If you want to learn about Whittaker’s Root Series Formula see my post “Whittaker’s Root Series: Going Transcendental”. That post has useful links to introductory material on the topic. You can also see my OEIS sequences, since a lot of them were obtained using Whittaker’s formula.

Whittaker’s Root Series Formula is obtained using the coefficients of a polynomial or an infinite power series. A few months ago I learned that the determinants of the matrices from the denominator of Whittaker’s formula are related to the coefficients of the series expansion of the reciprocal of the polynomial.

### Example

If we have the polynomial p(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+…+a_{1}x+a_{0}, Whittaker’s Root Series Formula is given by:

It seems that the determinants of the matrices from the denominator have the same absolute value of the coefficients of the series expansion of 1/p(x).

The paper “On determinants of Toeplitz-Hessenberg matrices arising in power series” by Alfred Inselberg is relevant to the topic, even though it’s not about Whittaker’s Root Series Formula. The paper discusses some matrices that are equivalent to the matrices from the denominator of Whittaker’s formula. However, we must add the condition that a_{0} =1.

As an example we can use p(x)=-x^3-x^2-x+1. The series expansion of 1/p(x) can be seen below. The coefficients are related to the Tribonacci numbers sequence or OEIS A000073.

The determinants of the matrices when applying Whittaker’s Root Series Formula to p(x)=-x^3-x^2-x+1 can be seen below.

If we let c_{n} be the coefficient of x^{n} in the power series q(x)=1/p(x), and M_{n} be the determinant of the nxn matrix found in the denominator of Whittaker’s Root Series Formula, then c_{n} =(-1)^{n} M_{n} .