Taylor's formula for general quantum calculus

Let I\subseteq\mathbb{R} be an interval and \beta\colon I\to I a strictly increasing continuous function with a unique fixed point s_۰\in I satisfying (t-s_۰)(\beta(t)-t)\le ۰ for all t\in I. Hamza et al. introduced the general quantum difference operator D_{\beta} by D_{\beta}f(t):=\frac{f(\beta(t))-f(t)}{\beta(t)-t} if t\ne s_۰ and D_{\beta}f(t):=f'(s_۰) if t=s_۰.   In this paper, we establish results concerning Taylor's formula associated with D_{\beta}. For this, we define two types of monomials and then present our main results. The obtained results are new in the literature and are useful for further research in the field.

کلیدواژه ها:

Quantum calculus ، quantum difference operator ، \beta-derivative ، \beta-integral ، Taylor's formula ، monomials