Shanks transformation

Definition

Let $(A_n)_{n \ge 0}$ be a sequence of real or complex numbers, and write $\Delta A_m = A_{m+1} - A_m$. For $n \ge 1$, the Shanks transformation of $(A_n)$ is $$S(A_n) = \frac{A_{n+1} A_{n-1} - A_n^2}{A_{n+1} - 2A_n + A_{n-1}},$$ defined when the denominator is not $0$. More generally, let $k \ge 1$ and $n \ge k$. The transformation of order $k$ is the quotient of two determinants: $$e_k(A_n) = \frac{\begin{vmatrix} A_{n-k} & \cdots & A_n \\ \Delta A_{n-k} & \cdots & \Delta A_n \\ \vdots & & \vdots \\ \Delta A_{n-1} & \cdots & \Delta A_{n+k-1} \end{vmatrix}}{\begin{vmatrix} 1 & \cdots & 1 \\ \Delta A_{n-k} & \cdots & \Delta A_n \\ \vdots & & \vdots \\ \Delta A_{n-1} & \cdots & \Delta A_{n+k-1} \end{vmatrix}}.$$ For $1 \le j \le k$, row $j$ of each determinant is $\Delta A_{n-k+j-1}, \dots, \Delta A_{n+j-1}$. The denominator must not be $0$. Thus $e_1(A_n) = S(A_n)$.