# Connection between the Chebyshev polynomials and the Faber polynomials

From a comment on this question:

@draks, there is a connection between the Chebyshev polynomials and the Faber polynomials (a.k.a. Shur polynomials), which 'invert" the cyclic partition polynomials (a.k.a. the refined partition polynomials of the first kind--signed OEIS A036039) to isolate the indeterminates in their definition.

Interesting. Do you have something where I can read on this?

@Draks, see oeis.org/A263916 and oeis.org/A127672 along with the Damianou and Damianou and Evripidou links.

The links don't help. Can anyone explain the connection?

The characteristic polynomial and the traces of the powers of a square matrix $A$ are related by

$$\operatorname{det}(I-x \; A)=\exp\left(-\sum_{m\geq 1} \frac{\operatorname{tr}(A^m) \; x^m}{m} \right)$$

$$=1 + \sum_{k \ge 1} P_k(b_1,..,b_k) \; x^k / k! = 1 + \sum_{k \geq 1} d_k \; x^k$$

where $b_k=-\operatorname{tr}(A^k)$ for $k>0$ and $P_m(b_1,..,b_m)$ are the cycle index partition polynomials of OEIS A036039 for the symmetric groups (see the Lang link there for a compilation). Presented umbrally,

$$\operatorname{det}(I - x \; A)=\exp[\ln(1 - b.\;x)]=e^{P. \; x}=1/(1-d.\;x) \; .$$

The Faber polynomials $F_m$ of A263916 can, in general, invert the the determinant polynomial, or the partition polynomials $P_n$, to generate the power sums, or traces:

$$F_m[P_1(b_1),P_2(b_1,b_2)/2!,..,P_m(b_1,..,b_m)/m!] = F_m(d_1,d_2,..,d_m) = -b_m = tr(A^m) \; .$$

Damianou points out that the adjacency matrix $A_n$ for the simple Lie algebra $B_n$ has the characteristic polynomial

$$det(xI - A_n) = 2\; T_n(x/2)= a_n(x)\; ,$$

the Chebyshev polynomials given by A127672, so with

$$det(I - x\;A_n) = x^n \; a_n(1/x) = 2 x^n \; T(1/(2x)) = 1 + \sum_{k = 1}^{n} d_k \; x^k \; ,$$

$$F_m(d_1,d_2,..,d_m) = -b_m = tr(A^m) \;,$$

unwrapping the determinant relation.

An example is afforded by the adjacency matrix for for the simple Lie algebra $B_3$, given by Damianou:

$$A_3=\begin{bmatrix} 0 & 1 & 0\\ 1 & 0 & 2\\ 0 & 1 & 0 \end{bmatrix} \; .$$

The characteristic polynomial $\operatorname{det}(xI-A_3)=a_3(x) = 2T_3(x/2)= x^3 -3x=$ has the zeroes $x_1=\sqrt{3},x_2=-\sqrt{3},x_3=0$, so $\operatorname{tr}(A_3^k)= x_1^k + x_2^k +x_3^k=3^{k/2} + (-1)^k 3^{k/2}$ (aerated A025192), which vanishes for odd $k$ and equals $2\; 3^{k/2}$ for even $k$. Then $P_k(b_1,..,b_k)$ vanishes except for $P_2(0,-6) = -6$, and $\operatorname{det}(I-uA_3)= 1-3u^2=u^3a_3(1/u) = 2x^3T_3(1/2x))$, so only $d_2 = -3$ is non-vanishing and $F_k[0,-3,0,..,0]= -b_k = \operatorname{tr}(A_3^k)$ for $k>0$.

• See A263916 for other connections of the Faber polynomials to the Chebyshev polynomials. – Tom Copeland Dec 4 '15 at 4:59
• Follow the OEIS links for relations to walks on paths of P_n and K_n graphs. – Tom Copeland Dec 8 '15 at 22:47