Some Identities of the Twisted 𝑞-Genocchi Numbers and Polynomials with Weight 𝛼 and 𝑞-Bernstein Polynomials with Weight 𝛼

H Y Lee,C S Ryoo,Nam Soon Jung

doi:10.1155/2011/123483

Abstract

Recently mathematicians have studied some interesting relations between <svg style="vertical-align:-2.29482pt;width:7.9124999px;" id="M5" height="9.875" version="1.1" viewBox="0 0 7.9124999 9.875" width="7.9124999" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,9.875)"> <g transform="translate(72,-64.1)"> <text transform="matrix(1,0,0,-1,-71.95,66.44)"> <tspan style="font-size: 12.50px; " x="0" y="0">𝑞</tspan> </text> </g> </g> </svg>-Genocchi numbers, <svg style="vertical-align:-2.29482pt;width:7.9124999px;" id="M6" height="9.875" version="1.1" viewBox="0 0 7.9124999 9.875" width="7.9124999" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,9.875)"> <g transform="translate(72,-64.1)"> <text transform="matrix(1,0,0,-1,-71.95,66.44)"> <tspan style="font-size: 12.50px; " x="0" y="0">𝑞</tspan> </text> </g> </g> </svg>-Euler numbers, polynomials, Bernstein polynomials, and <svg style="vertical-align:-2.29482pt;width:7.9124999px;" id="M7" height="9.875" version="1.1" viewBox="0 0 7.9124999 9.875" width="7.9124999" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,9.875)"> <g transform="translate(72,-64.1)"> <text transform="matrix(1,0,0,-1,-71.95,66.44)"> <tspan style="font-size: 12.50px; " x="0" y="0">𝑞</tspan> </text> </g> </g> </svg>-Bernstein polynomials. In this paper, we give some interesting identities of the twisted <svg style="vertical-align:-2.29482pt;width:7.9124999px;" id="M8" height="9.875" version="1.1" viewBox="0 0 7.9124999 9.875" width="7.9124999" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,9.875)"> <g transform="translate(72,-64.1)"> <text transform="matrix(1,0,0,-1,-71.95,66.44)"> <tspan style="font-size: 12.50px; " x="0" y="0">𝑞</tspan> </text> </g> </g> </svg>-Genocchi numbers, polynomials, and <svg style="vertical-align:-2.29482pt;width:7.9124999px;" id="M9" height="9.875" version="1.1" viewBox="0 0 7.9124999 9.875" width="7.9124999" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,9.875)"> <g transform="translate(72,-64.1)"> <text transform="matrix(1,0,0,-1,-71.95,66.44)"> <tspan style="font-size: 12.50px; " x="0" y="0">𝑞</tspan> </text> </g> </g> </svg>-Bernstein polynomials with weighted <svg style="vertical-align:-0.1254pt;width:8.9375px;" id="M10" height="7.1750002" version="1.1" viewBox="0 0 8.9375 7.1750002" width="8.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,7.175)"> <g transform="translate(72,-66.26)"> <text transform="matrix(1,0,0,-1,-71.95,66.44)"> <tspan style="font-size: 12.50px; " x="0" y="0">𝛼</tspan> </text> </g> </g> </svg>.

Highlights

Throughout this paper, let p be a fixed odd prime number
We investigate some properties for the twisted q-Genocchi numbers and polynomials with weight α
By Theorem 2.10 and 2.13 , we have the following corollary

Summary

Research Article

Mathematicians have studied some interesting relations between q-Genocchi numbers, q-Euler numbers, polynomials, Bernstein polynomials, and q-Bernstein polynomials. We give some interesting identities of the twisted q-Genocchi numbers, polynomials, and qBernstein polynomials with weighted α

Introduction

Zp x