Hankel Determinants of Certain Sequences Of Bernoulli Polynomials: A Direct Proof of an Inverse Matrix Entry from Statistics

TL;DR

This paper computes Hankel determinants of Bernoulli polynomial sequences linked to variance estimation in nonparametric regression, providing a direct proof for an inverse matrix entry, connecting deep polynomial properties with statistical matrices.

Contribution

It offers a novel direct proof for an inverse matrix entry derived from Hankel determinants of Bernoulli polynomials, bridging polynomial theory and statistical matrix analysis.

Findings

Explicit calculation of Hankel determinants for Bernoulli polynomial sequences

A direct proof of a specific inverse matrix entry from the Hankel matrix

Connection between Vandermonde matrices and statistical variance estimation

Abstract

We calculate the Hankel determinants of sequences of Bernoulli polynomials. This corresponding Hankel matrix comes from statistically estimating the variance in nonparametric regression. Besides its entries' natural and deep connection with Bernoulli polynomials, a special case of the matrix can be constructed from a corresponding Vandermonde matrix. As a result, instead of asymptotic analysis, we give a direct proof of calculating an entry of its inverse.