Change of basis and Stirling numbers

Polynomials form a vector space—the sum of two polynomials is a polynomial etc.—and the most natural basis for this vector space is powers of x:

1, x, x², x³, …

But the power basis is not the only possible basis, and often not the most useful basis in application.

Falling powers

In some applications the falling powers of x are a more useful basis. For positive integers n, the nth falling power of x is defined to be

x^{\underbar{\small{\emph{n}}}) = x(x-1)(x-2)\cdots(x-n+1)

Falling powers come up in combinatorics, in the calculus of finite differences, and in hypergeometric functions.

Change of basis

Since we have two bases for the vector space of polynomials, we can ask about the matrices that represent the change of basis from one to the other, and here’s where we see an interesting connection.

The entries of these matrices are numbers that come up in other applications, namely the Stirling numbers. You can think of Stirling numbers as variations on binomial coefficients. More on Stirling numbers here.

In summation notation, we have

\begin{align*} x^{\underbar{\small{\emph{n}}}} &= \sum_{k=0}^n S_1(n,k)x^{\text{\small{\emph{k}}}} \\ x^{\text{\small{\emph{n}}}} &= \sum_{k=0}^n S_2(n,k)x^{\underbar{\small\emph{n}}} \\ \end{align*}

where the S1 are the (signed) Stirling numbers of the 1st kind, and the S2 are the Stirling numbers of the 2nd kind.

(There are two conventions for defining Stirling numbers of the 1st kind, differing by a factor of (-1)n-k.)

Matrix form

This means the (ij)th element of matrix representing the change of basis from the power basis to the falling power basis is S1(i, j) and the (i, j)th entry of the matrix for the opposite change of basis is S2(i, j). These are lower triangular matrices because S1(i, j) and S2(i, j) are zero for j > i.

These are infinite matrices since there’s no limit to the degree of a polynomial. But if we limit our attention to polynomials of degree less than m, we take the upper left m by m submatrix of the infinite matrix. For example, if we look at polynomials of degree 4 or less, we have

\begin{bmatrix} x^\underbar{\tiny{0}} \\ x^\underbar{\tiny{1}} \\ x^\underbar{\tiny{2}} \\ x^\underbar{\tiny{3}} \\ x^\underbar{\tiny{4}} \\ \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & -1 & 1 & 0 & 0 \\ 0 & 2 & -3 & 1 & 0 \\ 0 & -6 & 11 & -6 & 1 \\ \end{bmatrix} \begin{bmatrix} x^\text{\tiny{0}} \\ x^\text{\tiny{1}} \\ x^\text{\tiny{2}} \\ x^\text{\tiny{3}} \\ x^\text{\tiny{4}} \\ \end{bmatrix}

to convert from powers to falling powers, and

\begin{bmatrix} x^\text{\tiny{0}} \\ x^\text{\tiny{1}} \\ x^\text{\tiny{2}} \\ x^\text{\tiny{3}} \\ x^\text{\tiny{4}} \\ \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 & 0 \\ 0 & 1 & 3 & 1 & 0 \\ 0 & 1 & 7 & 6 & 1 \\ \end{bmatrix} \begin{bmatrix} x^\underbar{\tiny{0}} \\ x^\underbar{\tiny{1}} \\ x^\underbar{\tiny{2}} \\ x^\underbar{\tiny{3}} \\ x^\underbar{\tiny{4}} \\ \end{bmatrix}

going from falling powers to powers.

Incidentally, if we filled a matrix with unsigned Stirling numbers of the 1st kind, we would have the change of basis matrix going from the power basis to rising powers defined by

x^{\overline{n}} = x(x+1)(x+2)\cdots(x+n+1)

It may be hard to see, but there’s a bar on top of the exponent n for rising powers whereas before we had a bar under the n for falling powers.

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3 thoughts on “Change of basis and Stirling numbers

  1. “These are lower triangular matrices because S1(i, j) and S2(i, j) for j > i.”
    I think you’re missing an “= 0”.

    We could also say that the matrices are lower triangular because an nth degree polynomial can be expressed in terms of the first n falling powers (and vice versa).

  2. Maybe use \overline instead of \bar for the rising powers? It’s easier to see when I quickly checked it like this:

    $x^{n} x^{\underline{n}} x^{\bar{n}} x^{\overline{n}}$

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