Abstract
Recently many mathematicians are working on Genocchi polynomials and Genocchi numbers. We define a new type of twisted <i >q</i>-Genocchi numbers and polynomials with weight <svg style="vertical-align:-0.1254pt;width:8.9375px;" id="M1" 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> and weak weight <svg style="vertical-align:-2.29482pt;width:8.8500004px;" id="M2" height="13.425" version="1.1" viewBox="0 0 8.8500004 13.425" width="8.8500004" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.425)"> <g transform="translate(72,-61.26)"> <text transform="matrix(1,0,0,-1,-71.95,63.6)"> <tspan style="font-size: 12.50px; " x="0" y="0">𝛽</tspan> </text> </g> </g> </svg> and give some interesting relations of the twisted <i >q</i>-Genocchi numbers and polynomials with weight <svg style="vertical-align:-0.1254pt;width:8.9375px;" id="M3" 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> and weak weight <svg style="vertical-align:-2.29482pt;width:8.8500004px;" id="M4" height="13.425" version="1.1" viewBox="0 0 8.8500004 13.425" width="8.8500004" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.425)"> <g transform="translate(72,-61.26)"> <text transform="matrix(1,0,0,-1,-71.95,63.6)"> <tspan style="font-size: 12.50px; " x="0" y="0">𝛽</tspan> </text> </g> </g> </svg>. Finally, we find relations between twisted <i >q</i>-Genocchi zeta function and twisted Hurwitz <i >q</i>-Genocchi zeta function.
Highlights
The Genocchi numbers and polynomials possess many interesting properties and arising in many areas of mathematics and physics
Many mathematicians have studied in the area of the q-Genocchi numbers and polynomials see 1–16
By Zp we denote the ring of p-adic rational integers, Qp denotes the field of p-adic rational numbers, Cp denotes the completion of algebraic closure of Qp, N denotes the set of natural numbers, Z denotes the ring of rational integers, Q denotes the field of rational numbers, C denotes the set of complex numbers, and Z N ∪ {0}
Summary
The Genocchi numbers and polynomials possess many interesting properties and arising in many areas of mathematics and physics. Throughout this paper we use the following notations. When one talks of q-extension, q is considered in many ways such as an indeterminate, a complex number q ∈ C, or p-adic number q ∈ Cp. If q ∈ C, one normally assume that |q| < 1. Throughout this paper we use the following notation: Abstract and Applied Analysis. F ∈ UD Zp f | f : Zp −→ Cp is uniformly differentiable function , 1.2 the fermionic p-adic q-integral on Zp is defined by Kim as follows: I−q f f x dμ−q x. For w ∈ Tp, we denote by φw : Zp → Cp the locally constant function x → wx
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