On the \(q\)-Genocchi numbers and polynomials with weight \(\alpha\) and weak weight \(\beta\) (Q443064)

The aim in this paper is to define \(q\)-Genocchi numbers \(G^{(\alpha,\beta)}_{n,q}\) and polynomials \(G^{(\alpha,\beta)}_{n,q}(x)\) with weight \(\alpha\) and weak weight \(\beta\). The authors investigate some properties which are related to \(q\)-Genocchi numbers and polynomials NEWLINEwith weight \(\alpha\) and weak weight \(\beta\). They also derive the existence of a specific interpolation function which interpolates \(q\)-Genocchi numbers and polynomials with weight \(\alpha\) and weak weight \(\beta\) at negative integers.