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In general, a function $f(\cdot)$ defined at points $x_1,x_2,\dots, x_k$, the $(k − 1)$th-order divided difference is defined by the recurrence relation: $$ f[x_1,x_2,\dots...x_k]=\frac{f[x_2,\dots......

$$\begin{equation} S(r,k) = f(r)S(0,k-1) + g(r)S(r+1,k-1) \end{equation}\ ,$$ where $r=0,1,2,\dots$ and $k=1,2,3,\dots$ . Also, $0<f(r)<1$ is an increasing function and $0<g(r)<1$ is a ...