Generalized Taylor’s Formula for Polynomials

Abstract

In the ordinary calculus, a function, f(x) that possesses derivatives of all Orders is analytic at x = a if it can be expressed as a power series about x = a. Taylor’s theorem teils us the power series is

$$ f(x) = \sum\limits_{n = 0}^\infty {f^{(n)} (a)} \frac{{(x - a)^n }} {{n!}}. $$

((2.1))

The Taylor expansion of an analytic function often allows us to extend the definition of the function to a larger and more interesting domain. For example, we can use the Taylor expansion of e^x to define the exponentials of complex numbers and Square matrices. We would also like to formulate a q-analogue of Taylor’s formula. But before doing so, let us first consider a more general Situation.