Even Order Pascal Tensors are Positive Definite

TL;DR

This paper proves that even order Pascal tensors are positive definite and odd order Pascal tensors are strongly completely positive, extending these properties to generalized Pascal tensors and fractional Hadamard power tensors, with determinant analysis.

Contribution

It introduces a novel induction proof method for positive definiteness and complete positivity in Pascal tensors and related families, based on matrix properties.

Findings

Even order Pascal tensors are positive definite.

Odd order Pascal tensors are strongly completely positive.

Determinant of the m-th order symmetric Pascal tensor equals (factorial(m-1))^m.

Abstract

In this paper, we show that even order Pascal tensors are positive definite, and odd order Pascal tensors are strongly completely positive. The significance of these is that our induction proof method also holds for some other families of completely positives tensors, whose construction satisfies certain rules, such an inherence property holds. We show that for all tensors in such a family, even order tensors would be positive definite, and odd order tensors would be strongly completely positive, as long as the matrices in this family are positive definite. In particular, we show that even order generalized Pascal tensors would be positive definite, and odd order generalized Pascal tensors would be strongly completely positive, as long as generalized Pascal matrices are positive definite. We also investigate even order positive definiteness and odd order strongly completely positivity for fractional Hadamard power tensors. Furthermore, we study determinants of Pascal tensors. We prove that the determinant of the $m$th order two dimensional symmetric Pascal tensor is equal to the $m$th power of the factorial of $m-1$.