A generalization of Layman-Lotockiǐ exponential interpolation (Q2266562)

For equally spaced data points f(i), \(i=0(1)n\), an exponential interpolation polynomial for f(z), namely \(\Sigma^ n_{i=0}a_ ir^ z_ i\), where \(r_ i\) are arbitrary quantities, is obtainable formally from the Newton divided difference series for the displacement operator \(E^ z=(1+\Delta)^ z\), regarding \(\Delta\) as the variable, for \(\Delta =s_ i=r_ i-1\). This generalization of Layman-Lotockiǐ exponential interpolation in which \(r_ i=i+1\), employs ''general factorial differences'' \(\prod^ i_{j=0}(\Delta -s_ j)f(0)\), \(i=0(1)n-1\). For the confluent case, where \(f^{(i)}(0)\) is given instead of f(i), \(i=0(1)n\), the results are similar, the derivative operator D replacing the \(\Delta\).