Derivation of a general mixed interpolation formula

A general procedure is developed to derive a mixed interpolation formula for approximating any (n + 1) times differentiable function f(x), for x \in [0,nh], by a function $f_n(x)$ of the type $f_n(x)=aU_1(kx) + bU_2(kx)+\sum^{n-2}_{i=0}cix^i$, interpolating points being the ones given by $x_j = jh,(h > 0),j = O,1, . . . . . n,$ where $U_1(kx)$ and $U_2 (kx)$ are the two linearly independent solutions of a suitable second order Ordinary Differential Equation (ODE) and k > 0 is an appropriately chosen parameter. The results for the particular case when $U_1(kx)$ and $U_2(kx)$ represent the trigonometric functions follow easily. An analysis of the error is also discussed and specific numerical examples are included for the sake of comparison with the known interpolation formulae. Tables showing the comparison of the maximum errors occuring in the use of the various interpolation formulae has also been presented for some specially chosen functions