Hankel Matrix

A Hankel matrix is a square matrix with constant skew diagonals. In other words, a Hankel matrix is a matrix in which the th entry depends only on the sum . Such matrices are sometimes known as persymmetric matrices or, in older literature, orthosymmetric matrices.

In the Wolfram Language, such a Hankel matrix can be generated for example by HankelMatrix[a, b, c, d, e, e, f, g, h, i], giving

(1)

An upper triangular Hankel matrix with first column and row can be specified in the Wolfram Language as HankelMatrix[c1, ..., cn], and HankelMatrix[n] where is an integer gives the matrix with first row and column equal to and with every element below the main skew diagonal equal to 0. The first few matrices are given by

			(2)
			(3)
			(4)

The elements of this Hankel matrix are given explicitly by

(5)

The determinant of is given by , where is the floor function, so the first few values are 1, , , 256, 3125, , , 16777216, ... (OEIS A000312).

Fauzan (2026) used determinants of Hankel matrices in an announced proof that zeta(5) is irrational.