A Generalization of the Calculus of Finite Differences to Include the Differential Calculus

As usually treated, the calculus of finite differences includes the differential calculus only as a limiting case. A difference quotient is defined for any set of values which are distinct, and when the theory is carried further to include the case of coincident values, the limit of the difference quotient is considered.t It is the purpose of the present paper to show how the difference quotient may be introduced so that its definition will hold for any set of values xl, x2, * * *, x,+p, whether these are distinct or not. For the purpose of notation we assume that xi appears ki times, where i takes on values from 1 to h, and 1 k= n+1. It is not absolutely necessary for our treatment, but will obviate the necessity of certain clumsy reasoning if we assume the following well known theorem: If XI, x2, *.., Xh are any distinct values and kA, k2, , kh are any positive or zero integers whose sum is n + 1, and if f(x,), f (xD), * * , f(k'14) (X), f (X2), * * *, f (h-1) (xh) are any real numbers, then there exists one and only one polynomial f (x) of degree n or less which together with its derivatives takes on the prescribed values. For convenience we shall denote this polynomial by the symbol