### Schur Polynomials, Banded Toeplitz Matrices and Widom's Formula

#### Abstract

We prove that for arbitrary partitions $\mathbf{\lambda} \subseteq \mathbf{\kappa},$ and integers $0\leq c<r\leq n,$ the sequence of Schur polynomials $S_{(\mathbf{\kappa} + k\cdot\mathbf{1}^c)/(\mathbf{\lambda} + k\cdot\mathbf{1}^r)}(x_1,\dots,x_n)$ for $k$ sufficiently large, satisfy a linear recurrence. The roots of the characteristic equation are given explicitly. These recurrences are also valid for certain sequences of minors of banded Toeplitz matrices.

In addition, we show that Widom's determinant formula from 1958 is a special case of a well-known identity for Schur polynomials.

In addition, we show that Widom's determinant formula from 1958 is a special case of a well-known identity for Schur polynomials.

#### Keywords

Banded Toeplitz matrices; Schur polynomials; Widom's determinant formula; sequence insertion; Young tableaux; recurrence