Taylor's formula for general quantum calculus

Document Type : Research Article

Authors

1 Department of Mathematics, Sorbonne University, Paris, France

2 Department of Mathematics, Ramniranjan Jhunjhunwala College,\\ Mumbai, Maharashtra 400 086, India

Abstract

Let $I\subseteq\mathbb{R}$ be an interval and $\beta\colon I\to I$ a strictly increasing continuous function with a unique fixed point $s_0\in I$ satisfying $(t-s_0)(\beta(t)-t)\le 0$ 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 $t\ne s_0$ and $D_{\beta}f(t):=f'(s_0)$ if $t=s_0$.   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.

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Main Subjects

References

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Journal of Mathematical Modeling
Volume 11, Issue 3
October 2023
Pages 491-505

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