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Abstract and Applied Analysis
VolumeΒ 2011, Article IDΒ 123483, 9 pages
http://dx.doi.org/10.1155/2011/123483
Research Article

Some Identities of the Twisted π‘ž-Genocchi Numbers and Polynomials with Weight 𝛼 and π‘ž-Bernstein Polynomials with Weight 𝛼

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

Received 7 July 2011; Accepted 22 August 2011

Academic Editor: JohnΒ Rassias

Copyright Β© 2011 H. Y. Lee 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

Recently mathematicians have studied some interesting relations between π‘ž-Genocchi numbers, π‘ž-Euler numbers, polynomials, Bernstein polynomials, and π‘ž-Bernstein polynomials. In this paper, we give some interesting identities of the twisted π‘ž-Genocchi numbers, polynomials, and π‘ž-Bernstein polynomials with weighted 𝛼.

1. Introduction

Throughout this paper, let 𝑝 be a fixed odd prime number. The symbols ℀𝑝, β„šπ‘, and ℂ𝑝 denote the ring of 𝑝-adic integers, the field of 𝑝-adic rational numbers, and the completion of algebraic closure of β„šπ‘. Let β„• be the set of natural numbers and let β„€+=β„•βˆͺ{0}. As a well-known definition, the 𝑝-adic absolute value is given by |π‘₯|𝑝=π‘βˆ’π‘Ÿ, where π‘₯=π‘π‘Ÿπ‘‘/𝑠 with (𝑑,𝑝)=(𝑠,𝑝)=(𝑑,𝑠)=1. When one talks of π‘ž-extension, π‘ž is variously considered as an indeterminate, a complex number π‘žβˆˆβ„‚, or a 𝑝-adic number π‘žβˆˆβ„‚π‘. In this paper we assume that π‘žβˆˆβ„‚π‘ with |1βˆ’π‘ž|𝑝<1.

We assume that UD(℀𝑝) is the space of the uniformly differentiable function on ℀𝑝. For π‘“βˆˆUD(℀𝑝), Kim defined the fermionic 𝑝-adic π‘ž-integral on ℀𝑝 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‘₯.(1.1) For π‘›βˆˆβ„•, let 𝑓𝑛(π‘₯)=𝑓(π‘₯+𝑛) be translation. As a well known equation, by (1.1), we have π‘žπ‘›ξ€œβ„€π‘π‘“(π‘₯+𝑛)π‘‘πœ‡βˆ’π‘ž(π‘₯)=(βˆ’1)π‘›πΌβˆ’π‘ž[2](𝑓)+π‘žπ‘›βˆ’1𝑙=0(βˆ’1)π‘›βˆ’1βˆ’π‘™π‘žπ‘™π‘“(𝑙),(1.2) compared [1–4]. 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(1.3)(cf. [1–16]). limπ‘žβ†’1[π‘₯]π‘ž=π‘₯ for any π‘₯ with |π‘₯|𝑝≀1 in the present 𝑝-adic case. To investigate relation of the twisted π‘ž-Genocchi numbers and polynomials with weight 𝛼 and the Bernstein polynomials with weight 𝛼, we will use useful property for [π‘₯]π‘žπ›Ό as follows;[π‘₯]π‘žπ›Ό[]=1βˆ’1βˆ’π‘₯π‘žβˆ’π›Ό,[]1βˆ’π‘₯π‘žβˆ’π›Ό[π‘₯]=1βˆ’π‘žπ›Ό.(1.4)

The twisted π‘ž-Genocchi numbers and polynomials with weight 𝛼 are defined by the generating function as follows, respectively: 𝐺(𝛼)𝑛,π‘ž,π‘€ξ€œ=π‘›β„€π‘πœ™π‘€([π‘₯]π‘₯)π‘žπ‘›βˆ’1π›Όπ‘‘πœ‡βˆ’π‘ž(π‘₯),(1.5)𝐺(𝛼)𝑛,π‘ž,π‘€ξ€œ(π‘₯)=π‘›β„€π‘πœ™π‘€[](𝑦)𝑦+π‘₯π‘žπ‘›βˆ’1π›Όπ‘‘πœ‡βˆ’π‘ž(𝑦).(1.6) In the special case, π‘₯=0, 𝐺(𝛼)𝑛,π‘ž,𝑀(0)=𝐺(𝛼)𝑛,π‘ž,𝑀 are called the 𝑛th twisted π‘ž-Genocchi numbers with weight 𝛼 (see [9]).

Let 𝐢𝑝𝑛={π‘€βˆ£π‘€π‘π‘›=1} be the cyclic group of order 𝑝𝑛 and let 𝑇𝑝=limπ‘›β†’βˆžπΆπ‘π‘›=ξšπ‘›β‰₯1𝐢𝑝𝑛,(1.7) see [9, 12–15].

Kim defined the π‘ž-Bernstein polynomials with weight 𝛼 of degree 𝑛 as follows: 𝐡(𝛼)𝑛,π‘˜βŽ›βŽœβŽœβŽπ‘›π‘˜βŽžβŽŸβŽŸβŽ [π‘₯](π‘₯)=π‘˜π‘žπ›Ό[]1βˆ’π‘₯π‘žπ‘›βˆ’π‘˜βˆ’π›Ό,where[]π‘₯∈0,1,𝑛,π‘˜βˆˆβ„€+,(1.8) compare [4, 7].

In this paper, we investigate some properties for the twisted π‘ž-Genocchi numbers and polynomials with weight 𝛼. By using these properties, we give some interesting identities on the twisted π‘ž-Genocchi polynomials with weight 𝛼 and π‘ž-Bernstein polynomials with weight 𝛼.

2. Some Identities on the Twisted π‘ž-Genocchi Polynomials with Weight 𝛼 and π‘ž-Bernstein Polynomials with Weight 𝛼

From (1.8), we can derive the following recurrence formula for the twisted π‘ž-Genocchi numbers with weight 𝛼: 𝐺(𝛼)0,π‘ž,𝑀=0,π‘žπ‘€πΊ(𝛼)𝑛,π‘ž,𝑀(1)+𝐺(𝛼)𝑛,π‘ž,𝑀=ξ‚»[2]π‘ž,if𝑛=1,0,if𝑛>1,(2.1)𝐺(𝛼)0,π‘ž,𝑀=0,π‘žπ‘€1+π‘žπ›ΌπΊ(𝛼)π‘ž,𝑀𝑛+π‘žπ›ΌπΊ(𝛼)𝑛,π‘ž,𝑀=ξ‚»π‘žπ›Ό[2]π‘ž,if𝑛=1,0,if𝑛>1,(2.2)π‘žπ›Όπ‘₯𝐺(𝛼)𝑛+1,π‘ž,𝑀[π‘₯](π‘₯)=π›Όπ‘ž+π‘žπ›Όπ‘₯𝐺(𝛼)π‘ž,𝑀𝑛+1(2.3) with usual convention about replacing (𝐺(𝛼)π‘ž,𝑀)𝑛 by 𝐺(𝛼)𝑛,π‘ž,𝑀.

By (1.5), we easily get 𝐺(𝛼)𝑛,π‘ž,𝑀[2](π‘₯)=π‘›π‘žξ‚΅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‘€π‘žπ›Όπ‘™+1.(2.4) By (2.4), we obtain the theorem below.

Theorem 2.1. Let π‘›βˆˆβ„€+. For π‘€βˆˆπ‘‡π‘, one has 𝐺(𝛼)𝑛,π‘ž,𝑀(π‘₯)=(βˆ’1)π‘›βˆ’1π‘€βˆ’1π‘žπ›Ό(1βˆ’π‘›)𝐺(𝛼)𝑛,π‘žβˆ’1,π‘€βˆ’1(1βˆ’π‘₯).(2.5)

By (2.1), (2.2), and (2.3) we note that 𝐺(𝛼)𝑛,π‘ž,𝑀=βˆ’π‘žπ‘€πΊ(𝛼)𝑛,π‘ž,𝑀(1)=βˆ’π‘›π‘€π‘žπΊ(𝛼)1,π‘ž,𝑀+𝑀2π‘žπ‘›2βˆ’π›Όξ“π‘™=2βŽ›βŽœβŽœβŽπ‘›π‘™βŽžβŽŸβŽŸβŽ π‘žπ›Όπ‘™πΊ(𝛼)𝑙,π‘ž,𝑀(1)=βˆ’π‘›π‘€π‘žπΊ(𝛼)1,π‘ž,𝑀+𝑀2π‘žπ‘›2βˆ’2𝛼𝑙=2βŽ›βŽœβŽœβŽπ‘›π‘™βŽžβŽŸβŽŸβŽ π‘žπ›Όπ‘™ξ‚€1+π‘žπ›ΌπΊ(𝛼)π‘ž,𝑀𝑙=βˆ’π‘›π‘€π‘žπΊ(𝛼)1,π‘ž,𝑀+𝑀2π‘ž2βˆ’2𝛼[2]π‘žπ›Ό+π‘ž2𝛼𝐺(𝛼)π‘ž,π‘€ξ‚π‘›βˆ’π‘›π‘€2π‘ž2𝐺(𝛼)1,π‘ž,𝑀=βˆ’π‘›π‘€π‘žπΊ(𝛼)1,π‘ž,𝑀+𝑀2π‘ž2𝐺(𝛼)𝑛,π‘ž,𝑀(2)βˆ’π‘›π‘€2π‘ž2𝐺(𝛼)1,π‘ž,𝑀.(2.6)

Therefore, by (2.6), we obtain the theorem below.

Theorem 2.2. For π‘›βˆˆβ„• with 𝑛>1, one has 𝐺(𝛼)𝑛,π‘ž,𝑀(2)=π‘€βˆ’2π‘žβˆ’2𝐺(𝛼)𝑛,π‘ž,𝑀+π‘€βˆ’1π‘žβˆ’1𝑛[2]π‘ž+𝑛[2]1+π‘žπ‘€π‘ž.1+π‘žπ‘€(2.7)

By (1.6) and Theorem 2.2, 𝐺(𝛼)𝑛+1,π‘ž,𝑀(2)=ξ€œπ‘›+1β„€π‘πœ™π‘€[](𝑦)𝑦+2π‘›π‘žπ›Όπ‘‘πœ‡βˆ’π‘ž=1(𝑦)([2]𝑛+1𝑛+1)π‘ž+(1+π‘žπ‘€π‘›+1)π‘€βˆ’1π‘žβˆ’1[2]π‘žξƒͺ1+π‘žπ‘€+π‘€βˆ’2π‘žβˆ’2𝐺(𝛼)𝑛+1,π‘ž,𝑀=[2]𝑛+1π‘ž1+π‘žπ‘€+π‘€βˆ’1π‘žβˆ’1[2]π‘ž1+π‘žπ‘€+π‘€βˆ’2π‘žβˆ’2𝐺(𝛼)𝑛+1,π‘ž,𝑀.𝑛+1(2.8) Hence, we obtain the corollary below.

Corollary 2.3. For π‘›βˆˆβ„•, one has ξ€œβ„€π‘πœ™π‘€[](𝑦)𝑦+2π‘›π‘žπ›Όπ‘‘πœ‡βˆ’π‘ž[2](𝑦)=π‘ž1+π‘žπ‘€+π‘€βˆ’1π‘žβˆ’1[2]π‘ž1+π‘žπ‘€+π‘€βˆ’2π‘žβˆ’2𝐺(𝛼)𝑛+1,π‘ž,𝑀.𝑛+1(2.9)

By fermionic integral on ℀𝑝,Theorems 2.1 and 2.2, we note thatξ€œβ„€π‘πœ™π‘€([]π‘₯)1βˆ’π‘₯π‘›π‘žβˆ’π›Όπ‘‘πœ‡βˆ’π‘ž(π‘₯)=(βˆ’1)π‘›π‘žπ›Όπ‘›ξ€œβ„€π‘πœ™π‘€([]π‘₯)π‘₯βˆ’1π‘›π‘žπ›Όπ‘‘πœ‡βˆ’π‘ž(π‘₯)=(βˆ’1)π‘›π‘žπ›Όπ‘›πΊ(𝛼)𝑛+1,π‘ž,𝑀(βˆ’1)𝑛+1=π‘€βˆ’1𝐺(𝛼)𝑛+1,π‘žβˆ’1,π‘€βˆ’1(2)𝑛+1=π‘€βˆ’1βŽ›βŽœβŽœβŽ[2]π‘žβˆ’11+π‘žβˆ’1π‘€βˆ’1[2]+π‘€π‘žπ‘žβˆ’11+π‘žβˆ’1π‘€βˆ’1+𝑀2π‘ž2𝐺(𝛼)𝑛+1,π‘žβˆ’1,π‘€βˆ’1⎞⎟⎟⎠=[2]𝑛+1π‘ž[2]1+π‘žπ‘€+π‘€π‘žπ‘ž1+π‘žπ‘€+π‘€π‘ž2𝐺(𝛼)𝑛+1,π‘žβˆ’1,π‘€βˆ’1.𝑛+1(2.10)

Therefore, we have the theorem below.

Theorem 2.4. For π‘›βˆˆβ„• with 𝑛>1, one has ξ€œβ„€π‘πœ™π‘€[](π‘₯)1βˆ’π‘₯π‘›π‘žβˆ’π›Όπ‘‘πœ‡βˆ’π‘ž[2](π‘₯)=π‘ž[2]1+π‘žπ‘€+π‘€π‘žπ‘ž1+π‘žπ‘€+π‘€π‘ž2𝐺(𝛼)𝑛+1,π‘žβˆ’1,π‘€βˆ’1.𝑛+1(2.11)

By (1.4), Theorem 2.4, we take the fermionic 𝑝-adic invariant integral on ℀𝑝 for one q-Bernstein polynomials as follows: ξ€œβ„€π‘πœ™π‘€(π‘₯)𝐡𝑛,π‘˜(π‘₯,π‘ž)π‘‘πœ‡βˆ’π‘žξ€œ(π‘₯)=β„€π‘πœ™π‘€βŽ›βŽœβŽœβŽπ‘›π‘˜βŽžβŽŸβŽŸβŽ [π‘₯](π‘₯)π‘˜π‘žπ›Ό[]1βˆ’π‘₯π‘žπ‘›βˆ’π‘˜βˆ’π›Όπ‘‘πœ‡βˆ’π‘ž=βŽ›βŽœβŽœβŽπ‘›π‘˜βŽžβŽŸβŽŸβŽ ξ€œ(π‘₯)β„€π‘πœ™π‘€([π‘₯]π‘₯)π‘˜π‘žπ›Όξ€·[π‘₯]1βˆ’π‘žπ›Όξ€Έπ‘›βˆ’π‘˜π‘‘πœ‡βˆ’π‘ž(=βŽ›βŽœβŽœβŽπ‘›π‘˜βŽžβŽŸβŽŸβŽ π‘₯)π‘›βˆ’π‘˜ξ“π‘™=0βŽ›βŽœβŽœβŽπ‘™βŽžβŽŸβŽŸβŽ π‘›βˆ’π‘˜(βˆ’1)𝑙𝐺(𝛼)π‘˜+𝑙+1,π‘ž,𝑀.π‘˜+𝑙+1(2.12)

By symmetry of π‘ž-Bernstein polynomials with weight 𝛼 of degree 𝑛, we get the following formula; ξ€œβ„€π‘πœ™π‘€(π‘₯)𝐡𝑛,π‘˜(π‘₯,π‘ž)π‘‘πœ‡βˆ’π‘ž(=ξ€œπ‘₯)β„€π‘πœ™π‘€βŽ›βŽœβŽœβŽπ‘›π‘˜βŽžβŽŸβŽŸβŽ [π‘₯](π‘₯)π‘žπ‘›βˆ’π‘˜π›Ό[]1βˆ’π‘₯π‘˜π‘žβˆ’π›Όπ‘‘πœ‡βˆ’π‘ž=ξ€œ(π‘₯)β„€π‘πœ™π‘€βŽ›βŽœβŽœβŽπ‘›π‘˜βŽžβŽŸβŽŸβŽ [](π‘₯)1βˆ’π‘₯π‘˜π‘žβˆ’π›Όξ€·[]1βˆ’1βˆ’π‘₯π‘žβˆ’π›Όξ€Έπ‘›βˆ’π‘˜π‘‘πœ‡βˆ’π‘ž=βŽ›βŽœβŽœβŽπ‘›π‘˜βŽžβŽŸβŽŸβŽ (π‘₯)π‘›βˆ’π‘˜ξ“π‘™=0βŽ›βŽœβŽœβŽπ‘™βŽžβŽŸβŽŸβŽ π‘›βˆ’π‘˜(βˆ’1)π‘›βˆ’π‘˜βˆ’π‘™ξ€œβ„€π‘πœ™π‘€[](π‘₯)1βˆ’π‘₯π‘žπ‘›βˆ’π‘™βˆ’π›Όπ‘‘πœ‡βˆ’π‘ž=βŽ›βŽœβŽœβŽπ‘›π‘˜βŽžβŽŸβŽŸβŽ (π‘₯)π‘›βˆ’π‘˜ξ“π‘™=0βŽ›βŽœβŽœβŽπ‘™βŽžβŽŸβŽŸβŽ π‘›βˆ’π‘˜(βˆ’1)π‘›βˆ’π‘˜βˆ’π‘™βŽ›βŽœβŽœβŽ[2]π‘ž[2]1+π‘žπ‘€+π‘€π‘žπ‘ž1+π‘žπ‘€+π‘€π‘ž2𝐺(𝛼)π‘›βˆ’π‘™+1,π‘žβˆ’1,π‘€βˆ’1⎞⎟⎟⎠.π‘›βˆ’π‘™+1(2.13)

Therefore, by (2.12) and (2.13), we have the theorem below.

Theorem 2.5. For π‘›βˆˆβ„• with 𝑛>1, one has π‘›βˆ’π‘˜ξ“π‘™=0βŽ›βŽœβŽœβŽπ‘™βŽžβŽŸβŽŸβŽ π‘›βˆ’π‘˜(βˆ’1)𝑙𝐺(𝛼)π‘˜+𝑙+1,π‘ž,𝑀=π‘˜+𝑙+1π‘›βˆ’π‘˜ξ“π‘™=0βŽ›βŽœβŽœβŽπ‘™βŽžβŽŸβŽŸβŽ (π‘›βˆ’π‘˜βˆ’1)π‘›βˆ’π‘˜βˆ’π‘™βŽ›βŽœβŽœβŽ[2]π‘ž[2]1+π‘žπ‘€+π‘€π‘žπ‘ž1+π‘žπ‘€+π‘€π‘ž2𝐺(𝛼)π‘›βˆ’π‘™+1,π‘žβˆ’1,π‘€βˆ’1⎞⎟⎟⎠.π‘›βˆ’π‘™+1(2.14) Also, we note that ξ€œβ„€π‘πœ™π‘€(π‘₯)𝐡𝑛,π‘˜(π‘₯,π‘ž)π‘‘πœ‡βˆ’π‘ž(=βŽ›βŽœβŽœβŽπ‘›π‘˜βŽžβŽŸβŽŸβŽ π‘₯)π‘›βˆ’π‘˜ξ“π‘™=0βŽ›βŽœβŽœβŽπ‘™βŽžβŽŸβŽŸβŽ π‘›βˆ’π‘˜(βˆ’1)𝑙𝐺(𝛼)π‘˜+𝑙+1,π‘ž,𝑀=βŽ›βŽœβŽœβŽπ‘›π‘˜βŽžβŽŸβŽŸβŽ ξ€œπ‘˜+𝑙+1β„€π‘πœ™π‘€[](π‘₯)1βˆ’π‘₯π‘žπ‘›βˆ’π‘˜βˆ’π›Ό[π‘₯]π‘˜π‘žπ›Όπ‘‘πœ‡βˆ’π‘ž=βŽ›βŽœβŽœβŽπ‘›π‘˜βŽžβŽŸβŽŸβŽ ξ€œ(π‘₯)β„€π‘πœ™π‘€[](π‘₯)1βˆ’π‘₯π‘žπ‘›βˆ’π‘˜βˆ’π›Όξ€·[]1βˆ’1βˆ’π‘₯π‘žβˆ’π›Όξ€Έπ‘˜π‘‘πœ‡βˆ’π‘ž=βŽ›βŽœβŽœβŽπ‘›π‘˜βŽžβŽŸβŽŸβŽ (π‘₯)π‘˜ξ“π‘™=0βŽ›βŽœβŽœβŽπ‘˜π‘™βŽžβŽŸβŽŸβŽ (βˆ’1)π‘˜βˆ’π‘™ξ€œβ„€π‘πœ™π‘€[](π‘₯)1βˆ’π‘₯π‘žπ‘›βˆ’π‘™βˆ’π›Όπ‘‘πœ‡βˆ’π‘ž=βŽ›βŽœβŽœβŽπ‘›π‘˜βŽžβŽŸβŽŸβŽ (π‘₯)π‘˜ξ“π‘™=0βŽ›βŽœβŽœβŽπ‘˜π‘™βŽžβŽŸβŽŸβŽ (βˆ’1)π‘˜βˆ’π‘™βŽ›βŽœβŽœβŽ[2]π‘ž[2]1+π‘žπ‘€+π‘€π‘žπ‘ž1+π‘žπ‘€+π‘€π‘ž2𝐺(𝛼)π‘›βˆ’π‘™+1,π‘žβˆ’1,π‘€βˆ’1⎞⎟⎟⎠.π‘›βˆ’π‘™+1(2.15)

Therefore, we have the theorem below.

Theorem 2.6. For 𝑛,π‘˜βˆˆβ„€+ with 𝑛>π‘˜+1, one has ξ€œβ„€π‘πœ™π‘€(π‘₯)π΅π‘˜,𝑛(π‘₯,π‘ž)π‘‘πœ‡βˆ’π‘ž(=βŽ›βŽœβŽœβŽπ‘›π‘˜βŽžβŽŸβŽŸβŽ π‘₯)π‘˜ξ“π‘™=0βŽ›βŽœβŽœβŽπ‘˜π‘™βŽžβŽŸβŽŸβŽ (βˆ’1)π‘˜βˆ’π‘™βŽ›βŽœβŽœβŽ[2]π‘ž[2]1+π‘žπ‘€+π‘€π‘žπ‘ž1+π‘žπ‘€+π‘€π‘ž2𝐺(𝛼)π‘›βˆ’π‘™+1,π‘žβˆ’1,π‘€βˆ’1⎞⎟⎟⎠.π‘›βˆ’π‘™+1(2.16)

By (2.11) and Theorem 2.6, we have the theorem below.

Theorem 2.7. Let 𝑛,π‘˜βˆˆβ„€+ with 𝑛>π‘˜+1. Then one has π‘›βˆ’π‘˜ξ“π‘™=0βŽ›βŽœβŽœβŽπ‘™βŽžβŽŸβŽŸβŽ π‘›βˆ’π‘˜(βˆ’1)𝑙𝐺(𝛼)π‘˜+𝑙+1,π‘ž,𝑀=π‘˜+𝑙+1π‘˜ξ“π‘™=0βŽ›βŽœβŽœβŽπ‘˜π‘™βŽžβŽŸβŽŸβŽ (βˆ’1)π‘˜βˆ’π‘™βŽ›βŽœβŽœβŽ[2]π‘ž[2]1+π‘žπ‘€+π‘€π‘žπ‘ž1+π‘žπ‘€+π‘€π‘ž2𝐺(𝛼)π‘›βˆ’π‘™+1,π‘žβˆ’1,π‘€βˆ’1⎞⎟⎟⎠.π‘›βˆ’π‘™+1(2.17)

Let 𝑛1,𝑛2,π‘˜βˆˆβ„€+ with 𝑛1+𝑛2>2π‘˜+1. Then we get ξ€œβ„€π‘πœ™π‘€(π‘₯)𝐡𝑛(𝛼)1,π‘˜(π‘₯,π‘ž)𝐡𝑛(𝛼)2,π‘˜(π‘₯,π‘ž)π‘‘πœ‡βˆ’π‘ž(=βŽ›βŽœβŽœβŽπ‘›π‘₯)1π‘˜βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘›2π‘˜βŽžβŽŸβŽŸβŽ 2π‘˜ξ“π‘™=0βŽ›βŽœβŽœβŽπ‘™βŽžβŽŸβŽŸβŽ 2π‘˜(βˆ’1)2π‘˜βˆ’π‘™ξ€œβ„€π‘πœ™π‘€[](π‘₯)1βˆ’π‘₯𝑛1+𝑛2π‘žβˆ’π‘™βˆ’π›Όπ‘‘πœ‡βˆ’π‘ž=βŽ›βŽœβŽœβŽπ‘›(π‘₯)1π‘˜βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘›2π‘˜βŽžβŽŸβŽŸβŽ 2π‘˜ξ“π‘™=0βŽ›βŽœβŽœβŽπ‘™βŽžβŽŸβŽŸβŽ 2π‘˜(βˆ’1)2π‘˜βˆ’π‘™βŽ›βŽœβŽœβŽ[2]π‘ž[2]1+π‘žπ‘€+π‘€π‘žπ‘ž1+π‘žπ‘€+π‘€π‘ž2𝐺𝑛(𝛼)1+𝑛2βˆ’π‘™+1,π‘žβˆ’1,π‘€βˆ’1𝑛1+𝑛2⎞⎟⎟⎠.βˆ’π‘™+1(2.18)

Therefore, we obtain the theorem below.

Theorem 2.8. For 𝑛1,𝑛2,π‘˜βˆˆβ„€+, one has ξ€œβ„€π‘πœ™π‘€(π‘₯)𝐡𝑛(𝛼)1,π‘˜(π‘₯,π‘ž)𝐡𝑛(𝛼)2,π‘˜(π‘₯,π‘ž)π‘‘πœ‡βˆ’π‘ž(=βŽ›βŽœβŽœβŽπ‘›π‘₯)1π‘˜βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘›2π‘˜βŽžβŽŸβŽŸβŽ 2π‘˜ξ“π‘™=0βŽ›βŽœβŽœβŽπ‘™βŽžβŽŸβŽŸβŽ 2π‘˜(βˆ’1)2π‘˜βˆ’π‘™βŽ›βŽœβŽœβŽ[2]π‘ž[2]1+π‘žπ‘€+π‘€π‘žπ‘ž1+π‘žπ‘€+π‘€π‘ž2𝐺𝑛(𝛼)1+𝑛2βˆ’π‘™+1,π‘žβˆ’1,π‘€βˆ’1𝑛1+𝑛2⎞⎟⎟⎠=⎧βŽͺβŽͺ⎨βŽͺβŽͺ⎩[2]βˆ’π‘™+1π‘ž[2]1+π‘žπ‘€+π‘€π‘žπ‘ž1+π‘žπ‘€+π‘€π‘ž2𝐺𝑛(𝛼)1+𝑛2βˆ’π‘™+1,π‘žβˆ’1,π‘€βˆ’1𝑛1+𝑛2,βˆ’π‘™+1ifπ‘˜=0,π‘€π‘ž2𝑛1π‘˜π‘›ξ‚Άξ‚΅2π‘˜ξ‚Ά2π‘˜ξ“π‘™=0𝑙2π‘˜(βˆ’1)2π‘˜βˆ’π‘™πΊπ‘›1+𝑛2βˆ’π‘™+1,π‘žβˆ’1,π‘€βˆ’1𝑛1+𝑛2,βˆ’π‘™+1ifπ‘˜>0,(2.19)

By simple calculation, we easily see thatξ€œβ„€π‘πœ™π‘€(π‘₯)𝐡𝑛(𝛼)1,π‘˜(π‘₯,π‘ž)𝐡𝑛(𝛼)2,π‘˜(π‘₯,π‘ž)π‘‘πœ‡βˆ’π‘ž(=βŽ›βŽœβŽœβŽπ‘›π‘₯)1π‘˜βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘›2π‘˜βŽžβŽŸβŽŸβŽ π‘›1+𝑛2βˆ’2π‘˜ξ“π‘™=0(βˆ’1)π‘™βŽ›βŽœβŽœβŽπ‘›1+𝑛2π‘™βŽžβŽŸβŽŸβŽ ξ€œβˆ’2π‘˜β„€π‘πœ™π‘€[π‘₯](π‘₯)π‘ž2π‘˜+π‘™π›Όπ‘‘πœ‡βˆ’π‘ž=βŽ›βŽœβŽœβŽπ‘›(π‘₯)1π‘˜βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘›2π‘˜βŽžβŽŸβŽŸβŽ π‘›1+𝑛2βˆ’2π‘˜ξ“π‘™=0(βˆ’1)π‘™βŽ›βŽœβŽœβŽπ‘›1+𝑛2π‘™βŽžβŽŸβŽŸβŽ πΊβˆ’2π‘˜(𝛼)2π‘˜+𝑙+1,π‘ž,𝑀,2π‘˜+𝑙+1where𝑛1,𝑛2,π‘˜βˆˆβ„€+.(2.20) Therefore, by (2.20) and Theorem 2.8, we obtain the theorem below.

Theorem 2.9. Let 𝑛1,𝑛2,π‘˜βˆˆβ„€+ with 𝑛1+𝑛2>2π‘˜+1. Then one has 2π‘˜ξ“π‘™=0βŽ›βŽœβŽœβŽπ‘™βŽžβŽŸβŽŸβŽ 2π‘˜(βˆ’1)2π‘˜βˆ’π‘™βŽ›βŽœβŽœβŽ[2]π‘ž[2]1+π‘žπ‘€+π‘€π‘žπ‘ž1+π‘žπ‘€+π‘€π‘ž2𝐺𝑛(𝛼)1+𝑛2βˆ’π‘™+1,π‘žβˆ’1,π‘€βˆ’1𝑛1+𝑛2⎞⎟⎟⎠=βˆ’π‘™+1𝑛1+𝑛2βˆ’2π‘˜ξ“π‘™=0(βˆ’1)π‘™βŽ›βŽœβŽœβŽπ‘›1+𝑛2π‘™βŽžβŽŸβŽŸβŽ πΊβˆ’2π‘˜(𝛼)2π‘˜+𝑙+1,π‘ž,𝑀.2π‘˜+𝑙+1(2.21)

For 𝑛1,𝑛2,…,𝑛𝑠,π‘˜βˆˆβ„€+, 𝑛1+𝑛2+β‹―+𝑛𝑠>π‘ π‘˜+1, and let βˆ‘π‘ π‘–=1𝑛𝑖=π‘š, then by the symmetry of π‘ž-Bernstein polynomials with weight 𝛼, we see that ξ€œβ„€π‘πœ™π‘€(π‘₯)𝑠𝑖=1𝐡(𝛼)π‘˜,𝑛𝑖(π‘₯,π‘ž)π‘‘πœ‡βˆ’π‘ž=(π‘₯)𝑠𝑖=1βŽ›βŽœβŽœβŽπ‘›π‘–π‘˜βŽžβŽŸβŽŸβŽ π‘ π‘˜ξ“π‘™=0βŽ›βŽœβŽœβŽπ‘™βŽžβŽŸβŽŸβŽ (π‘ π‘˜βˆ’1)π‘ π‘˜βˆ’π‘™ξ€œβ„€π‘πœ™π‘€([]π‘₯)1βˆ’π‘₯π‘žπ‘šβˆ’π‘™βˆ’π›Όπ‘‘πœ‡βˆ’π‘ž(=π‘₯)𝑠𝑖=1βŽ›βŽœβŽœβŽπ‘›π‘–π‘˜βŽžβŽŸβŽŸβŽ π‘ π‘˜ξ“π‘™=0βŽ›βŽœβŽœβŽπ‘™βŽžβŽŸβŽŸβŽ π‘ π‘˜(βˆ’1)π‘ π‘˜βˆ’π‘™βŽ›βŽœβŽœβŽ[2]π‘ž[2]1+π‘žπ‘€+π‘€π‘žπ‘ž1+π‘žπ‘€+π‘€π‘ž2𝐺(𝛼)π‘šβˆ’π‘™+1,π‘žβˆ’1,π‘€βˆ’1⎞⎟⎟⎠.π‘šβˆ’π‘™+1(2.22)

Therefore, we have the theorem below.

Theorem 2.10. For 𝑛1,𝑛2,…,𝑛𝑠,π‘˜βˆˆβ„€+ with 𝑛1+𝑛2+β‹―+𝑛𝑠>π‘ π‘˜+1, one has ξ€œβ„€π‘πœ™π‘€(π‘₯)𝑠𝑖=1𝐡(𝛼)π‘˜,𝑛𝑖(π‘₯,π‘ž)π‘‘πœ‡βˆ’π‘ž=(π‘₯)𝑠𝑖=1βŽ›βŽœβŽœβŽπ‘›π‘–π‘˜βŽžβŽŸβŽŸβŽ π‘ π‘˜ξ“π‘™=0βŽ›βŽœβŽœβŽπ‘™βŽžβŽŸβŽŸβŽ (π‘ π‘˜βˆ’1)π‘ π‘˜βˆ’π‘™βŽ›βŽœβŽœβŽ[2]π‘ž[2]1+π‘žπ‘€+π‘€π‘žπ‘ž+1+π‘žπ‘€wπ‘ž2𝐺(𝛼)π‘šβˆ’π‘™+1,π‘žβˆ’1,π‘€βˆ’1⎞⎟⎟⎠,π‘šβˆ’π‘™+1(2.23) where 𝑛1+β‹―+𝑛𝑠=π‘š.

In the same manner as in (2.15), we can get the following relation:ξ€œβ„€π‘πœ™π‘€(π‘₯)𝑠𝑖=1𝐡(𝛼)π‘˜,𝑛𝑖(π‘₯,π‘ž)π‘‘πœ‡βˆ’π‘ž=(π‘₯)𝑠𝑖=1βŽ›βŽœβŽœβŽπ‘›π‘–π‘˜βŽžβŽŸβŽŸβŽ ξ€œβ„€π‘πœ™π‘€([π‘₯]π‘₯)π‘žπ‘ π‘˜π›Όπ‘šβˆ’π‘ π‘˜ξ“π‘™=0(βˆ’1)π‘™βŽ›βŽœβŽœβŽπ‘™βŽžβŽŸβŽŸβŽ (π‘šβˆ’π‘ π‘˜βˆ’1)𝑙[π‘₯]π‘™π‘žπ›Όπ‘‘πœ‡βˆ’π‘ž(=π‘₯)𝑠𝑖=1βŽ›βŽœβŽœβŽπ‘›π‘–π‘˜βŽžβŽŸβŽŸβŽ π‘šβˆ’π‘ π‘˜ξ“π‘™=0(βˆ’1)π‘™βŽ›βŽœβŽœβŽπ‘™βŽžβŽŸβŽŸβŽ πΊπ‘šβˆ’π‘ π‘˜(𝛼)π‘ π‘˜+𝑙+1,π‘ž,𝑀,π‘ π‘˜+𝑙+1(2.24) where 𝑛1,𝑛2,…,𝑛𝑠,π‘˜βˆˆβ„€+ with π‘š=𝑛1+𝑛2+β‹―+𝑛𝑠>π‘ π‘˜+1.

By Theorem 2.10 and (2.13), we have the following corollary.

Corollary 2.11. Let π‘šβˆˆβ„•. For 𝑛1,𝑛2,…,𝑛𝑠,π‘˜βˆˆβ„€+ with 𝑛1+β‹―+𝑛𝑠>π‘šπ‘˜+1, one has π‘ π‘˜ξ“π‘™=0βŽ›βŽœβŽœβŽπ‘™βŽžβŽŸβŽŸβŽ π‘ π‘˜(βˆ’1)π‘ π‘˜βˆ’π‘™βŽ›βŽœβŽœβŽ[2]π‘ž[2]1+π‘žπ‘€+π‘€π‘žπ‘ž1+π‘žπ‘€+π‘€π‘ž2𝐺(𝛼)π‘šβˆ’π‘™+1,π‘žβˆ’1,π‘€βˆ’1⎞⎟⎟⎠=π‘šβˆ’π‘™+1π‘šβˆ’π‘ π‘˜ξ“π‘™=0(βˆ’1)π‘™βŽ›βŽœβŽœβŽπ‘™βŽžβŽŸβŽŸβŽ πΊπ‘šβˆ’π‘ π‘˜(𝛼)π‘ π‘˜+𝑙+1,π‘ž,𝑀,π‘ π‘˜+𝑙+1(2.25) where 𝑛1+β‹―+𝑛𝑠=π‘š.

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