Hankel Matrix

A Hankel matrix has 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 also called persymmetric matrices (Tamm 2001) or orthosymmetric matrices (Graham and Sloane 1984). These names also have a different usage referring to reflection symmetry about the skew diagonal, rather than constant values along skew diagonals (Petersen and Pedersen 2012).

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)

A Hankel matrix with first column and row and zeros below its main skew diagonal 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, .... The absolute values are given by OEIS A000312.

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