In the first part of this paper, the authors look at applications of the following lemma: Let \[ f(z) = \sum_{n \geq 0} \alpha_nz^n \] be analytic for \(| z| < 1\), and assume that for some positive integer \(p\) and a fixed complex number \(\alpha\) we have that \[ \sum_{n \geq 0} (n+1,p)(\alpha_{n+p} - \alpha_{n+p-1}) \] converges, and \[ \lim_{n \to \infty} (n+1,p)(\alpha_{n+p} - \alpha) = 0. \] Then \[ \frac{1}{p} \lim_{z \to 1^{-}} \left(\frac{d^p}{dz^p}[(1-z)f(z)] \right) = \sum_{n \geq 0} (n+1,p-1)(\alpha - \alpha_{n+p-1}). \] The authors’ notation \((a,n)\) is for the product \(a(a+1) \cdots (a+n-1)\). For instance, the lemma is applied to the Heine transformation, and the result is the following, where the standard \(q\)-series notation is employed: For each integer \(p \geq 1\), \[ \sum_{n \geq 0} (n+1,p-1)\left(\frac{(b)_{\infty}}{(c)_{\infty}} - \frac{(b)_{n+p-1}}{(c)_{n+p-1}} \right) = -(p-1)! \frac{(b)_{\infty}}{(c)_{\infty}} \sum_{n \geq 1} \frac{(c/b)_n(bq^{p-1})^n}{(q)_n(1-q^n)^p}. \] In the second part of the paper, the authors look at applications of a second lemma:
Let \(f\) and \(g\) be two functions defined by the series \[ f(x) = \sum_{n \geq 0}f_nx^n \] and \[ g(x) = \sum_{n \geq 0}g_nx^n, \] and assume that the first series converges absolutely and that the series \[ \sum_{n \geq 0}\sum_{k \geq 0} | g_nf_kq^{kn}x^k| \] converges. Then \[ \sum_{n \geq 0}f_ng(q^n)x^n = \sum_{n \geq 0} g_nf(q^nx). \] These applications are to identities of which the following is a typical example: \[ \sum_{n \geq 0} \left( \frac{(q)_{\infty}}{(q)_n} - \frac{(q)_{\infty}^2}{(q)^2_n} \right) = \sum_{n \geq 1} \frac{(-1)^{n+1}q^{n(n+1)/2}}{1-q^n}. \]