Some Determinantal Identities

TL;DR

This paper explores various determinantal identities related to binomial coefficients, Stirling numbers, and Fibonacci numbers, providing new formulas and applications including inverse matrices and recurrence relations.

Contribution

It introduces new determinantal identities and formulas for inverse matrices, sums of powers, and higher-order Fibonacci numbers, expanding the theoretical framework of these mathematical objects.

Findings

Derived three determinantal identities involving binomial coefficients and Stirling numbers.

Obtained the inverse of the Vandermonde matrix explicitly.

Proved a determinantal identity for higher-order Fibonacci numbers.

Abstract

Some applications of a result, which is proved recently, is considered. We first prove three determinantal identities concerning the binomial coefficient and Stirling numbers of the first and the second kind. We also easily obtain the inverse of the Vandermonde matrix. Then we derive a recurrence formula for sums of powers, which is similar to the well-known Newton identity. In the last section, we consider some sequences given by a homologous linear recurrence equation. A determinantal identity for the Fibonacci numbers of higher order is proved. We finish with an expression of the generalized Vandermonde determinant in terms of the standard Vandermonde determinant and elementary symmetric polynomials.