q-Taylor’s Formula for Formal Power Series and Heine’s Binomial Formula

Abstract

We now begin to apply what we have learned so far, particularly q-Taylor’s formula (4.1), to study identities involving infinite sums and products. In order to do this, we first have to remark that the generalized Taylor formula (2.2) about a = 0, and hence the q-Taylor formula (4.1) about c = 0, apply not only to polynomials, but also to formal power series. A formal power series, of the form

$$ f(x) = \sum\limits_{k = 0}^\infty {c_k x^k } , $$

may be thought of as a polynomial of infinite degree. It is “formal” because often we do not worry about whether the series converges or not, and we can operate on (for example, differentiate) the series formally. We have to assume a and c to be zero in order to avoid divergence problems. Of course, f(0) = c₀ by definition.