Sequences of Polynomials

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Sequences of Polynomials

``Few things are harder to put up with than the annoyance of a good example. ''
Mark Twain (1835-1910)

In a similar vein, structures are found in the coefficients of sequences of polynomials. The first example in Figure 6 shows the binomial coefficients ${ n \choose m }$ mod 3, or equivalently Pascal's Triangle mod 3. For the sake of what follows, it is convenient to think of the ith row as the coefficients of the polynomial (1+x)ⁱ taken modulo three. This apparently fractal pattern has been the object of much careful study [8].

**Figure 6:** Eighty rows of Pascal's Triangle mod 3
$\begin{figure} \centerline{ \psfig {figure=FIGBIN80BW.ps,width=2in} }\end{figure}$

Figure 7 shows the coefficients of the first eighty Cebyshev polynomials mod 3 laid out like the binomial coefficients of Figure 6. Recall that the nth Cebyshev polynomial T_n is defined by $T_n(x):= \cos (n \arccos x)$ . They have the explicit representation :

$\begin{displaymath} T_n(x)= \frac{n}{2} \sum_{k=0}^{\lfloor n/2 \rfloor} (-1)^k \frac{(n-k-1)!}{k!(n-2k)!}(2x)^{n-2k},\end{displaymath}$

and satisfy the recursion

$\begin{displaymath} T_n(x) = 2xT_{n-1}(x)- T_{n-2}(x)\,, \quad n = 2,3, \ldots.\end{displaymath}$

Note that the expression for T_n(x) resembles the ${ n \choose m }$ form of the binomial coefficients and its recursion relation is similar to that for the Pascal's Triangle.

Figure 8 shows the Stirling numbers of the second kind mod 3 organized as a triangle as well. Recall that Stirling numbers of the second kind are defined by

$\begin{displaymath} S(n,m) := \frac{1}{m!} \sum_{k=0}^{m} {m\choose k} (-1)^{m-k} k^n \end{displaymath}$

and give number of ways of partitioning a set of n elements into m non-empty subsets. Once again the form of ${ n \choose m }$ appears in its expression.

**Figure 7:** Eighty Cebychev Polynomials mod 3
$\begin{figure} \centerline{ \psfig {figure=FIGCEB80BW.ps,width=2in} }\end{figure}$

While the forms for each of the polynomials are relatively well-known, it is apparent that they are graphically related to each other (and distinguishable from each other). Each is a variant on the binomial coefficients.

**Figure 8:** Eighty rows of Stirling Numbers of the second kind mod 3
$\begin{figure} \centerline{ \psfig {figure=FIGSTIR2BW.ps,width=2in} }\end{figure}$

It is possible to find similar sorts of structure in virtually any sequence of polynomials: Legendre polynomials; Euler polynomials; sequences of Padé denominators to the exponential or to $(1-x)^\alpha$ with $\alpha$ rational. Then, selecting any moduli, a distinct pattern will emerge. These intriguing images hint at an underlying structure within the polynomials themselves and demand some explanation. While conjectures exist for their origin, an incontrovertible proof for the theorems suggested by these pictures is not yet in hand. And when there finally is a proof, might it be offered in some visual form?

Next: Quasi-Rationals Up: The Structure of Numbers Previous: Binary Expansions

loki@cecm.sfu.ca