On Lagrange-Hermite Interpolation

where Dt d/dt and aj,m is a Kronecker symbol. These conditions are used by Householder [5, pp. 193-195] to derive the formulas for p = 1, 2. The formula for p = 3 is given by Salzer [9]. The solution for n = 0 is given by Taylor's formula. Many authors have reported on the case where p depends on i. General prescriptions for a solution in this more general case may be found in Fort [2, pp. 85-88], Greville [3], Hermite [4], Krylov [6, pp. 45-49], Kuntzmann [7, pp. 167-169], and Spitzbart [12]; but these prescriptions do not determine the structure of the interpolating polynomial. By restricting ourselves to the case where p is independent of i, which is the most important case in practice, we can determine the structure. Salzer [10] discovered some of the properties of P ,, (t) by semiempirical means. We shall obtain, by a partial fraction expansion, a solution of surprising simplicity. [See (3.6), (3.7), or (3.8).] The solution depends upon the Bell polynomials which we now discuss.