Specialized data points

The selection of particular collections of data points leads to specialized forms of Lagrange's interpolation formula, many of which are of common enough ocurrence to deserve special names and treatments.

By choosing x_i=-i, the interpolation points become integers, running backwards from zero. For N points the master polynomial will be

$$x^{(N)}=x(x+1) \cdots (x+N-1),$$ (26)

the subscript intended to imply something like a power, but more like factorial, but with increasing factors. In common with powers, the exponent N tells the number of terms in the product, and x⁽⁰⁾ can be taken as 1. Evidently, these ``factorials" can be rewritten as polynomials,

$$x^{(N)}= \sum_{i=0}^{i=N}S_i^Nx^i,$$ (27)

Their coefficients are called Stirling numbers of the first kind. The inverse relation,

$$x^N= \sum_{i=0}^{i=N}{\cal S}_i^Nx^{(i)},$$ (28)

defines Stirling numbers of the second kind. The two kinds of numbers evidently satisfy an orthogonality relation, expressing the fact that they are elements of mutually inverse matrices.

By defining a difference operator for functions:

$$\Delta f(x)$$ = f(x) - f(x-1), (29)

we can observe that

$$\Delta x^{(N)}$$ = x^(N)-(x-1)^(N)

$$x(x-1) \cdots (x+N-1) - (x-1)x \cdots (x-1+N-1)$$ =

$$x(x-1) \cdots (x+N-1-1)((x+N-1)-(x-1))$$ =

= Nx^(N-1), (30)

which in remarkably similar to the formula for the derivative of powers. Note that the operator takes backward differences, subtracting the function at a lesser value of its argument from its present value.

Another useful identity, which is a direct consequence of the definition, is

x^(N+1)=x^(N)(x+n), (31)

It is convenient to introduce a second kind of factorial,

$$x_{(N)}=x(x-1)\cdots(x-N+1),$$ (32)

wherein x is decremented rather than incremented in successive factors. Its relation to the first is

x_(N)=(-1)^N(-x)^(N), (33)

The Stirling matrix relates the ``factorial powers'' to normal powers, but a different relationship is called for when the interpolation polynomials for a fixed number of points are referred to normal polynomials. The point is that in this context, what is required is the inverse of the Vandermonde matrix, whose elements are coefficients of the individual Lagrange interpolation polynomials, not the coefficients of the one single master polynomial. Thus we would need the coefficients of

$$\frac{(x-0)}{(i-0)} \frac{(x-1)}{(i-1)} \frac{(x-2)}{(i-2)} \cdot... ...-i)} \cdots \frac{(x-n+2)}{(i-n+2)} \frac{(x-n+1)}{(i-n+1)} \frac{(x-n)}{(i-n)}$$ v_i(x) =

$$\frac{x_{(i)}}{i_{(i)}} \frac{(x-(i+1))_{(n-(i+1))}}{(-1)_{(n-(i+1))}}.$$ = (34)