π‘₯]π‘›βˆ’π‘™π‘žπ›ΌπΈ(β„Ž,𝛼)𝑙,π‘ž,𝑀=ξ‚€[π‘₯]π‘žπ›Ό+π‘žπ›Όπ‘₯𝐸(β„Ž,𝛼)π‘ž,𝑀𝑛,(2.6) with usual convention about replacing (𝐸(β„Ž,𝛼)π‘ž,𝑀)𝑛 with 𝐸(β„Ž,𝛼)𝑛,π‘ž,𝑀.
Let 𝐹(β„Ž,𝛼)π‘ž,𝑀(𝑑,π‘₯)=βˆ‘βˆžπ‘›=0𝐸(β„Ž,𝛼)𝑛,π‘ž,𝑀(π‘₯)𝑑𝑛/𝑛!. Then we see that 𝐹(β„Ž,𝛼)π‘ž,𝑀(𝑑,π‘₯)=[2]π‘žβˆžξ“π‘š=0(βˆ’1)π‘šπ‘€π‘šπ‘žπ‘šβ„Žπ‘’[π‘₯+π‘š]π‘žπ›Όπ‘‘.(2.7) In the special case, π‘₯=0, let 𝐹(β„Ž,𝛼)π‘ž,𝑀(𝑑,0)=𝐹(β„Ž,𝛼)π‘ž,𝑀(𝑑).
By (2.1), we get 𝐸(β„Ž,𝛼)𝑛,π‘žβˆ’1,π‘€βˆ’1(1βˆ’π‘₯)=(βˆ’1)π‘›π‘€π‘žπ›Όπ‘›+β„Žβˆ’1𝐸(β„Ž,𝛼)𝑛,π‘ž,𝑀(π‘₯).(2.8) From (2.3) and (2.7), we note that π‘€π‘žβ„ŽπΉ(β„Ž,𝛼)π‘ž,𝑀(𝑑,1)+𝐹(β„Ž,𝛼)π‘ž,𝑀(𝑑)=[2]π‘ž.(2.9) By (2.9), we get the following recurrence formula: 𝐸(β„Ž,𝛼)0,π‘ž,𝑀=[2]π‘ž1+π‘žβ„Žπ‘€,π‘žβ„Žπ‘€πΈ(β„Ž,𝛼)𝑛,π‘ž,𝑀(1)+𝐸(β„Ž,𝛼)𝑛,π‘ž,𝑀=0if𝑛>0.(2.10)

By (2.10) and Theorem 2.1, we obtain the following theorem.

Theorem 2.2. For π‘›βˆˆβ„€+ and π‘€βˆˆπ‘‡π‘, we have 𝐸(β„Ž,𝛼)0,π‘ž,𝑀=[2]π‘ž1+π‘žβ„Žπ‘€,π‘žβ„Žπ‘€ξ‚€π‘žπ›ΌπΈ(β„Ž,𝛼)π‘ž,𝑀+1𝑛+𝐸(β„Ž,𝛼)𝑛,π‘ž,𝑀=0if𝑛>0,(2.11) with usual convention about replacing (𝐸(β„Ž,𝛼)π‘ž,𝑀)𝑛 with 𝐸(β„Ž,𝛼)𝑛,π‘ž,𝑀.
By (2.4), Theorem 2.1, and Theorem 2.2, we have π‘ž2β„Žπ‘€2𝐸(β„Ž,𝛼)𝑛,π‘ž,𝑀(2)βˆ’[2]π‘ž1+π‘žβ„Žπ‘€π‘ž2β„Žπ‘€2βˆ’[2]π‘ž1+π‘žβ„Žπ‘€π‘žβ„Žπ‘€=π‘ž2β„Žπ‘€2𝑛𝑙=0βŽ›βŽœβŽœβŽπ‘›π‘™βŽžβŽŸβŽŸβŽ π‘žπ›Όπ‘™ξ‚€π‘žπ›ΌπΈ(β„Ž,𝛼)π‘ž,𝑀+1ξ‚π‘™βˆ’[2]π‘ž1+π‘žβ„Žπ‘€π‘ž2β„Žπ‘€2βˆ’[2]π‘ž1+π‘žβ„Žπ‘€π‘žβ„Žπ‘€=π‘ž2β„Žπ‘€2𝑛𝑙=1βŽ›βŽœβŽœβŽπ‘›π‘™βŽžβŽŸβŽŸβŽ π‘žπ›Όπ‘™ξ‚€π‘žπ›ΌπΈ(β„Ž,𝛼)π‘ž,𝑀+1ξ‚π‘™βˆ’[2]π‘ž1+π‘žβ„Žπ‘€π‘žβ„Žπ‘€=βˆ’π‘žβ„Žπ‘€π‘›ξ“π‘™=1βŽ›βŽœβŽœβŽπ‘›π‘™βŽžβŽŸβŽŸβŽ π‘žπ›Όπ‘™πΈ(β„Ž,𝛼)𝑙,π‘ž,π‘€βˆ’[2]π‘ž1+π‘žβ„Žπ‘€π‘žβ„Žπ‘€=βˆ’π‘žβ„Žπ‘€π‘›ξ“π‘™=0βŽ›βŽœβŽœβŽπ‘›π‘™βŽžβŽŸβŽŸβŽ π‘žπ›Όπ‘™πΈ(β„Ž,𝛼)𝑙,π‘ž,𝑀=βˆ’π‘žβ„Žπ‘€πΈ(β„Ž,𝛼)𝑛,π‘ž,𝑀(1)=𝐸(β„Ž,𝛼)𝑛,π‘ž,𝑀if𝑛>0.(2.12)

Therefore, we obtain the following theorem.

Theorem 2.3. For π‘›βˆˆβ„•, we have 𝐸(β„Ž,𝛼)𝑛,π‘ž,𝑀(2)=ξ‚΅1π‘ž2β„Žπ‘€2𝐸(β„Ž,𝛼)𝑛,π‘ž,𝑀+[2]π‘ž1+π‘žβ„Žπ‘€+ξ‚΅1π‘žβ„Žπ‘€ξ‚Ά[2]π‘ž1+π‘žβ„Žπ‘€.(2.13)
By (2.8), we see that π‘žβ„Žβˆ’1π‘€ξ€œβ„€π‘[1βˆ’π‘₯]π‘›π‘žβˆ’π›Όπ‘ž(β„Žβˆ’1)π‘₯𝑀π‘₯π‘‘πœ‡βˆ’π‘ž(π‘₯)=(βˆ’1)π‘›π‘žπ›Όπ‘›+β„Žβˆ’1π‘€ξ€œβ„€π‘[π‘₯βˆ’1]π‘›π‘žπ›Όπ‘ž(β„Žβˆ’1)π‘₯𝑀π‘₯π‘‘πœ‡βˆ’π‘ž(π‘₯)=(βˆ’1)π‘›π‘žπ›Όπ‘›+β„Žβˆ’1𝑀𝐸(β„Ž,𝛼)𝑛,π‘ž,𝑀(βˆ’1)=𝐸(β„Ž,𝛼)𝑛,π‘žβˆ’1,π‘€βˆ’1(2).(2.14)

Therefore, we obtain the following theorem.

Theorem 2.4. For π‘›βˆˆβ„€+, we have π‘žβ„Žβˆ’1π‘€ξ€œβ„€π‘[1βˆ’π‘₯]π‘›π‘žβˆ’π›Όπ‘ž(β„Žβˆ’1)π‘₯𝑀π‘₯π‘‘πœ‡βˆ’π‘ž(π‘₯)=𝐸(β„Ž,𝛼)𝑛,π‘žβˆ’1,π‘€βˆ’1(2).(2.15) Let π‘›βˆˆβ„•. By Theorems 2.3 and 2.4, we get π‘žβ„Žβˆ’1π‘€ξ€œβ„€π‘[1βˆ’π‘₯]π‘›π‘žβˆ’π›Όπ‘ž(β„Žβˆ’1)π‘₯𝑀π‘₯π‘‘πœ‡βˆ’π‘ž(π‘₯)=π‘ž2β„Žπ‘€2𝐸(β„Ž,𝛼)𝑛,π‘žβˆ’1,π‘€βˆ’1+π‘žβ„Žβˆ’1𝑀[2]π‘ž1+π‘žβ„Žπ‘€ξ‚Ά+π‘ž2β„Žβˆ’1𝑀2ξ‚΅[2]π‘ž1+π‘žβ„Žπ‘€ξ‚Ά.(2.16)
From (2.16), we have ξ€œβ„€π‘[1βˆ’π‘₯]π‘›π‘žβˆ’1π‘ž(β„Žβˆ’1)π‘₯𝑀π‘₯π‘‘πœ‡βˆ’π‘ž(π‘₯)=π‘žβ„Ž+1𝑀𝐸(β„Ž,𝛼)𝑛,π‘žβˆ’1,π‘€βˆ’1+ξ‚΅[2]π‘ž1+π‘žβ„Žπ‘€ξ‚Ά+π‘žβ„Žπ‘€ξ‚΅[2]π‘ž1+π‘žβ„Žπ‘€ξ‚Ά.(2.17)

Therefore, we obtain the following corollary.

Corollary 2.5. For π‘›βˆˆβ„•, we have ξ€œβ„€π‘[1βˆ’π‘₯]π‘›π‘žβˆ’π›Όπ‘ž(β„Žβˆ’1)π‘₯𝑀π‘₯π‘‘πœ‡βˆ’π‘ž(π‘₯)=π‘žβ„Ž+1𝑀𝐸(β„Ž,𝛼)𝑛,π‘žβˆ’1,π‘€βˆ’1+[2]π‘ž.(2.18)
Kim defined the π‘ž-Bernstein polynomials with weight 𝛼 of degree 𝑛 as below.
For π‘₯βˆˆβ„€π‘, the 𝑝-adic π‘ž-Bernstein polynomials with weight 𝛼 of degree 𝑛 are given by 𝐡(𝛼)π‘˜,𝑛(π‘₯,π‘ž)=βŽ›βŽœβŽœβŽπ‘›π‘˜βŽžβŽŸβŽŸβŽ [π‘₯]π‘˜π‘žπ›Ό[1βˆ’π‘₯]π‘›βˆ’π‘˜π‘žβˆ’π›Ό,where𝑛,π‘˜βˆˆβ„€+.(2.19) compare [5, 10, 22] By (2.19), we get the symmetry of π‘ž-Bernstein polynomials as follows: 𝐡(𝛼)π‘˜,𝑛(π‘₯,π‘ž)=𝐡(𝛼)π‘›βˆ’π‘˜,𝑛1βˆ’π‘₯,π‘žβˆ’1ξ€Έ,(2.20) see [8]. Thus, by Corollary 2.5, (2.19), and (2.20), we see that ξ€œβ„€π‘π΅(𝛼)π‘˜,𝑛(π‘₯,π‘ž)π‘ž(β„Žβˆ’1)π‘₯𝑀π‘₯π‘‘πœ‡βˆ’π‘ž(π‘₯)=ξ€œβ„€π‘π΅(𝛼)π‘›βˆ’π‘˜,𝑛1βˆ’π‘₯,π‘žβˆ’1ξ€Έπ‘ž(β„Žβˆ’1)π‘₯𝑀π‘₯π‘‘πœ‡βˆ’π‘ž(π‘₯)=βŽ›βŽœβŽœβŽπ‘›π‘˜βŽžβŽŸβŽŸβŽ π‘˜ξ“π‘™=0βŽ›βŽœβŽœβŽπ‘˜π‘™βŽžβŽŸβŽŸβŽ (βˆ’1)π‘˜+π‘™ξ€œβ„€π‘[1βˆ’π‘₯]π‘›βˆ’π‘™π‘žβˆ’π›Όπ‘ž(β„Žβˆ’1)π‘₯𝑀π‘₯π‘‘πœ‡βˆ’π‘ž(π‘₯)=βŽ›βŽœβŽœβŽπ‘›π‘˜βŽžβŽŸβŽŸβŽ π‘˜ξ“π‘™=0βŽ›βŽœβŽœβŽπ‘˜π‘™βŽžβŽŸβŽŸβŽ (βˆ’1)π‘˜+π‘™ξ‚€π‘žβ„Ž+1𝑀𝐸(β„Ž,𝛼)π‘›βˆ’π‘™,π‘žβˆ’1,π‘€βˆ’1+[2]π‘žξ‚.(2.21)
For 𝑛,π‘˜βˆˆβ„€+ with 𝑛>π‘˜, we have ξ€œβ„€π‘π΅(𝛼)π‘˜,𝑛(π‘₯,π‘ž)π‘ž(β„Žβˆ’1)π‘₯𝑀π‘₯π‘‘πœ‡βˆ’π‘ž(π‘₯)=βŽ›βŽœβŽœβŽπ‘›π‘˜βŽžβŽŸβŽŸβŽ π‘˜ξ“π‘™=0βŽ›βŽœβŽœβŽπ‘˜π‘™βŽžβŽŸβŽŸβŽ (βˆ’1)π‘˜+π‘™ξ‚€π‘žβ„Ž+1𝑀𝐸(β„Ž,𝛼)π‘›βˆ’π‘™,π‘žβˆ’1,π‘€βˆ’1+[2]π‘žξ‚=⎧βŽͺ⎨βŽͺβŽ©π‘žβ„Ž+1𝑀𝐸(β„Ž,𝛼)𝑛,π‘žβˆ’1,π‘€βˆ’1+[2]π‘žifπ‘˜=0,π‘žβ„Ž+1π‘€ξ‚΅π‘›π‘˜ξ‚Άπ‘˜ξ“π‘™=0ξ‚΅π‘˜π‘™ξ‚Ά(βˆ’1)π‘˜+𝑙𝐸(β„Ž,𝛼)π‘›βˆ’π‘™,π‘žβˆ’1,π‘€βˆ’1ifπ‘˜>0.(2.22) Let us take the fermionic π‘ž-integral on ℀𝑝 for the π‘ž-Bernstein polynomials with weight 𝛼 of degree 𝑛 as follows: ξ€œβ„€π‘π΅(𝛼)π‘˜,𝑛(π‘₯,π‘ž)π‘ž(β„Žβˆ’1)π‘₯𝑀π‘₯π‘‘πœ‡βˆ’π‘ž(π‘₯)=βŽ›βŽœβŽœβŽπ‘›π‘˜βŽžβŽŸβŽŸβŽ ξ€œβ„€π‘[π‘₯]π‘˜π‘ž[1βˆ’π‘₯]π‘›βˆ’π‘˜π‘žβˆ’π›Όπ‘ž(β„Žβˆ’1)π‘₯𝑀π‘₯π‘‘πœ‡βˆ’π‘ž(π‘₯)=βŽ›βŽœβŽœβŽπ‘›π‘˜βŽžβŽŸβŽŸβŽ π‘›βˆ’π‘˜ξ“π‘™=0βŽ›βŽœβŽœβŽπ‘›βˆ’π‘˜π‘™βŽžβŽŸβŽŸβŽ (βˆ’1)𝑙𝐸(β„Ž,𝛼)𝑙+π‘˜,π‘ž,𝑀.(2.23)

Therefore, by (2.22) and (2.23), we obtain the following theorem.

Theorem 2.6. Let 𝑛,π‘˜βˆˆβ„€+ with 𝑛>π‘˜. Then we have ξ€œβ„€π‘π΅(𝛼)π‘˜,𝑛(π‘₯,π‘ž)π‘ž(β„Žβˆ’1)π‘₯𝑀π‘₯π‘‘πœ‡βˆ’π‘ž(π‘₯)=⎧βŽͺ⎨βŽͺβŽ©π‘žβ„Ž+1𝑀𝐸(β„Ž,𝛼)𝑛,π‘žβˆ’1,π‘€βˆ’1+[2]π‘žifπ‘˜=0,π‘žβ„Ž+1π‘€ξ‚΅π‘›π‘˜ξ‚Άπ‘˜ξ“π‘™=0ξ‚΅π‘˜π‘™ξ‚Ά(βˆ’1)π‘˜+𝑙𝐸(β„Ž,𝛼)π‘›βˆ’π‘™,π‘žβˆ’1,π‘€βˆ’1ifπ‘˜>0.(2.24) Moreover, π‘›βˆ’π‘˜ξ“π‘™=0βŽ›βŽœβŽœβŽπ‘›βˆ’π‘˜π‘™βŽžβŽŸβŽŸβŽ (βˆ’1)𝑙E(β„Ž,𝛼)𝑙+π‘˜,π‘ž,𝑀=⎧βŽͺ⎨βŽͺβŽ©π‘žβ„Ž+1𝑀𝐸(β„Ž,𝛼)𝑛,π‘žβˆ’1,π‘€βˆ’1+[2]π‘žifπ‘˜=0,π‘žβ„Ž+1π‘€π‘˜ξ“π‘™=0ξ‚΅π‘˜π‘™ξ‚Ά(βˆ’1)π‘˜+𝑙𝐸(β„Ž,𝛼)π‘›βˆ’π‘™,π‘žβˆ’1,π‘€βˆ’1ifπ‘˜>0.(2.25)
Let 𝑛1,𝑛2,π‘˜βˆˆβ„€+ with 𝑛1+𝑛2>2π‘˜. Then we get ξ€œβ„€π‘π΅(𝛼)π‘˜,𝑛1(π‘₯,π‘ž)𝐡(𝛼)π‘˜,𝑛2(π‘₯,π‘ž)π‘ž(β„Žβˆ’1)π‘₯𝑀π‘₯π‘‘πœ‡βˆ’π‘ž(π‘₯)=βŽ›βŽœβŽœβŽπ‘›1π‘˜βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘›2π‘˜βŽžβŽŸβŽŸβŽ 2π‘˜ξ“π‘™=0βŽ›βŽœβŽœβŽ2π‘˜π‘™βŽžβŽŸβŽŸβŽ (βˆ’1)𝑙+2π‘˜ξ€œβ„€π‘[1βˆ’π‘₯]𝑛1+𝑛2βˆ’π‘™π‘žβˆ’π›Όπ‘ž(β„Žβˆ’1)π‘₯𝑀π‘₯π‘‘πœ‡βˆ’π‘ž(π‘₯)=βŽ›βŽœβŽœβŽπ‘›1π‘˜βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘›2π‘˜βŽžβŽŸβŽŸβŽ 2π‘˜ξ“π‘™=0βŽ›βŽœβŽœβŽ2π‘˜π‘™βŽžβŽŸβŽŸβŽ (βˆ’1)𝑙+2π‘˜ξ‚€π‘žβ„Ž+1𝑀𝐸(β„Ž,𝛼)𝑛1+𝑛2βˆ’π‘™,π‘žβˆ’1,π‘€βˆ’1+[2]π‘žξ‚=⎧βŽͺ⎨βŽͺβŽ©π‘žβ„Ž+1𝑀𝐸(β„Ž,𝛼)𝑛1+𝑛2,π‘žβˆ’1,π‘€βˆ’1+[2]π‘žifπ‘˜=0,𝑛1π‘˜ξ‚Άξ‚΅π‘›2π‘˜ξ‚Ά2π‘˜ξ“π‘™=0ξ‚΅2π‘˜π‘™ξ‚Ά(βˆ’1)2π‘˜+π‘™ξ‚€π‘žβ„Ž+1𝑀𝐸(β„Ž,𝛼)𝑛1+𝑛2βˆ’π‘™,π‘žβˆ’1,π‘€βˆ’1+[2]π‘žξ‚ifπ‘˜β‰ 0.(2.26)

Therefore, by (2.26), we obtain the following theorem.

Theorem 2.7. For 𝑛1,𝑛2,π‘˜βˆˆβ„€+ with 𝑛1+𝑛2>2π‘˜, we have ξ€œβ„€π‘π΅(𝛼)π‘˜,𝑛1(π‘₯,π‘ž)𝐡(𝛼)π‘˜,𝑛2(π‘₯,π‘ž)π‘ž(β„Žβˆ’1)π‘₯𝑀π‘₯π‘‘πœ‡βˆ’π‘ž(π‘₯)=⎧βŽͺ⎨βŽͺβŽ©π‘žβ„Ž+1𝑀𝐸(β„Ž,𝛼)𝑛1+𝑛2,π‘žβˆ’1,π‘€βˆ’1+[2]π‘žifπ‘˜=0,π‘žβ„Ž+1𝑀𝑛1π‘˜ξ‚Άξ‚΅π‘›2π‘˜ξ‚Ά2π‘˜ξ“π‘™=0ξ‚΅2π‘˜π‘™ξ‚Ά(βˆ’1)2π‘˜+𝑙𝐸(β„Ž,𝛼)𝑛1+𝑛2βˆ’π‘™,π‘žβˆ’1,π‘€βˆ’1ifπ‘˜β‰ 0.(2.27)
From the binomial theorem, we can derive the following equation: ξ€œβ„€π‘π΅(𝛼)π‘˜,𝑛1(π‘₯,π‘ž)𝐡(𝛼)π‘˜,𝑛2(π‘₯,π‘ž)π‘ž(β„Žβˆ’1)π‘₯𝑀π‘₯π‘‘πœ‡βˆ’π‘ž(π‘₯)=βŽ›βŽœβŽœβŽπ‘›1π‘˜βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘›2π‘˜βŽžβŽŸβŽŸβŽ π‘›1+𝑛2βˆ’2π‘˜ξ“π‘™=0(βˆ’1)π‘™βŽ›βŽœβŽœβŽπ‘›1+𝑛2βˆ’2π‘˜π‘™βŽžβŽŸβŽŸβŽ ξ€œβ„€π‘[π‘₯]2π‘˜+π‘™π‘žπ‘ž(β„Žβˆ’1)π‘₯𝑀π‘₯π‘‘πœ‡βˆ’π‘ž(π‘₯)=βŽ›βŽœβŽœβŽπ‘›1π‘˜βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘›2π‘˜βŽžβŽŸβŽŸβŽ π‘›1+𝑛2βˆ’2π‘˜ξ“π‘™=0(βˆ’1)π‘™βŽ›βŽœβŽœβŽπ‘›1+𝑛2βˆ’2π‘˜π‘™βŽžβŽŸβŽŸβŽ πΈ(β„Ž,𝛼)2π‘˜+𝑙,π‘ž,𝑀.(2.28)

Thus, by (2.28) and Theorem 2.7, we obtain the following corollary.

Corollary 2.8. Let 𝑛1,𝑛2,π‘˜βˆˆβ„€+ with 𝑛1+𝑛2>2π‘˜. Then we have 𝑛1+𝑛2βˆ’2π‘˜ξ“π‘™=0(βˆ’1)π‘™βŽ›βŽœβŽœβŽπ‘›1+𝑛2βˆ’2π‘˜π‘™βŽžβŽŸβŽŸβŽ πΈ(β„Ž,𝛼)2π‘˜+𝑙,π‘ž=⎧βŽͺ⎨βŽͺβŽ©π‘žβ„Ž+1𝑀𝐸(β„Ž,𝛼)𝑛1+𝑛2,π‘žβˆ’π›Ό,π‘€βˆ’1+[2]π‘žifπ‘˜=0,π‘žβ„Ž+1𝑀2π‘˜ξ“π‘™=0ξ‚΅2π‘˜π‘™ξ‚Ά(βˆ’1)2π‘˜+𝑙𝐸(β„Ž,𝛼)𝑛1+𝑛2βˆ’π‘™,π‘žβˆ’π›Ό,π‘€βˆ’1ifπ‘˜>0.(2.29)
For π‘₯βˆˆβ„€π‘ and π‘ βˆˆβ„• with 𝑠β‰₯2, let 𝑛1,𝑛2,…,𝑛𝑠,π‘˜βˆˆβ„€+ with 𝑛1+β‹―+𝑛𝑠>π‘ π‘˜. Then we take the fermionic 𝑝-adic π‘ž-integral on ℀𝑝 for the π‘ž-Bernstein polynomials with weight 𝛼 of degree 𝑛 as follows: ξ€œβ„€π‘π΅(𝛼)π‘˜,𝑛1(π‘₯,π‘ž)⋯𝐡(𝛼)π‘˜,𝑛𝑠(π‘₯,π‘ž)ξ„Ώξ…€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…Œπ‘ -timesπ‘ž(β„Žβˆ’1)π‘₯𝑀π‘₯π‘‘πœ‡βˆ’π‘ž(π‘₯)=βŽ›βŽœβŽœβŽπ‘›1π‘˜βŽžβŽŸβŽŸβŽ β‹―βŽ›βŽœβŽœβŽπ‘›π‘ π‘˜βŽžβŽŸβŽŸβŽ ξ€œβ„€π‘[π‘₯]π‘ π‘˜π‘ž[1βˆ’π‘₯]𝑛1+β‹―+π‘›π‘ βˆ’π‘ π‘˜π‘žβˆ’π›Όπ‘ž(β„Žβˆ’1)π‘₯𝑀π‘₯π‘‘πœ‡βˆ’π‘ž(π‘₯)=βŽ›βŽœβŽœβŽπ‘›1π‘˜βŽžβŽŸβŽŸβŽ β‹―βŽ›βŽœβŽœβŽπ‘›sπ‘˜βŽžβŽŸβŽŸβŽ π‘ π‘˜ξ“π‘™=0βŽ›βŽœβŽœβŽπ‘ π‘˜π‘™βŽžβŽŸβŽŸβŽ (βˆ’1)𝑙+π‘ π‘˜ξ€œβ„€π‘[1βˆ’π‘₯]𝑛1+β‹―+π‘›π‘ βˆ’π‘™π‘žβˆ’π›Όπ‘ž(β„Žβˆ’1)π‘₯𝑀π‘₯π‘‘πœ‡βˆ’π‘ž(π‘₯)=βŽ›βŽœβŽœβŽπ‘›1π‘˜βŽžβŽŸβŽŸβŽ β‹―βŽ›βŽœβŽœβŽπ‘›π‘ π‘˜βŽžβŽŸβŽŸβŽ π‘ π‘˜ξ“π‘™=0βŽ›βŽœβŽœβŽπ‘ π‘˜π‘™βŽžβŽŸβŽŸβŽ (βˆ’1)𝑙+π‘ π‘˜ξ‚€π‘žβ„Ž+1𝑀𝐸(β„Ž,𝛼)𝑛1+β‹―+π‘›π‘ βˆ’π‘™,π‘žβˆ’1,π‘€βˆ’1+[2]π‘žξ‚=⎧βŽͺ⎨βŽͺβŽ©π‘žβ„Ž+1𝑀𝐸(β„Ž,𝛼)𝑛1+β‹―+𝑛𝑠,π‘žβˆ’1,π‘€βˆ’1+[2]π‘žifπ‘˜=0,π‘žβ„Ž+1𝑀𝑛1π‘˜ξ‚Άβ‹―ξ‚΅π‘›π‘ π‘˜ξ‚Άπ‘ π‘˜ξ“π‘™=0ξ‚΅π‘ π‘˜π‘™ξ‚Ά(βˆ’1)𝑙+π‘ π‘˜πΈ(β„Ž,𝛼)𝑛1+β‹―+π‘›π‘ βˆ’π‘™,π‘žβˆ’1,π‘€βˆ’1ifπ‘˜>0.(2.30)

Therefore, by (2.30), we obtain the following theorem.

Theorem 2.9. For π‘ βˆˆβ„• with 𝑠β‰₯2, let 𝑛1,𝑛2,…,𝑛𝑠,π‘˜βˆˆβ„€+ with 𝑛1+β‹―+𝑛𝑠>π‘ π‘˜. Then we get ξ€œβ„€π‘π΅(𝛼)π‘˜,𝑛1(π‘₯,π‘ž)⋯𝐡(𝛼)π‘˜,𝑛𝑠(π‘₯,π‘ž)ξ„Ώξ…€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…Œπ‘ -timesπ‘ž(β„Žβˆ’1)π‘₯𝑀π‘₯π‘‘πœ‡βˆ’π‘ž(π‘₯)=⎧βŽͺ⎨βŽͺβŽ©π‘žβ„Ž+1𝑀𝐸(β„Ž,𝛼)𝑛1+β‹―+𝑛𝑠,π‘žβˆ’1,π‘€βˆ’1+[2]π‘žifπ‘˜=0,π‘žβ„Ž+1𝑀𝑛1π‘˜ξ‚Άβ‹―ξ‚΅π‘›π‘ π‘˜ξ‚Άπ‘ π‘˜ξ“π‘™=0ξ‚΅π‘ π‘˜π‘™ξ‚Ά(βˆ’1)𝑙+π‘ π‘˜πΈ(β„Ž,𝛼)𝑛1+β‹―+π‘›π‘ βˆ’π‘™,π‘žβˆ’1,π‘€βˆ’1ifπ‘˜>0.(2.31)
By the definition of π‘ž-Bernstein polynomials with weight 𝛼 and the binomial theorem, we easily get ξ€œβ„€π‘π΅(𝛼)π‘˜,𝑛1(π‘₯,π‘ž)⋯𝐡(𝛼)π‘˜,𝑛𝑠(π‘₯,π‘ž)ξ„Ώξ…€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…Œπ‘ -timesπ‘ž(β„Žβˆ’1)π‘₯𝑀π‘₯π‘‘πœ‡βˆ’π‘ž(π‘₯)=βŽ›βŽœβŽœβŽπ‘›1π‘˜βŽžβŽŸβŽŸβŽ β‹―βŽ›βŽœβŽœβŽπ‘›π‘ π‘˜βŽžβŽŸβŽŸβŽ π‘›1+β‹―+π‘›π‘ βˆ’π‘ π‘˜ξ“π‘™=0(βˆ’1)π‘™βŽ›βŽœβŽœβŽπ‘›1+β‹―+π‘›π‘ βˆ’π‘ π‘˜π‘™βŽžβŽŸβŽŸβŽ ξ€œβ„€π‘[π‘₯]π‘ π‘˜+π‘™π‘žπ‘ž(β„Žβˆ’1)π‘₯𝑀π‘₯π‘‘πœ‡βˆ’π‘ž(π‘₯)=βŽ›βŽœβŽœβŽπ‘›1π‘˜βŽžβŽŸβŽŸβŽ β‹―βŽ›βŽœβŽœβŽπ‘›π‘ π‘˜βŽžβŽŸβŽŸβŽ π‘›1+β‹―+π‘›π‘ βˆ’π‘ π‘˜ξ“π‘™=0(βˆ’1)π‘™βŽ›βŽœβŽœβŽπ‘›1+β‹―+π‘›π‘ βˆ’π‘ π‘˜π‘™βŽžβŽŸβŽŸβŽ πΈ(β„Ž,𝛼)π‘ π‘˜+𝑙,π‘ž,𝑀.(2.32)

Therefore, we have the following corollary.

Corollary 2.10. For π‘€βˆˆπ‘‡π‘,π‘ βˆˆβ„• with 𝑠β‰₯2, let 𝑛1,𝑛2,…,𝑛𝑠,π‘˜βˆˆβ„€+ with 𝑛1+β‹―+𝑛𝑠>π‘ π‘˜. Then we have 𝑛1+β‹―+nπ‘ βˆ’π‘ π‘˜ξ“π‘™=0(βˆ’1)π‘™βŽ›βŽœβŽœβŽπ‘›1+β‹―+π‘›π‘ βˆ’π‘ π‘˜π‘™βŽžβŽŸβŽŸβŽ πΈ(β„Ž,𝛼)π‘ π‘˜+𝑙,π‘ž,𝑀=⎧βŽͺ⎨βŽͺβŽ©π‘žβ„Ž+1𝑀𝐸(β„Ž,𝛼)𝑛1+β‹―+𝑛𝑠,π‘žβˆ’1,π‘€βˆ’1+[2]π‘žifπ‘˜=0,π‘žβ„Ž+1π‘€π‘ π‘˜ξ“π‘™=0ξ‚΅π‘ π‘˜π‘™ξ‚Ά(βˆ’1)𝑙+π‘ π‘˜πΈ(β„Ž,𝛼)𝑛1+β‹―+π‘›π‘ βˆ’π‘™,π‘žβˆ’1,π‘€βˆ’1ifπ‘˜>0.(2.33)

References

  1. L.-C. Jang, W.-J. Kim, and Y. Simsek, β€œA study on the p-adic integral representation on β„€p associated with Bernstein and Bernoulli polynomials,” Advances in Difference Equations, Article ID 163217, 6 pages, 2010. View at Google Scholar
  2. T. Kim, β€œq-Euler numbers and polynomials associated with p-adic q-integrals,” Journal of Nonlinear Mathematical Physics, vol. 14, no. 1, pp. 15–27, 2007. View at Publisher Β· 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 β„€p,” Russian Journal of Mathematical Physics, vol. 16, no. 4, pp. 484–491, 2009. View at Publisher Β· View at Google Scholar
  4. T. Kim, β€œBarnes-type multiple q-zeta functions and q-Euler polynomials,” Journal of Physics. A. Mathematical and Theoretical, vol. 43, no. 25, Article ID 255201, 11 pages, 2010. View at Publisher Β· View at Google Scholar
  5. 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 Β· View at Google Scholar
  6. T. Kim, β€œq-Volkenborn integration,” Russian Journal of Mathematical Physics, vol. 9, no. 3, pp. 288–299, 2002. View at Google Scholar Β· View at Zentralblatt MATH
  7. T. Kim, β€œq-Bernoulli numbers and polynomials associated with Gaussian binomial coefficients,” Russian Journal of Mathematical Physics, vol. 15, no. 1, pp. 51–57, 2008. View at Google Scholar Β· View at Zentralblatt MATH
  8. 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 Β· View at Google Scholar
  9. T. Kim, B. Lee, J. Choi, and Y. H. Kim, β€œA new approach of q-Euler numbers and polynomials,” Proceedings of the Jangjeon Mathematical Society, vol. 14, no. 1, pp. 7–14, 2011. View at Google Scholar
  10. T. Kim, J. Choi, and Y.-H. Kim, β€œSome identities on the q-Bernstein polynomials, q-Stirling numbers and q-Bernoulli numbers,” Advanced Studies in Contemporary Mathematics, vol. 20, no. 3, pp. 335–341, 2010. View at Google Scholar
  11. T. Kim, β€œSome identities for the Bernoulli, the Euler and the Genocchi numbers and polynomials,” Advanced Studies in Contemporary Mathematics, vol. 20, no. 1, pp. 23–28, 2010. View at Google Scholar Β· View at Zentralblatt MATH
  12. T. Kim, β€œThe modified q-Euler numbers and polynomials,” Advanced Studies in Contemporary Mathematics, vol. 16, no. 2, pp. 161–170, 2008. View at Google Scholar
  13. T. Kim, β€œNote on the Euler numbers and polynomials,” Advanced Studies in Contemporary Mathematics, vol. 17, no. 2, pp. 131–136, 2008. View at Google Scholar Β· View at Zentralblatt MATH
  14. H. Ozden and Y. Simsek, β€œA new extension of q-Euler numbers and polynomials related to their interpolation functions,” Applied Mathematics Letters, vol. 21, no. 9, pp. 934–939, 2008. View at Publisher Β· View at Google Scholar
  15. H. Ozden and Y. Simsek, β€œInterpolation function of the h, q-extension of twisted Euler numbers,” Computers & Mathematics with Applications, vol. 56, no. 4, pp. 898–908, 2008. View at Publisher Β· View at Google Scholar
  16. H. Ozden, Y. Simsek, and I. N. Cangul, β€œEuler polynomials associated with p-adic q-Euler measure,” General Mathematics, vol. 15, no. 2, pp. 24–37, 2007. View at Google Scholar
  17. S.-H. Rim, J.-H. Jin, E.-J. Moon, and S.-J. Lee, β€œOn multiple interpolation functions of the q-Genocchi polynomials,” Journal of Inequalities and Applications, Article ID 351419, 13 pages, 2010. View at Google Scholar
  18. S.-H. Rim, S. J. Lee, E. J. Moon, and J. H. Jin, β€œOn the q-Genocchi numbers and polynomials associated with q-zeta function,” Proceedings of the Jangjeon Mathematical Society, vol. 12, no. 3, pp. 261–267, 2009. View at Google Scholar
  19. C. S. Ryoo, β€œOn the generalized Barnes type multiple q-Euler polynomials twisted by ramified roots of unity,” Proceedings of the Jangjeon Mathematical Society, vol. 13, no. 2, pp. 255–263, 2010. View at Google Scholar
  20. C. S. Ryoo, β€œA note on the weighted q-Euler numbers and polynomials,” Advanced Studies in Contemporary Mathematics, vol. 21, pp. 47–54, 2011. View at Google Scholar
  21. Y. Simsek, O. Yurekli, and V. Kurt, β€œOn interpolation functions of the twisted generalized Frobinuous-Euler numbers,” Advanced Studies in Contemporary Mathematics, vol. 14, pp. 49–68, 2007. View at Google Scholar
  22. Y. Simsek and M. Acikgoz, β€œA new generating function of (q-) Bernstein-type polynomials and their interpolation function,” Abstract and Applied Analysis, vol. 2010, Article ID 769095, 12 pages, 2010. View at Publisher Β· View at Google Scholar