Levin-type transformations

Definition

Let $\sum_{j \ge 0} a_j$ be a series with partial sums $s_n = \sum_{j=0}^{n} a_j$. Let $(\omega_n)$ be a sequence of nonzero numbers, the remainder estimates, let $\beta > 0$, and let $k \ge 1$ be an integer. The Levin transformation is $$\mathcal{L}_k^{(n)} = \frac{\displaystyle\sum_{j=0}^{k} (-1)^j \binom{k}{j} (\beta + n + j)^{k-1} \frac{s_{n+j}}{\omega_{n+j}}}{\displaystyle\sum_{j=0}^{k} (-1)^j \binom{k}{j} (\beta + n + j)^{k-1} \frac{1}{\omega_{n+j}}},$$ defined when the denominator is not $0$. The choice $\omega_n = a_n$ gives Levin's $t$-transformation, and $\omega_n = (\beta + n) a_n$ gives Levin's $u$-transformation. Transformations of the same construction with other choices of remainder estimates or of the factors $(\beta + n + j)^{k-1}$ are called Levin-type transformations.