On the quantum differentiation of smooth real-valued functions

Abstract

Calculating the value of \(C^{k\in\{1,\infty\}}\) class of smoothness real-valued function's derivative in point of \(\mathbb{R}^+\) in radius of convergence of its Taylor polynomial (or series), applying an analog of Newton's binomial theorem and \(q\)-difference operator. \((P,q)\)-power difference introduced in section 5. Additionally, by means of Newton's interpolation formula, the discrete analog of Taylor series, interpolation using \(q\)-difference and \(p,q\)-power difference is shown.