Positivity properties of the matrix $\left[(i+j)^{i+j}\right]$

TL;DR

This paper proves that a matrix constructed from positive real numbers with entries $(p_i+p_j)^{p_i+p_j}$ is infinitely divisible, nonsingular, and totally positive, revealing important positivity properties.

Contribution

It introduces and proves positivity properties of a new class of matrices based on sums of positive real numbers raised to their sums.

Findings

Matrix is infinitely divisible

Matrix is nonsingular

Matrix is totally positive

Abstract

Let $p_1<p_2<\cdots<p_n$ be positive real numbers. It is shown that the matrix whose $i,j$ entry is $(p_i+p_j)^{p_i+p_j}$ is infinitely divisible, nonsingular and totally positive.